Notation by work

Quantum and Circuits

All notation · Glossary · Source coverage

1494 symbol spellings in 4211 mathematical passages across 26 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.

The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.

All 1494 entries are available below.

*Notation table · 62 passages

Search aliases: *

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Notation table · Read passage · c_main.tex, line 25
β(f),β^*(f) · Complete-cover degree and its affine-output optimization.
Exact mathematical passage
```
\beta(f),\beta^*(f)
```

CNotation table · 76 passages

Search aliases: C

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Notation table · Read passage · c_main.tex, line 17
midrule C(f) · Minimum fan-in-two AND/OR/NOT gate count, allowing fan-out.
Exact mathematical passage
```
C(f)
```

fNotation table · 142 passages

Search aliases: f

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Notation table · Read passage · c_main.tex, line 17
midrule C(f) · Minimum fan-in-two AND/OR/NOT gate count, allowing fan-out.
Exact mathematical passage
```
C(f)
```

INotation table · 113 passages

Search aliases: I

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Notation table · Read passage · c_main.tex, line 22
I_f · Boolean functions supported only on positive inputs of f.
Exact mathematical passage
```
I_f
```

I_fNotation table · 8 passages

Search aliases: I_f

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Notation table · Read passage · c_main.tex, line 22
I_f · Boolean functions supported only on positive inputs of f.
Exact mathematical passage
```
I_f
```

\KTrestNotation table · 9 passages

Search aliases: KTrest

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Notation table · Read passage · c_main.tex, line 20
KTrest(f) · Minimum such Toffoli count, allowing any finite zero workspace.
Exact mathematical passage
```
\KTrest(f)
```

LNotation table · 154 passages

Search aliases: L

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Notation table · Read passage · c_main.tex, line 18
L(f) · Minimum total NOT/CNOT/Toffoli count of a restored implementation.
Exact mathematical passage
```
L(f)
```

MNotation table · 89 passages

Search aliases: M

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Notation table · Read passage · c_main.tex, line 21
M(f) · Minimum AND count in an XOR--AND computation, with affine operations uncharged.
Exact mathematical passage
```
M(f)
```

nNotation table · 616 passages

Search aliases: n

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Notation table · Read passage · g_main.tex, line 3445
Exact group sizes. The finite state spaces are |P_n| = 2^n(n+1)/2|GL(n,2)|, and |Sp(2n,2)| = 2^n^2∏_j=1^n(2^2j-1). For n=1,2,3 this gives: @rcc@ n · |P_n| · |Sp(2n,2)|
Exact mathematical passage
```
n
```

PNotation table · 306 passages

Search aliases: P

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Notation table · Read passage · c_main.tex, line 26
[P]_2 · Homogeneous degree-two component of the multilinear polynomial P in the specified coordinates.
Exact mathematical passage
```
[P]_2
```

P_nNotation table · 10 passages

Search aliases: P_n

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Notation table · Read passage · g_main.tex, line 3445
Exact group sizes. The finite state spaces are |P_n| = 2^n(n+1)/2|GL(n,2)|, and |Sp(2n,2)| = 2^n^2∏_j=1^n(2^2j-1). For n=1,2,3 this gives: @rcc@ n · |P_n| · |Sp(2n,2)|
Exact mathematical passage
```
|P_n|
```

\SpNotation table · 13 passages

Search aliases: Sp

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Notation table · Read passage · g_main.tex, line 3445
Exact group sizes. The finite state spaces are |P_n| = 2^n(n+1)/2|GL(n,2)|, and |Sp(2n,2)| = 2^n^2∏_j=1^n(2^2j-1). For n=1,2,3 this gives: @rcc@ n · |P_n| · |Sp(2n,2)|
Exact mathematical passage
```
|\Sp(2n,2)|
```

TNotation table · 123 passages

Search aliases: T

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Notation table · Read passage · c_main.tex, line 19
T · Number of Toffoli gates in a specified implementation.
Exact mathematical passage
```
T
```

vNotation table · 211 passages

Search aliases: v

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Notation table · Read passage · c_main.tex, line 24
β_v(f) · Minimum degree of a member of I_f that is one at v.
Exact mathematical passage
```
\beta_v(f)
```

|Notation table · 119 passages

Search aliases: |

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Notation table · Read passage · g_main.tex, line 3445
Exact group sizes. The finite state spaces are |P_n| = 2^n(n+1)/2|GL(n,2)|, and |Sp(2n,2)| = 2^n^2∏_j=1^n(2^2j-1). For n=1,2,3 this gives: @rcc@ n · |P_n| · |Sp(2n,2)|
Exact mathematical passage
```
|P_n|
```

\alphaNotation table · 57 passages

Search aliases: α alpha

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Notation table · Read passage · c_main.tex, line 23
α_1(f) · Minimum degree of one nonzero member of I_f.
Exact mathematical passage
```
\alpha_1(f)
```

\alpha_1Notation table · 14 passages

Search aliases: α_1 alpha

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Notation table · Read passage · c_main.tex, line 23
α_1(f) · Minimum degree of one nonzero member of I_f.
Exact mathematical passage
```
\alpha_1(f)
```

\betaNotation table · 86 passages

Search aliases: β beta

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Notation table · Read passage · c_main.tex, line 24
β_v(f) · Minimum degree of a member of I_f that is one at v.
Exact mathematical passage
```
\beta_v(f)
```

\beta^*Notation table · 28 passages

Search aliases: β^* beta

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Notation table · Read passage · c_main.tex, line 25
β(f),β^*(f) · Complete-cover degree and its affine-output optimization.
Exact mathematical passage
```
\beta(f),\beta^*(f)
```

\beta_vNotation table · 7 passages

Search aliases: β_v beta

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Notation table · Read passage · c_main.tex, line 24
β_v(f) · Minimum degree of a member of I_f that is one at v.
Exact mathematical passage
```
\beta_v(f)
```

+Definition passage · 426 passages

Search aliases: +

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Definition passage: Gate model · Read passage · g_main.tex, line 42
,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
+\infty
```

=Definition passage · 1108 passages

Search aliases: =

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Definition passage: Gate model · Read passage · g_main.tex, line 27
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
\mathfrak M=(\mathcal X,\mathcal S,c,\circ),
```

ADefinition passage · 280 passages

Search aliases: A

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1211
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
A\in\U_n(R)
```

aDefinition passage · 216 passages

Search aliases: a

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1468
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{p,q}(-a)\,h_m\cdots h_1\,\tau_{p,q}(a),
\qquad p<q,
```

\mathrm{amb}Definition passage · 17 passages

Search aliases: amb mathrm

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 49
[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\mathfrak M_{\mathrm{res}}\subseteq\mathfrak M_{\mathrm{amb}}
```

bDefinition passage · 126 passages

Search aliases: b

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1473
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{q,x}(b),\quad \tau_{p,x}(b),\quad
\tau_{y,p}(b),\quad \tau_{y,q}(b)
```

\mathbf cDefinition passage · 4 passages

Search aliases: c mathbf

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Definition passage: Resource vector · Read passage · g_main.tex, line 82
[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.
Exact mathematical passage
```
\mathbf c(s)=(c_1(s),\ldots,c_r(s))\in\R_{\ge0}^r
```

cDefinition passage · 171 passages

Search aliases: c

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Definition passage: Gate model · Read passage · g_main.tex, line 27
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
\mathfrak M=(\mathcal X,\mathcal S,c,\circ),
```

c_1Definition passage · 3 passages

Search aliases: c_1

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Definition passage: Resource vector · Read passage · g_main.tex, line 82
[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.
Exact mathematical passage
```
\mathbf c(s)=(c_1(s),\ldots,c_r(s))\in\R_{\ge0}^r
```

c_rDefinition passage · 1 passage

Search aliases: c_r

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Definition passage: Resource vector · Read passage · g_main.tex, line 82
[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.
Exact mathematical passage
```
\mathbf c(s)=(c_1(s),\ldots,c_r(s))\in\R_{\ge0}^r
```

\CXDefinition passage · 19 passages

Search aliases: CX

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Definition passage: Parabolic and ambient Clifford metrics · Read passage · g_main.tex, line 3412
[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.
Exact mathematical passage
```
\mathcal S_P
=
\{S_i,\CZ_{ij},\CX_{i\to j}\},
```

\CX_{i\to j}Definition passage · 3 passages

Search aliases: CX_i→ j CX to

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Definition passage: Parabolic and ambient Clifford metrics · Read passage · g_main.tex, line 3412
[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.
Exact mathematical passage
```
\mathcal S_P
=
\{S_i,\CZ_{ij},\CX_{i\to j}\},
```

\CZDefinition passage · 9 passages

Search aliases: CZ

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Definition passage: Parabolic and ambient Clifford metrics · Read passage · g_main.tex, line 3412
[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.
Exact mathematical passage
```
\mathcal S_P
=
\{S_i,\CZ_{ij},\CX_{i\to j}\},
```

\CZ_{ij}Definition passage · 3 passages

Search aliases: CZ_ij CZ

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Definition passage: Parabolic and ambient Clifford metrics · Read passage · g_main.tex, line 3412
[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.
Exact mathematical passage
```
\mathcal S_P
=
\{S_i,\CZ_{ij},\CX_{i\to j}\},
```

DDefinition passage · 111 passages

Search aliases: D

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Definition passage: Extremal distortion · Read passage · g_main.tex, line 132
Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.
Exact mathematical passage
```
D_n=\sup_{U\in\mathcal T_n}\Delta(U)
```

dDefinition passage · 220 passages

Search aliases: d

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Definition passage: Excursion profile · Read passage · g_main.tex, line 330
Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a forbidden dependency, item the number of simultaneously open catalytic intervals, item the intersection graph of those intervals, item the maximum al…
Exact mathematical passage
```
d(X_tH,H)
```

D_nDefinition passage · 15 passages

Search aliases: D_n

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Definition passage: Extremal distortion · Read passage · g_main.tex, line 132
Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.
Exact mathematical passage
```
D_n=\sup_{U\in\mathcal T_n}\Delta(U)
```

\DeltaDefDefinition passage · 18 passages

Search aliases: DeltaDef

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1218
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
\DeltaDef(A)=\ellDown(A)-\ellAll(A)\ge0.
```

\downarrowDefinition passage · 10 passages

Search aliases: downarrow

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1212
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
\ellDown(A)=|A|_{S_n^\downarrow(R)},
\qquad
\ellAll(A)=|A|_{S_n^*(R)}.
```

\mathcal EDefinition passage · 1 passage

Search aliases: E mathcal

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Definition passage: Catalytic excess · Read passage · g_main.tex, line 317
[Catalytic excess] For a normal form with r matched excursions and C useful cross interactions, define E=C-2r when each consumer contributes one unit and each opening or closing costs one unit. Positive excess is necessary in the simplest transvection model, but it is not sufficient. An all-cross configuration may admit a mirror word of equal restricted length. Temporal asymmetry or phase holonomy is also required.
Exact mathematical passage
```
\mathcal E=C-2r
```

\ellAllDefinition passage · 62 passages

Search aliases: ellAll

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1212
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
\ellDown(A)=|A|_{S_n^\downarrow(R)},
\qquad
\ellAll(A)=|A|_{S_n^*(R)}.
```

\ellAmbDefinition passage · 15 passages

Search aliases: ellAmb

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 53
[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\ellRes(U)=\ell_{\mathfrak M_{\mathrm{res}}}(U),
\qquad
\ellAmb(U)=\ell_{\mathfrak M_{\mathrm{amb}}}(U).
```

\ellBudgetDefinition passage · 8 passages

Search aliases: ellBudget

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Definition passage: Upper-gate budget · Read passage · g_main.tex, line 2221
[Upper-gate budget] For q≥0 and A∈U_n(F), define ellBudgetq(A) = min{|w|:[w]=A, u(w)≤ q}. Thus ellBudget0(A)=ellDown(A), ellBudgetq+1(A)≤ellBudgetq(A), ellAll(A)≤ellBudgetq(A).
Exact mathematical passage
```
\ellBudget{q}(A)
=
\min\{|w|:[w]=A,\ u(w)\le q\}.
```

\ellDownDefinition passage · 64 passages

Search aliases: ellDown

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1212
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
\ellDown(A)=|A|_{S_n^\downarrow(R)},
\qquad
\ellAll(A)=|A|_{S_n^*(R)}.
```

\ellResDefinition passage · 17 passages

Search aliases: ellRes

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 53
[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\ellRes(U)=\ell_{\mathfrak M_{\mathrm{res}}}(U),
\qquad
\ellAmb(U)=\ell_{\mathfrak M_{\mathrm{amb}}}(U).
```

\FDefinition passage · 131 passages

Search aliases: F

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Definition passage: Interval tree · Read passage · g_main.tex, line 1946
[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).
Exact mathematical passage
```
\label{g:eq:tree-potential}
\Phi_T(A)
=
\sum_{v\in\operatorname{Int}(T)}
\rank A[R_v,L_v],
\qquad A\in\U_n(\F).
```

GDefinition passage · 105 passages

Search aliases: G

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Definition passage: Packet schedule · Read passage · s_main.tex, line 423
[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).
Exact mathematical passage
```
U=G_m\cdots G_1
```

G_1Definition passage · 12 passages

Search aliases: G_1

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Definition passage: Packet schedule · Read passage · s_main.tex, line 423
[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).
Exact mathematical passage
```
U=G_m\cdots G_1
```

G_mDefinition passage · 10 passages

Search aliases: G_m

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Definition passage: Packet schedule · Read passage · s_main.tex, line 423
[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).
Exact mathematical passage
```
U=G_m\cdots G_1
```

G_tDefinition passage · 18 passages

Search aliases: G_t

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Definition passage: Packet schedule · Read passage · s_main.tex, line 431
[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).
Exact mathematical passage
```
G_tP_{t-1}=P_tG_tP_{t-1}
\qquad(1\le t\le m).
\label{s:eq:dynamic-packet-condition}
```

HDefinition passage · 124 passages

Search aliases: H

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Definition passage: Excursion profile · Read passage · g_main.tex, line 330
Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a forbidden dependency, item the number of simultaneously open catalytic intervals, item the intersection graph of those intervals, item the maximum al…
Exact mathematical passage
```
d(X_tH,H)
```

hDefinition passage · 82 passages

Search aliases: h

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1468
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{p,q}(-a)\,h_m\cdots h_1\,\tau_{p,q}(a),
\qquad p<q,
```

h_1Definition passage · 7 passages

Search aliases: h_1

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1468
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{p,q}(-a)\,h_m\cdots h_1\,\tau_{p,q}(a),
\qquad p<q,
```

H_iDefinition passage · 18 passages

Search aliases: H_i

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Definition passage: Parabolic and ambient Clifford metrics · Read passage · g_main.tex, line 3418
[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.
Exact mathematical passage
```
\mathcal S_C
=
\mathcal S_P\cup\{H_i\}.
```

h_iDefinition passage · 9 passages

Search aliases: h_i

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1472
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
h_i
```

h_mDefinition passage · 7 passages

Search aliases: h_m

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1468
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{p,q}(-a)\,h_m\cdots h_1\,\tau_{p,q}(a),
\qquad p<q,
```

\HhDefinition passage · 23 passages

Search aliases: Hh

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Definition passage: Packet schedule · Read passage · s_main.tex, line 426
[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).
Exact mathematical passage
```
\Hh
```

iDefinition passage · 190 passages

Search aliases: i

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Definition passage: Flagged matroid tour · Read passage · g_main.tex, line 755
[Flagged matroid tour] Let M be a representable matroid with designated initial basis, target basis, and consumer states. A flagged tour is a sequence of basis exchanges or transvections in which the ith carrier remains inside a specified filtration layer. An ambient tour drops this constraint. The six-row atom realizes the graphic regular matroid M(K_4). Its ambient state tour is shorter than every compatible flag lift. Regularity explains why the same dependence pattern survives every field an…
Exact mathematical passage
```
i
```

\infDefinition passage · 6 passages

Search aliases: inf

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Definition passage: Gate model · Read passage · g_main.tex, line 39
te model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
\ell_{\mathfrak M}(U)=\inf\{c(w):w=U\}.
```

\operatorname{Int}Definition passage · 1 passage

Search aliases: Int operatorname

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Definition passage: Interval tree · Read passage · g_main.tex, line 1946
[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).
Exact mathematical passage
```
\label{g:eq:tree-potential}
\Phi_T(A)
=
\sum_{v\in\operatorname{Int}(T)}
\rank A[R_v,L_v],
\qquad A\in\U_n(\F).
```

jDefinition passage · 100 passages

Search aliases: j

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Definition passage: Gate model · Read passage · g_main.tex, line 35
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
c(w)=\sum_{j=1}^L c(s_j).
```

L_vDefinition passage · 2 passages

Search aliases: L_v

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Definition passage: Interval tree · Read passage · g_main.tex, line 1945
[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).
Exact mathematical passage
```
L_v
```

\limDefinition passage · 3 passages

Search aliases: lim

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Definition passage: Stable tensor cost · Read passage · g_main.tex, line 661
[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.
Exact mathematical passage
```
\tau_\otimes(U)
=
\lim_{r\to\infty}
\frac{\ellAmb(U^{\otimes r})}{r m^{r-1}},
```

\lim_{r\to\infty}Definition passage · 3 passages

Search aliases: lim_r→∞ lim to infty

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Definition passage: Stable tensor cost · Read passage · g_main.tex, line 661
[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.
Exact mathematical passage
```
\tau_\otimes(U)
=
\lim_{r\to\infty}
\frac{\ellAmb(U^{\otimes r})}{r m^{r-1}},
```

\mathfrak MDefinition passage · 6 passages

Search aliases: M mathfrak

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Definition passage: Gate model · Read passage · g_main.tex, line 27
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
\mathfrak M=(\mathcal X,\mathcal S,c,\circ),
```

mDefinition passage · 276 passages

Search aliases: m

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Definition passage: Tensor synergy · Read passage · g_main.tex, line 650
[Tensor synergy] Define Syn(U,V) = nellAmb(U)+mellAmb(V)-ellAmb(U⊗ V). Positive synergy means that the joint ambient realization improves on independent lifted circuits. Restricted tree-tight metrics have zero tensor synergy. Ambient synergy is a possible source of collective activation.
Exact mathematical passage
```
\Syn(U,V)
=
n\ellAmb(U)+m\ellAmb(V)-\ellAmb(U\otimes V).
```

m^{r-1}Definition passage · 10 passages

Search aliases: m^r-1

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Definition passage: Stable tensor cost · Read passage · g_main.tex, line 661
[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.
Exact mathematical passage
```
\tau_\otimes(U)
=
\lim_{r\to\infty}
\frac{\ellAmb(U^{\otimes r})}{r m^{r-1}},
```

\mathfrak M_{\mathrm{amb}}Definition passage · 2 passages

Search aliases: M_amb mathfrak mathrm

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 49
[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\mathfrak M_{\mathrm{res}}\subseteq\mathfrak M_{\mathrm{amb}}
```

\mathfrak M_{\mathrm{res}}Definition passage · 2 passages

Search aliases: M_res mathfrak mathrm

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 49
[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\mathfrak M_{\mathrm{res}}\subseteq\mathfrak M_{\mathrm{amb}}
```

\minDefinition passage · 25 passages

Search aliases: min

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Definition passage: Extremal distortion · Read passage · g_main.tex, line 141
Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.
Exact mathematical passage
```
\tau=\min\{n:D_n>0\}.
```

\overline{\mathcal W}Definition passage · 2 passages

Search aliases: overline W overline mathcal

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Definition passage: Border cost · Read passage · g_main.tex, line 174
Approximate synthesis and border defects. For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= underlineℓ_res(U)- underlineℓ_amb(U). This is the circuit analogue of the distinction between tensor rank and border rank. An exact defect can disappe…
Exact mathematical passage
```
\overline{\mathcal W}_{L}
```

\overline{\mathcal W}_{L}Definition passage · 2 passages

Search aliases: overline W_L overline mathcal

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Definition passage: Border cost · Read passage · g_main.tex, line 174
Approximate synthesis and border defects. For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= underlineℓ_res(U)- underlineℓ_amb(U). This is the circuit analogue of the distinction between tensor rank and border rank. An exact defect can disappe…
Exact mathematical passage
```
\overline{\mathcal W}_{L}
```

\mathcal PDefinition passage · 4 passages

Search aliases: P mathcal

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Definition passage: Resource vector · Read passage · g_main.tex, line 86
[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.
Exact mathematical passage
```
\mathcal P_{\mathfrak M}(U)
=
\left\{
\sum_{j=1}^L\mathbf c(s_j):s_L\cdots s_1=U
\right\}.
```

pDefinition passage · 102 passages

Search aliases: p

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1468
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{p,q}(-a)\,h_m\cdots h_1\,\tau_{p,q}(a),
\qquad p<q,
```

\mathcal P_{\mathfrak M}Definition passage · 2 passages

Search aliases: P_ M mathcal mathfrak

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Definition passage: Resource vector · Read passage · g_main.tex, line 86
[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.
Exact mathematical passage
```
\mathcal P_{\mathfrak M}(U)
=
\left\{
\sum_{j=1}^L\mathbf c(s_j):s_L\cdots s_1=U
\right\}.
```

P_0Definition passage · 6 passages

Search aliases: P_0

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Definition passage: Packet schedule · Read passage · s_main.tex, line 427
[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).
Exact mathematical passage
```
P_0,P_1,\ldots,P_m.
```

P_1Definition passage · 2 passages

Search aliases: P_1

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Definition passage: Packet schedule · Read passage · s_main.tex, line 427
[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).
Exact mathematical passage
```
P_0,P_1,\ldots,P_m.
```

P_mDefinition passage · 18 passages

Search aliases: P_m

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Definition passage: Packet schedule · Read passage · s_main.tex, line 427
[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).
Exact mathematical passage
```
P_0,P_1,\ldots,P_m.
```

P_tDefinition passage · 13 passages

Search aliases: P_t

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Definition passage: Packet schedule · Read passage · s_main.tex, line 431
[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).
Exact mathematical passage
```
G_tP_{t-1}=P_tG_tP_{t-1}
\qquad(1\le t\le m).
\label{s:eq:dynamic-packet-condition}
```

P_{t-1}Definition passage · 7 passages

Search aliases: P_t-1

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Definition passage: Packet schedule · Read passage · s_main.tex, line 431
[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).
Exact mathematical passage
```
G_tP_{t-1}=P_tG_tP_{t-1}
\qquad(1\le t\le m).
\label{s:eq:dynamic-packet-condition}
```

qDefinition passage · 186 passages

Search aliases: q

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Definition passage: Excursion profile · Read passage · g_main.tex, line 329
Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a forbidden dependency, item the number of simultaneously open catalytic intervals, item the intersection graph of those intervals, item the maximum al…
Exact mathematical passage
```
q
```

RDefinition passage · 147 passages

Search aliases: R

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Definition passage: Extremal distortion · Read passage · g_main.tex, line 136
Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.
Exact mathematical passage
```
R_n=\sup_{U\in\mathcal T_n}
\frac{\ellRes(U)}{\ellAmb(U)}.
```

\RDefinition passage · 8 passages

Search aliases: R

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Definition passage: Gate model · Read passage · g_main.tex, line 30
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
c:\mathcal S\to\R_{\ge0}
```

rDefinition passage · 193 passages

Search aliases: r

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Definition passage: Resource vector · Read passage · g_main.tex, line 82
[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.
Exact mathematical passage
```
\mathbf c(s)=(c_1(s),\ldots,c_r(s))\in\R_{\ge0}^r
```

R_nDefinition passage · 1 passage

Search aliases: R_n

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Definition passage: Extremal distortion · Read passage · g_main.tex, line 136
Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.
Exact mathematical passage
```
R_n=\sup_{U\in\mathcal T_n}
\frac{\ellRes(U)}{\ellAmb(U)}.
```

R_vDefinition passage · 2 passages

Search aliases: R_v

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Definition passage: Interval tree · Read passage · g_main.tex, line 1945
[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).
Exact mathematical passage
```
R_v
```

\R_{\ge0}Definition passage · 1 passage

Search aliases: R_≥0 R ge

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Definition passage: Gate model · Read passage · g_main.tex, line 30
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
c:\mathcal S\to\R_{\ge0}
```

\R_{\ge0}^rDefinition passage · 2 passages

Search aliases: R_≥0^r R ge

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Definition passage: Resource vector · Read passage · g_main.tex, line 82
[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.
Exact mathematical passage
```
\mathbf c(s)=(c_1(s),\ldots,c_r(s))\in\R_{\ge0}^r
```

\rankDefinition passage · 9 passages

Search aliases: rank

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Definition passage: Interval tree · Read passage · g_main.tex, line 1946
[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).
Exact mathematical passage
```
\label{g:eq:tree-potential}
\Phi_T(A)
=
\sum_{v\in\operatorname{Int}(T)}
\rank A[R_v,L_v],
\qquad A\in\U_n(\F).
```

\mathrm{res}Definition passage · 17 passages

Search aliases: res mathrm

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 49
[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\mathfrak M_{\mathrm{res}}\subseteq\mathfrak M_{\mathrm{amb}}
```

SDefinition passage · 143 passages

Search aliases: S

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1212
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
\ellDown(A)=|A|_{S_n^\downarrow(R)},
\qquad
\ellAll(A)=|A|_{S_n^*(R)}.
```

\mathcal SDefinition passage · 13 passages

Search aliases: S mathcal

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Definition passage: Gate model · Read passage · g_main.tex, line 27
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
\mathfrak M=(\mathcal X,\mathcal S,c,\circ),
```

sDefinition passage · 136 passages

Search aliases: s

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Definition passage: Gate model · Read passage · g_main.tex, line 31
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
w=s_L\cdots s_1
```

s_1Definition passage · 26 passages

Search aliases: s_1

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Definition passage: Gate model · Read passage · g_main.tex, line 31
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
w=s_L\cdots s_1
```

\mathcal S_{\mathrm{amb}}Definition passage · 1 passage

Search aliases: S_amb mathcal mathrm

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Definition passage: Cost-preserving equivalence · Read passage · g_main.tex, line 160
[Cost-preserving equivalence] Targets U and V are equivalent if an automorphism φ of the ambient realization system satisfies φ(U)=V, φ( S_res)= S_res, φ( S_amb)= S_amb, and preserves the gate costs. Under this equivalence, both lengths and the defect are orbit invariants. A useful threshold theorem should classify minimal defective orbits, not merely list raw matrices.
Exact mathematical passage
```
\varphi(U)=V,
\qquad
\varphi(\mathcal S_{\mathrm{res}})=\mathcal S_{\mathrm{res}},
\qquad
\varphi(\mathcal S_{\mathrm{amb}})=\mathcal S_{\mathrm{amb}},
```

\mathcal S_CDefinition passage · 2 passages

Search aliases: S_C mathcal

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Definition passage: Parabolic and ambient Clifford metrics · Read passage · g_main.tex, line 3418
[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.
Exact mathematical passage
```
\mathcal S_C
=
\mathcal S_P\cup\{H_i\}.
```

S_iDefinition passage · 6 passages

Search aliases: S_i

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Definition passage: Parabolic and ambient Clifford metrics · Read passage · g_main.tex, line 3412
[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.
Exact mathematical passage
```
\mathcal S_P
=
\{S_i,\CZ_{ij},\CX_{i\to j}\},
```

s_jDefinition passage · 10 passages

Search aliases: s_j

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Definition passage: Gate model · Read passage · g_main.tex, line 35
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
c(w)=\sum_{j=1}^L c(s_j).
```

s_LDefinition passage · 6 passages

Search aliases: s_L

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Definition passage: Gate model · Read passage · g_main.tex, line 31
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
w=s_L\cdots s_1
```

S_n^*Definition passage · 5 passages

Search aliases: S_n^*

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1212
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
\ellDown(A)=|A|_{S_n^\downarrow(R)},
\qquad
\ellAll(A)=|A|_{S_n^*(R)}.
```

S_n^\downarrowDefinition passage · 4 passages

Search aliases: S_n^downarrow downarrow

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1212
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
\ellDown(A)=|A|_{S_n^\downarrow(R)},
\qquad
\ellAll(A)=|A|_{S_n^*(R)}.
```

\mathcal S_PDefinition passage · 3 passages

Search aliases: S_P mathcal

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Definition passage: Parabolic and ambient Clifford metrics · Read passage · g_main.tex, line 3412
[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.
Exact mathematical passage
```
\mathcal S_P
=
\{S_i,\CZ_{ij},\CX_{i\to j}\},
```

\mathcal S_{\mathrm{res}}Definition passage · 1 passage

Search aliases: S_res mathcal mathrm

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Definition passage: Cost-preserving equivalence · Read passage · g_main.tex, line 160
[Cost-preserving equivalence] Targets U and V are equivalent if an automorphism φ of the ambient realization system satisfies φ(U)=V, φ( S_res)= S_res, φ( S_amb)= S_amb, and preserves the gate costs. Under this equivalence, both lengths and the defect are orbit invariants. A useful threshold theorem should classify minimal defective orbits, not merely list raw matrices.
Exact mathematical passage
```
\varphi(U)=V,
\qquad
\varphi(\mathcal S_{\mathrm{res}})=\mathcal S_{\mathrm{res}},
\qquad
\varphi(\mathcal S_{\mathrm{amb}})=\mathcal S_{\mathrm{amb}},
```

\supDefinition passage · 8 passages

Search aliases: sup

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Definition passage: Extremal distortion · Read passage · g_main.tex, line 132
Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.
Exact mathematical passage
```
D_n=\sup_{U\in\mathcal T_n}\Delta(U)
```

\sup_{U\in\mathcal T_n}Definition passage · 3 passages

Search aliases: sup_U∈ T_n sup in mathcal

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Definition passage: Extremal distortion · Read passage · g_main.tex, line 132
Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.
Exact mathematical passage
```
D_n=\sup_{U\in\mathcal T_n}\Delta(U)
```

\SynDefinition passage · 1 passage

Search aliases: Syn

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Definition passage: Tensor synergy · Read passage · g_main.tex, line 650
[Tensor synergy] Define Syn(U,V) = nellAmb(U)+mellAmb(V)-ellAmb(U⊗ V). Positive synergy means that the joint ambient realization improves on independent lifted circuits. Restricted tree-tight metrics have zero tensor synergy. Ambient synergy is a possible source of collective activation.
Exact mathematical passage
```
\Syn(U,V)
=
n\ellAmb(U)+m\ellAmb(V)-\ellAmb(U\otimes V).
```

\mathcal TDefinition passage · 10 passages

Search aliases: T mathcal

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 52
[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\mathcal T
```

tDefinition passage · 191 passages

Search aliases: t

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Definition passage: Excursion profile · Read passage · g_main.tex, line 330
Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a forbidden dependency, item the number of simultaneously open catalytic intervals, item the intersection graph of those intervals, item the maximum al…
Exact mathematical passage
```
d(X_tH,H)
```

\mathcal T_nDefinition passage · 8 passages

Search aliases: T_n mathcal

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Definition passage: Extremal distortion · Read passage · g_main.tex, line 132
Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.
Exact mathematical passage
```
D_n=\sup_{U\in\mathcal T_n}\Delta(U)
```

UDefinition passage · 161 passages

Search aliases: U

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Definition passage: Gate model · Read passage · g_main.tex, line 38
begindefinition[Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
U\in\mathcal X
```

\UDefinition passage · 66 passages

Search aliases: U

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1211
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
A\in\U_n(R)
```

uDefinition passage · 126 passages

Search aliases: u

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Definition passage: Upper-gate budget · Read passage · g_main.tex, line 2221
[Upper-gate budget] For q≥0 and A∈U_n(F), define ellBudgetq(A) = min{|w|:[w]=A, u(w)≤ q}. Thus ellBudget0(A)=ellDown(A), ellBudgetq+1(A)≤ellBudgetq(A), ellAll(A)≤ellBudgetq(A).
Exact mathematical passage
```
\ellBudget{q}(A)
=
\min\{|w|:[w]=A,\ u(w)\le q\}.
```

U^{\otimes r}Definition passage · 2 passages

Search aliases: U^⊗ r otimes

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Definition passage: Stable tensor cost · Read passage · g_main.tex, line 661
[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.
Exact mathematical passage
```
\tau_\otimes(U)
=
\lim_{r\to\infty}
\frac{\ellAmb(U^{\otimes r})}{r m^{r-1}},
```

U_nDefinition passage · 3 passages

Search aliases: U_n

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Definition passage: Critical catalyst penalty · Read passage · g_main.tex, line 423
[Critical catalyst penalty] For a target family U_n, define λ_c(U_n) = sup{λ:ℓ_λ,amb(U_n)<ellRes(U_n)}. Determining λ_c exactly is a refined synthesis problem. It can be more physically meaningful than unweighted gate count.
Exact mathematical passage
```
U_n
```

\U_nDefinition passage · 46 passages

Search aliases: U_n U

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1211
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
A\in\U_n(R)
```

\underline\DeltaDefinition passage · 1 passage

Search aliases: underlineΔ underline Delta

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Definition passage: Border cost · Read passage · g_main.tex, line 179
pproximate synthesis and border defects For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= underlineℓ_res(U)- underlineℓ_amb(U). This is the circuit analogue of the distinction between tensor rank and border rank. An exact defect can disappear…
Exact mathematical passage
```
\underline\Delta(U)=
\underline\ell_{\mathrm{res}}(U)-
\underline\ell_{\mathrm{amb}}(U).
```

\underline\ellDefinition passage · 3 passages

Search aliases: underlineℓ underline ell

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Definition passage: Border cost · Read passage · g_main.tex, line 175
Approximate synthesis and border defects. For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= underlineℓ_res(U)- underlineℓ_amb(U). This is the circuit analogue of the distinction between tensor rank and border rank. An exact defect can disappe…
Exact mathematical passage
```
\underline\ell(U)=\inf\{L:U\in\overline{\mathcal W}_{L}\}.
```

\underline\ell_{\mathrm{amb}}Definition passage · 1 passage

Search aliases: underlineℓ_amb underline ell mathrm

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Definition passage: Border cost · Read passage · g_main.tex, line 179
pproximate synthesis and border defects For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= underlineℓ_res(U)- underlineℓ_amb(U). This is the circuit analogue of the distinction between tensor rank and border rank. An exact defect can disappear…
Exact mathematical passage
```
\underline\Delta(U)=
\underline\ell_{\mathrm{res}}(U)-
\underline\ell_{\mathrm{amb}}(U).
```

\underline\ell_{\mathrm{res}}Definition passage · 1 passage

Search aliases: underlineℓ_res underline ell mathrm

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Definition passage: Border cost · Read passage · g_main.tex, line 179
pproximate synthesis and border defects For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= underlineℓ_res(U)- underlineℓ_amb(U). This is the circuit analogue of the distinction between tensor rank and border rank. An exact defect can disappear…
Exact mathematical passage
```
\underline\Delta(U)=
\underline\ell_{\mathrm{res}}(U)-
\underline\ell_{\mathrm{amb}}(U).
```

VDefinition passage · 47 passages

Search aliases: V

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Definition passage: Cost-preserving equivalence · Read passage · g_main.tex, line 159
[Cost-preserving equivalence] Targets U and V are equivalent if an automorphism φ of the ambient realization system satisfies φ(U)=V, φ( S_res)= S_res, φ( S_amb)= S_amb, and preserves the gate costs. Under this equivalence, both lengths and the defect are orbit invariants. A useful threshold theorem should classify minimal defective orbits, not merely list raw matrices.
Exact mathematical passage
```
V
```

wDefinition passage · 87 passages

Search aliases: w

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Definition passage: Gate model · Read passage · g_main.tex, line 31
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
w=s_L\cdots s_1
```

XDefinition passage · 83 passages

Search aliases: X

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Definition passage: Excursion profile · Read passage · g_main.tex, line 327
Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a forbidden dependency, item the number of simultaneously open catalytic intervals, item the intersection graph of those intervals, item the maximum al…
Exact mathematical passage
```
X_0,X_1,\ldots,X_L
```

\mathcal XDefinition passage · 3 passages

Search aliases: X mathcal

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Definition passage: Gate model · Read passage · g_main.tex, line 27
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
\mathfrak M=(\mathcal X,\mathcal S,c,\circ),
```

xDefinition passage · 242 passages

Search aliases: x

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1473
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{q,x}(b),\quad \tau_{p,x}(b),\quad
\tau_{y,p}(b),\quad \tau_{y,q}(b)
```

X_0Definition passage · 1 passage

Search aliases: X_0

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Definition passage: Excursion profile · Read passage · g_main.tex, line 327
Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a forbidden dependency, item the number of simultaneously open catalytic intervals, item the intersection graph of those intervals, item the maximum al…
Exact mathematical passage
```
X_0,X_1,\ldots,X_L
```

X_1Definition passage · 1 passage

Search aliases: X_1

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Definition passage: Excursion profile · Read passage · g_main.tex, line 327
Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a forbidden dependency, item the number of simultaneously open catalytic intervals, item the intersection graph of those intervals, item the maximum al…
Exact mathematical passage
```
X_0,X_1,\ldots,X_L
```

X_LDefinition passage · 1 passage

Search aliases: X_L

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Definition passage: Excursion profile · Read passage · g_main.tex, line 327
Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a forbidden dependency, item the number of simultaneously open catalytic intervals, item the intersection graph of those intervals, item the maximum al…
Exact mathematical passage
```
X_0,X_1,\ldots,X_L
```

X_tDefinition passage · 6 passages

Search aliases: X_t

Link to this entry ·

Definition passage: Excursion profile · Read passage · g_main.tex, line 330
Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a forbidden dependency, item the number of simultaneously open catalytic intervals, item the intersection graph of those intervals, item the maximum al…
Exact mathematical passage
```
d(X_tH,H)
```

yDefinition passage · 35 passages

Search aliases: y

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1473
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{q,x}(b),\quad \tau_{p,x}(b),\quad
\tau_{y,p}(b),\quad \tau_{y,q}(b)
```

|_{S_n^*(R)}Definition passage · 1 passage

Search aliases: |_S_n^*(R)

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1212
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
\ellDown(A)=|A|_{S_n^\downarrow(R)},
\qquad
\ellAll(A)=|A|_{S_n^*(R)}.
```

|_{S_n^\downarrow(R)}Definition passage · 1 passage

Search aliases: |_S_n^downarrow(R) downarrow

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Definition passage: Intrinsic and ambient lengths · Read passage · g_main.tex, line 1212
[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.
Exact mathematical passage
```
\ellDown(A)=|A|_{S_n^\downarrow(R)},
\qquad
\ellAll(A)=|A|_{S_n^*(R)}.
```

\DeltaDefinition passage · 22 passages

Search aliases: Δ Delta

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 59
ed realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\Delta(U)=\ellRes(U)-\ellAmb(U).
```

\varepsilonDefinition passage · 8 passages

Search aliases: ε varepsilon

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Definition passage: Approximate symmetry collapse · Read passage · s_main.tex, line 16
[Approximate symmetry collapse] A uniform circuit family exhibits an approximate symmetry collapse to error ε(n) when a polynomial-size classical representation reproduces its acceptance probability to additive error ε(n), with the error below the promise gap required by the decision problem.
Exact mathematical passage
```
\varepsilon(n)
```

\kappaDefinition passage · 11 passages

Search aliases: κ kappa

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Definition passage: Stable tensor cost · Read passage · g_main.tex, line 668
[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.
Exact mathematical passage
```
\kappa_\otimes(U)
=
\ellRes(U)-\tau_\otimes(U).
```

\kappa_\otimesDefinition passage · 8 passages

Search aliases: κ_⊗ kappa otimes

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Definition passage: Stable tensor cost · Read passage · g_main.tex, line 668
[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.
Exact mathematical passage
```
\kappa_\otimes(U)
=
\ellRes(U)-\tau_\otimes(U).
```

\lambdaDefinition passage · 76 passages

Search aliases: λ lambda

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Definition passage: Critical catalyst penalty · Read passage · g_main.tex, line 424
[Critical catalyst penalty] For a target family U_n, define λ_c(U_n) = sup{λ:ℓ_λ,amb(U_n)<ellRes(U_n)}. Determining λ_c exactly is a refined synthesis problem. It can be more physically meaningful than unweighted gate count.
Exact mathematical passage
```
\lambda_c(U_n)
=
\sup\{\lambda:\ell_{\lambda,\mathrm{amb}}(U_n)<\ellRes(U_n)\}.
```

\lambda_cDefinition passage · 4 passages

Search aliases: λ_c lambda

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Definition passage: Critical catalyst penalty · Read passage · g_main.tex, line 424
[Critical catalyst penalty] For a target family U_n, define λ_c(U_n) = sup{λ:ℓ_λ,amb(U_n)<ellRes(U_n)}. Determining λ_c exactly is a refined synthesis problem. It can be more physically meaningful than unweighted gate count.
Exact mathematical passage
```
\lambda_c(U_n)
=
\sup\{\lambda:\ell_{\lambda,\mathrm{amb}}(U_n)<\ellRes(U_n)\}.
```

\tauDefinition passage · 45 passages

Search aliases: τ tau

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Definition passage: Extremal distortion · Read passage · g_main.tex, line 141
Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.
Exact mathematical passage
```
\tau=\min\{n:D_n>0\}.
```

\tau_{p,q}Definition passage · 3 passages

Search aliases: τ_p,q tau

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1468
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{p,q}(-a)\,h_m\cdots h_1\,\tau_{p,q}(a),
\qquad p<q,
```

\tau_{p,x}Definition passage · 2 passages

Search aliases: τ_p,x tau

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1473
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{q,x}(b),\quad \tau_{p,x}(b),\quad
\tau_{y,p}(b),\quad \tau_{y,q}(b)
```

\tau_{q,x}Definition passage · 2 passages

Search aliases: τ_q,x tau

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1473
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{q,x}(b),\quad \tau_{p,x}(b),\quad
\tau_{y,p}(b),\quad \tau_{y,q}(b)
```

\tau_{y,p}Definition passage · 2 passages

Search aliases: τ_y,p tau

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1473
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{q,x}(b),\quad \tau_{p,x}(b),\quad
\tau_{y,p}(b),\quad \tau_{y,q}(b)
```

\tau_{y,q}Definition passage · 2 passages

Search aliases: τ_y,q tau

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Definition passage: Simple one-catalyst word · Read passage · g_main.tex, line 1473
[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.
Exact mathematical passage
```
\tau_{q,x}(b),\quad \tau_{p,x}(b),\quad
\tau_{y,p}(b),\quad \tau_{y,q}(b)
```

\tau_\otimesDefinition passage · 2 passages

Search aliases: τ_⊗ tau otimes

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Definition passage: Stable tensor cost · Read passage · g_main.tex, line 661
[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.
Exact mathematical passage
```
\tau_\otimes(U)
=
\lim_{r\to\infty}
\frac{\ellAmb(U^{\otimes r})}{r m^{r-1}},
```

\PhiDefinition passage · 30 passages

Search aliases: Φ Phi

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Definition passage: Interval tree · Read passage · g_main.tex, line 1946
[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).
Exact mathematical passage
```
\label{g:eq:tree-potential}
\Phi_T(A)
=
\sum_{v\in\operatorname{Int}(T)}
\rank A[R_v,L_v],
\qquad A\in\U_n(\F).
```

\varphiDefinition passage · 14 passages

Search aliases: φ varphi

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Definition passage: Cost-preserving equivalence · Read passage · g_main.tex, line 159
[Cost-preserving equivalence] Targets U and V are equivalent if an automorphism φ of the ambient realization system satisfies φ(U)=V, φ( S_res)= S_res, φ( S_amb)= S_amb, and preserves the gate costs. Under this equivalence, both lengths and the defect are orbit invariants. A useful threshold theorem should classify minimal defective orbits, not merely list raw matrices.
Exact mathematical passage
```
\varphi
```

\Phi_TDefinition passage · 5 passages

Search aliases: Φ_T Phi

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Definition passage: Interval tree · Read passage · g_main.tex, line 1946
[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).
Exact mathematical passage
```
\label{g:eq:tree-potential}
\Phi_T(A)
=
\sum_{v\in\operatorname{Int}(T)}
\rank A[R_v,L_v],
\qquad A\in\U_n(\F).
```

\ldotsDefinition passage · 82 passages

Search aliases: … ldots

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Definition passage: Resource vector · Read passage · g_main.tex, line 82
[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.
Exact mathematical passage
```
\mathbf c(s)=(c_1(s),\ldots,c_r(s))\in\R_{\ge0}^r
```

\ellDefinition passage · 64 passages

Search aliases: ℓ ell

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Definition passage: Gate model · Read passage · g_main.tex, line 39
te model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
\ell_{\mathfrak M}(U)=\inf\{c(w):w=U\}.
```

\ell_{\mathfrak M}Definition passage · 1 passage

Search aliases: ℓ_ M ell mathfrak

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Definition passage: Gate model · Read passage · g_main.tex, line 39
te model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
\ell_{\mathfrak M}(U)=\inf\{c(w):w=U\}.
```

\ell_{\mathfrak M_{\mathrm{amb}}}Definition passage · 1 passage

Search aliases: ℓ_ M_amb ell mathfrak mathrm

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 53
[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\ellRes(U)=\ell_{\mathfrak M_{\mathrm{res}}}(U),
\qquad
\ellAmb(U)=\ell_{\mathfrak M_{\mathrm{amb}}}(U).
```

\ell_{\mathfrak M_{\mathrm{res}}}Definition passage · 1 passage

Search aliases: ℓ_ M_res ell mathfrak mathrm

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 53
[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\ellRes(U)=\ell_{\mathfrak M_{\mathrm{res}}}(U),
\qquad
\ellAmb(U)=\ell_{\mathfrak M_{\mathrm{amb}}}(U).
```

\ell_{\lambda,\mathrm{amb}}Definition passage · 1 passage

Search aliases: ℓ_λ,amb ell lambda mathrm

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Definition passage: Critical catalyst penalty · Read passage · g_main.tex, line 424
[Critical catalyst penalty] For a target family U_n, define λ_c(U_n) = sup{λ:ℓ_λ,amb(U_n)<ellRes(U_n)}. Determining λ_c exactly is a refined synthesis problem. It can be more physically meaningful than unweighted gate count.
Exact mathematical passage
```
\lambda_c(U_n)
=
\sup\{\lambda:\ell_{\lambda,\mathrm{amb}}(U_n)<\ellRes(U_n)\}.
```

\toDefinition passage · 74 passages

Search aliases: → to

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Definition passage: Gate model · Read passage · g_main.tex, line 30
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
c:\mathcal S\to\R_{\ge0}
```

\inDefinition passage · 215 passages

Search aliases: ∈ in

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Definition passage: Gate model · Read passage · g_main.tex, line 38
begindefinition[Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
U\in\mathcal X
```

\sumDefinition passage · 160 passages

Search aliases: ∑ sum

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Definition passage: Gate model · Read passage · g_main.tex, line 35
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
c(w)=\sum_{j=1}^L c(s_j).
```

\sum_{j=1}^LDefinition passage · 2 passages

Search aliases: ∑_j=1^L sum

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Definition passage: Gate model · Read passage · g_main.tex, line 35
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
c(w)=\sum_{j=1}^L c(s_j).
```

\sum_{v\in\operatorname{Int}(T)}Definition passage · 1 passage

Search aliases: ∑_v∈Int(T) sum in operatorname

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Definition passage: Interval tree · Read passage · g_main.tex, line 1946
[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).
Exact mathematical passage
```
\label{g:eq:tree-potential}
\Phi_T(A)
=
\sum_{v\in\operatorname{Int}(T)}
\rank A[R_v,L_v],
\qquad A\in\U_n(\F).
```

\circDefinition passage · 4 passages

Search aliases: ∘ circ

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Definition passage: Gate model · Read passage · g_main.tex, line 27
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
\mathfrak M=(\mathcal X,\mathcal S,c,\circ),
```

\inftyDefinition passage · 18 passages

Search aliases: ∞ infty

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Definition passage: Gate model · Read passage · g_main.tex, line 42
,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
+\infty
```

\cupDefinition passage · 1 passage

Search aliases: ∪ cup

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Definition passage: Parabolic and ambient Clifford metrics · Read passage · g_main.tex, line 3418
[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.
Exact mathematical passage
```
\mathcal S_C
=
\mathcal S_P\cup\{H_i\}.
```

\leDefinition passage · 183 passages

Search aliases: ≤ le

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Definition passage: Upper-gate budget · Read passage · g_main.tex, line 2221
[Upper-gate budget] For q≥0 and A∈U_n(F), define ellBudgetq(A) = min{|w|:[w]=A, u(w)≤ q}. Thus ellBudget0(A)=ellDown(A), ellBudgetq+1(A)≤ellBudgetq(A), ellAll(A)≤ellBudgetq(A).
Exact mathematical passage
```
\ellBudget{q}(A)
=
\min\{|w|:[w]=A,\ u(w)\le q\}.
```

\geDefinition passage · 128 passages

Search aliases: ≥ ge

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Definition passage: Gate model · Read passage · g_main.tex, line 30
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
c:\mathcal S\to\R_{\ge0}
```

\subseteqDefinition passage · 25 passages

Search aliases: ⊆ subseteq

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Definition passage: Nested realization pair · Read passage · g_main.tex, line 49
[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).
Exact mathematical passage
```
\mathfrak M_{\mathrm{res}}\subseteq\mathfrak M_{\mathrm{amb}}
```

\otimesDefinition passage · 54 passages

Search aliases: ⊗ otimes

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Definition passage: Tensor synergy · Read passage · g_main.tex, line 650
[Tensor synergy] Define Syn(U,V) = nellAmb(U)+mellAmb(V)-ellAmb(U⊗ V). Positive synergy means that the joint ambient realization improves on independent lifted circuits. Restricted tree-tight metrics have zero tensor synergy. Ambient synergy is a possible source of collective activation.
Exact mathematical passage
```
\Syn(U,V)
=
n\ellAmb(U)+m\ellAmb(V)-\ellAmb(U\otimes V).
```

\cdotsDefinition passage · 46 passages

Search aliases: ⋯ cdots

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Definition passage: Gate model · Read passage · g_main.tex, line 31
Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.
Exact mathematical passage
```
w=s_L\cdots s_1
```

A_nIntroduction passage · 20 passages

Search aliases: A_n

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Introduction passage · Read passage · g_main.tex, line 1056
aths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be chosen so that ellExc00(A_n)-ellExc11(A_n) ≥ lfloorfracn-52rfloor. Thus a rank-one excursion across one adjacent flag layer can create a linear saving.
Exact mathematical passage
```
\ellExc{0}{0}(A_n)-\ellExc{1}{1}(A_n)
\ge
\left\lfloor\frac{n-5}{2}\right\rfloor.
```

BIntroduction passage · 104 passages

Search aliases: B

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Introduction passage · Read passage · c_visual.tex, line 655
guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s) (∑_t B(t)z_T^t). Each factor is traced on its own variable block. The union trace is then their product. This is an ordinary factorization of a finit…
Exact mathematical passage
```
B
```

b^+Introduction passage · 5 passages

Search aliases: b^+

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Introduction passage · Read passage · g_main.tex, line 1055
The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be chosen so that ellExc00(A_n)-ellExc11(A_n) ≥ lfloorfracn-52rfloor. Thus a rank-one excursion across one adjacent flag layer can create a line…
Exact mathematical passage
```
b^+(X)
```

\mathbb CIntroduction passage · 7 passages

Search aliases: C mathbb

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Introduction passage · Read passage · q_main.tex, line 113
A signal is a vector x:I_q→ C. We write Id_q for the identity matrix of dimension q, and I for an identity matrix when its dimension is clear. Our inner product is conjugate-linear in the first argument:
Exact mathematical passage
```
x:I_q\to\mathbb C
```

\DegIntroduction passage · 5 passages

Search aliases: Deg

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Introduction passage · Read passage · c_lean.tex, line 36
ocsectionThe formal endpoint Let c be a finite legal classical NOT/CNOT/Toffoli circuit. For n input bits and a zero-initialized ancillas, assume for every x and both initial target values b that run(c)(x,0^a,b)=(x,0^a,bx f(x)). Write T=TC(c) for its number of Toffoli gates. The Block 02 joint endpoint is Degree and complete-cover bound Deg(T+1,f) and 2β^*(f)≤ T+3. pathrefCircuitObstructions.restored_degree_and_cover_bound The first conclusion is an explicit certificate that f has a Boolean poly…
Exact mathematical passage
```
\boxed{\Deg(T+1,f)\quad\text{and}\quad 2\beta^*(f)\le T+3.}
```

eIntroduction passage · 134 passages

Search aliases: e

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Introduction passage · Read passage · c_main.tex, line 451
tionExtracting the quadratic grade [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The map v↦ v+a is a fixed-point-free involution of V, so the displayed terms form n/2 disjo…
Exact mathematical passage
```
P_m(0)=P_m(e_v)=0
```

e_uIntroduction passage · 8 passages

Search aliases: e_u

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Introduction passage · Read passage · c_main.tex, line 452
labelc:thm:grade-cost Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The map v↦ v+a is a fixed-point-free involution of V, so the displayed terms form n/2 disjoint pairs. Proposition~[source reference] supplie…
Exact mathematical passage
```
 [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].
```

e_vIntroduction passage · 10 passages

Search aliases: e_v

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Introduction passage · Read passage · c_main.tex, line 451
tionExtracting the quadratic grade [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The map v↦ v+a is a fixed-point-free involution of V, so the displayed terms form n/2 disjo…
Exact mathematical passage
```
P_m(0)=P_m(e_v)=0
```

\ellExcIntroduction passage · 6 passages

Search aliases: ellExc

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Introduction passage · Read passage · g_main.tex, line 1055
The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be chosen so that ellExc00(A_n)-ellExc11(A_n) ≥ lfloorfracn-52rfloor. Thus a rank-one excursion across one adjacent flag layer can create a line…
Exact mathematical passage
```
\ellExc{r}{b}
```

FIntroduction passage · 88 passages

Search aliases: F

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Introduction passage · Read passage · q_main.tex, line 50
Here D is a diagonal zero-or-one matrix, F is unitary, and † denotes conjugate transpose. Consequently P is an orthogonal projector. The transformation F is reversible. The selection P need not be. A complete decomposition consists of both Px and (I-P)x, whose sum recovers x.
Exact mathematical passage
```
F
```

\F_2^mIntroduction passage · 1 passage

Search aliases: F_2^m F

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Introduction passage · Read passage · c_visual.tex, line 416
guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. Its restored nonlinear cost is 2m-3.
Exact mathematical passage
```
 V=\F_2^m\setminus\{0,a\},\qquad n=2^m-2.
```

I_NIntroduction passage · 5 passages

Search aliases: I_N

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Introduction passage · Read passage · q_main.tex, line 604
Write widehatx_r for the unscaled M-point transform of lane r. Then for k∈ I_N,
Exact mathematical passage
```
k\in I_N
```

I_qIntroduction passage · 6 passages

Search aliases: I_q

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Introduction passage · Read passage · q_main.tex, line 113
A signal is a vector x:I_q→ C. We write Id_q for the identity matrix of dimension q, and I for an identity matrix when its dimension is clear. Our inner product is conjugate-linear in the first argument:
Exact mathematical passage
```
x:I_q\to\mathbb C
```

\operatorname{Id}Introduction passage · 4 passages

Search aliases: Id operatorname

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Introduction passage · Read passage · q_main.tex, line 113
A signal is a vector x:I_q→ C. We write Id_q for the identity matrix of dimension q, and I for an identity matrix when its dimension is clear. Our inner product is conjugate-linear in the first argument:
Exact mathematical passage
```
\operatorname{Id}_q
```

\operatorname{Id}_qIntroduction passage · 2 passages

Search aliases: Id_q operatorname

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Introduction passage · Read passage · q_main.tex, line 113
A signal is a vector x:I_q→ C. We write Id_q for the identity matrix of dimension q, and I for an identity matrix when its dimension is clear. Our inner product is conjugate-linear in the first argument:
Exact mathematical passage
```
\operatorname{Id}_q
```

kIntroduction passage · 223 passages

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Introduction passage · Read passage · q_main.tex, line 604
Write widehatx_r for the unscaled M-point transform of lane r. Then for k∈ I_N,
Exact mathematical passage
```
k\in I_N
```

L_jIntroduction passage · 2 passages

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Introduction passage · Read passage · c_visual.tex, line 420
guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. Its restored nonlinear cost is 2m-3.
Exact mathematical passage
```
 L_j(x)=\sum_{v:v_j=1}x_v,\qquad
 P_m(x)=\prod_{j=1}^m L_j(x).
```

\lfloorIntroduction passage · 20 passages

Search aliases: lfloor

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Introduction passage · Read passage · g_main.tex, line 1056
aths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be chosen so that ellExc00(A_n)-ellExc11(A_n) ≥ lfloorfracn-52rfloor. Thus a rank-one excursion across one adjacent flag layer can create a linear saving.
Exact mathematical passage
```
\ellExc{0}{0}(A_n)-\ellExc{1}{1}(A_n)
\ge
\left\lfloor\frac{n-5}{2}\right\rfloor.
```

\min_{a:\,P_a=P}Introduction passage · 1 passage

Search aliases: min_a: P_a=P min

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Introduction passage · Read passage · g_main.tex, line 3978
Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives ambient Clifford conjugation must therefore be invariant under the relevant symplectic action or must explicitly charge the frame change.
Exact mathematical passage
```
\ell_T(P)=
\min_{a:\,P_a=P}
\operatorname{wt}(\operatorname{odd}(a)).
```

NIntroduction passage · 141 passages

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Introduction passage · Read passage · q_main.tex, line 604
Write widehatx_r for the unscaled M-point transform of lane r. Then for k∈ I_N,
Exact mathematical passage
```
k\in I_N
```

\operatorname{odd}Introduction passage · 2 passages

Search aliases: odd operatorname

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Introduction passage · Read passage · g_main.tex, line 3977
Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives ambient Clifford conjugation must therefore be invariant under the relevant symplectic action or must explicitly charge the frame change.
Exact mathematical passage
```
\operatorname{odd}(a)
```

P_aIntroduction passage · 1 passage

Search aliases: P_a

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Introduction passage · Read passage · g_main.tex, line 3978
Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives ambient Clifford conjugation must therefore be invariant under the relevant symplectic action or must explicitly charge the frame change.
Exact mathematical passage
```
\ell_T(P)=
\min_{a:\,P_a=P}
\operatorname{wt}(\operatorname{odd}(a)).
```

QIntroduction passage · 64 passages

Search aliases: Q

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Introduction passage · Read passage · c_main.tex, line 441
Extracting the quadratic grade. [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The map v↦ v+a is a fixed-point-free involution of V, so the displayed terms form n/2 disjoint…
Exact mathematical passage
```
Q_m=[P_m]_2
```

Q_mIntroduction passage · 8 passages

Search aliases: Q_m

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Introduction passage · Read passage · c_main.tex, line 441
Extracting the quadratic grade. [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The map v↦ v+a is a fixed-point-free involution of V, so the displayed terms form n/2 disjoint…
Exact mathematical passage
```
Q_m=[P_m]_2
```

r^+Introduction passage · 4 passages

Search aliases: r^+

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Introduction passage · Read passage · g_main.tex, line 1055
The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be chosen so that ellExc00(A_n)-ellExc11(A_n) ≥ lfloorfracn-52rfloor. Thus a rank-one excursion across one adjacent flag layer can create a line…
Exact mathematical passage
```
r^+(X)=\rank X^+
```

\operatorname{Real}Introduction passage · 2 passages

Search aliases: Real operatorname

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Introduction passage · Read passage · g_main.tex, line 4677
Realization fibers. Let P denote packet data and let Real_res(P) be the space of restricted realizations compatible with P. The packet may completely classify the discrete object while the realization fiber remains empty, positive dimensional, disconnected, or singular.
Exact mathematical passage
```
\operatorname{Real}_{\mathrm{res}}(P)
```

\operatorname{Real}_{\mathrm{res}}Introduction passage · 1 passage

Search aliases: Real_res operatorname mathrm

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Introduction passage · Read passage · g_main.tex, line 4677
Realization fibers. Let P denote packet data and let Real_res(P) be the space of restricted realizations compatible with P. The packet may completely classify the discrete object while the realization fiber remains empty, positive dimensional, disconnected, or singular.
Exact mathematical passage
```
\operatorname{Real}_{\mathrm{res}}(P)
```

\rfloorIntroduction passage · 20 passages

Search aliases: rfloor

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Introduction passage · Read passage · g_main.tex, line 1056
aths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be chosen so that ellExc00(A_n)-ellExc11(A_n) ≥ lfloorfracn-52rfloor. Thus a rank-one excursion across one adjacent flag layer can create a linear saving.
Exact mathematical passage
```
\ellExc{0}{0}(A_n)-\ellExc{1}{1}(A_n)
\ge
\left\lfloor\frac{n-5}{2}\right\rfloor.
```

\operatorname{run}Introduction passage · 3 passages

Search aliases: run operatorname

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Introduction passage · Read passage · c_lean.tex, line 31
the original rights record and the original release files. There is no new publication-novelty or unrestricted hardness claim. The formal endpoint. addcontentslinetocsectionThe formal endpoint Let c be a finite legal classical NOT/CNOT/Toffoli circuit. For n input bits and a zero-initialized ancillas, assume for every x and both initial target values b that run(c)(x,0^a,b)=(x,0^a,bx f(x)). Write T=TC(c) for its number of Toffoli gates. The Block 02 joint endpoint is Degree and complete-cover bou…
Exact mathematical passage
```
\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\x f(x)).
```

s_vIntroduction passage · 15 passages

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Introduction passage · Read passage · g_main.tex, line 3278
Connected higher-dimensional qudit families. The snapshot target A(u;s_1,…,s_v) from the core chapter is defined over any coefficient ring for which the displayed additions make sense. Over Z_d, it becomes a modular SUM circuit. Its explicit ambient word uses exactly two reverse-directed SUM gates and has length u+v+2. The interval-tree proof of exact restricted length is most transparent over fields, but the displayed upper-bound and resource-amortization mechanism remain valid over every Z_d.
Exact mathematical passage
```
A(u;s_1,\ldots,s_v)
```

T_{ab\to t}Introduction passage · 2 passages

Search aliases: T_ab→ t to

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Introduction passage · Read passage · g_main.tex, line 3791
Conjugation formulas. Let T_ab→ t denote Toffoli. Conjugating an affine gate by T can produce nonlinear permutations. For example, if X_t flips the target, T_ab→ tX_tT_ab→ t=X_t, so no gain occurs. If a CNOT changes a control or exports the target, the conjugated map can contain several quadratic terms. The exact formula is easiest in algebraic normal form: substitute the Toffoli-updated coordinate into the affine map, then apply the inverse substitution.
Exact mathematical passage
```
T_{ab\to t}
```

\TCIntroduction passage · 4 passages

Search aliases: TC

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Introduction passage · Read passage · c_lean.tex, line 34
here is no new publication-novelty or unrestricted hardness claim. The formal endpoint. addcontentslinetocsectionThe formal endpoint Let c be a finite legal classical NOT/CNOT/Toffoli circuit. For n input bits and a zero-initialized ancillas, assume for every x and both initial target values b that run(c)(x,0^a,b)=(x,0^a,bx f(x)). Write T=TC(c) for its number of Toffoli gates. The Block 02 joint endpoint is Degree and complete-cover bound Deg(T+1,f) and 2β^*(f)≤ T+3. pathrefCircuitObstructions.r…
Exact mathematical passage
```
T=\TC(c)
```

u_nIntroduction passage · 4 passages

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Introduction passage · Read passage · g_main.tex, line 2132
[Proof of [source reference]] For n≥7, put v_n=lfloorfracn-12rfloor, u_n=n-v_n-2, and take the phases 0,1,…,v_n-1. The inequality v_n≤ u_n+1 holds, so the snapshot target is defined. By [source reference], ellDown(A_n)=u_n+2v_n=n+v_n-2, while ellAll(A_n)≤ u_n+v_n+2=n. Hence ellDown(A_n)-ellAll(A_n) ≥ v_n-2 = lfloorfracn-52rfloor. Connectivity follows from the preceding remark.
Exact mathematical passage
```
v_n=\left\lfloor\frac{n-1}{2}\right\rfloor,
\qquad
u_n=n-v_n-2,
```

v_jIntroduction passage · 2 passages

Search aliases: v_j

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Introduction passage · Read passage · c_visual.tex, line 420
guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. Its restored nonlinear cost is 2m-3.
Exact mathematical passage
```
 L_j(x)=\sum_{v:v_j=1}x_v,\qquad
 P_m(x)=\prod_{j=1}^m L_j(x).
```

v_nIntroduction passage · 6 passages

Search aliases: v_n

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Introduction passage · Read passage · g_main.tex, line 2132
[Proof of [source reference]] For n≥7, put v_n=lfloorfracn-12rfloor, u_n=n-v_n-2, and take the phases 0,1,…,v_n-1. The inequality v_n≤ u_n+1 holds, so the snapshot target is defined. By [source reference], ellDown(A_n)=u_n+2v_n=n+v_n-2, while ellAll(A_n)≤ u_n+v_n+2=n. Hence ellDown(A_n)-ellAll(A_n) ≥ v_n-2 = lfloorfracn-52rfloor. Connectivity follows from the preceding remark.
Exact mathematical passage
```
v_n=\left\lfloor\frac{n-1}{2}\right\rfloor,
\qquad
u_n=n-v_n-2,
```

\widehat{x_r}Introduction passage · 5 passages

Search aliases: widehatx_r widehat

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Introduction passage · Read passage · q_main.tex, line 603
Write widehatx_r for the unscaled M-point transform of lane r. Then for k∈ I_N,
Exact mathematical passage
```
\widehat{x_r}
```

\operatorname{wt}Introduction passage · 1 passage

Search aliases: wt operatorname

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Introduction passage · Read passage · g_main.tex, line 3978
Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives ambient Clifford conjugation must therefore be invariant under the relevant symplectic action or must explicitly charge the frame change.
Exact mathematical passage
```
\ell_T(P)=
\min_{a:\,P_a=P}
\operatorname{wt}(\operatorname{odd}(a)).
```

\xIntroduction passage · 6 passages

Search aliases: x

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Introduction passage · Read passage · c_lean.tex, line 31
the original rights record and the original release files. There is no new publication-novelty or unrestricted hardness claim. The formal endpoint. addcontentslinetocsectionThe formal endpoint Let c be a finite legal classical NOT/CNOT/Toffoli circuit. For n input bits and a zero-initialized ancillas, assume for every x and both initial target values b that run(c)(x,0^a,b)=(x,0^a,bx f(x)). Write T=TC(c) for its number of Toffoli gates. The Block 02 joint endpoint is Degree and complete-cover bou…
Exact mathematical passage
```
\operatorname{run}(c)(x,0^a,b)=(x,0^a,b\x f(x)).
```

X^+Introduction passage · 5 passages

Search aliases: X^+

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Introduction passage · Read passage · g_main.tex, line 1055
The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be chosen so that ellExc00(A_n)-ellExc11(A_n) ≥ lfloorfracn-52rfloor. Thus a rank-one excursion across one adjacent flag layer can create a line…
Exact mathematical passage
```
X^+
```

X_{ij}Introduction passage · 2 passages

Search aliases: X_ij

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Introduction passage · Read passage · g_main.tex, line 1055
The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be chosen so that ellExc00(A_n)-ellExc11(A_n) ≥ lfloorfracn-52rfloor. Thus a rank-one excursion across one adjacent flag layer can create a line…
Exact mathematical passage
```
X_{ij}\ne0
```

X_sIntroduction passage · 1 passage

Search aliases: X_s

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Introduction passage · Read passage · g_main.tex, line 3313
Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t, Z_s · ↦ Z_s, · Z_t · ↦ Z_s^-aZ_t. Thus the modular linear action embeds into a symplectic transformation. The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency changes both X propagation and the contragredient Z propagation, then restores them.
Exact mathematical passage
```
X_s&\mapsto X_sX_t^a,
&
X_t&\mapsto X_t,\\
Z_s&\mapsto Z_s,
&
Z_t&\mapsto Z_s^{-a}Z_t.
```

X_t^aIntroduction passage · 1 passage

Search aliases: X_t^a

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Introduction passage · Read passage · g_main.tex, line 3313
Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t, Z_s · ↦ Z_s, · Z_t · ↦ Z_s^-aZ_t. Thus the modular linear action embeds into a symplectic transformation. The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency changes both X propagation and the contragredient Z propagation, then restores them.
Exact mathematical passage
```
X_s&\mapsto X_sX_t^a,
&
X_t&\mapsto X_t,\\
Z_s&\mapsto Z_s,
&
Z_t&\mapsto Z_s^{-a}Z_t.
```

x_uIntroduction passage · 5 passages

Search aliases: x_u

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Introduction passage · Read passage · c_main.tex, line 452
labelc:thm:grade-cost Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The map v↦ v+a is a fixed-point-free involution of V, so the displayed terms form n/2 disjoint pairs. Proposition~[source reference] supplie…
Exact mathematical passage
```
 [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a].
```

x_vIntroduction passage · 10 passages

Search aliases: x_v

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Introduction passage · Read passage · c_main.tex, line 441
Extracting the quadratic grade. [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The map v↦ v+a is a fixed-point-free involution of V, so the displayed terms form n/2 disjoint…
Exact mathematical passage
```
x_v
```

x_{v+a}Introduction passage · 2 passages

Search aliases: x_v+a

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Introduction passage · Read passage · c_main.tex, line 442
Extracting the quadratic grade. [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The map v↦ v+a is a fixed-point-free involution of V, so the displayed terms form n/2 disjoint…
Exact mathematical passage
```
 Q_m=\sum_{\{v,v+a\}\subseteq V}x_vx_{v+a},
```

ZIntroduction passage · 17 passages

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Introduction passage · Read passage · g_main.tex, line 3312
Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t,
Exact mathematical passage
```
Z
```

\ZIntroduction passage · 14 passages

Search aliases: Z

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Introduction passage · Read passage · g_main.tex, line 3278
Connected higher-dimensional qudit families. The snapshot target A(u;s_1,…,s_v) from the core chapter is defined over any coefficient ring for which the displayed additions make sense. Over Z_d, it becomes a modular SUM circuit. Its explicit ambient word uses exactly two reverse-directed SUM gates and has length u+v+2. The interval-tree proof of exact restricted length is most transparent over fields, but the displayed upper-bound and resource-amortization mechanism remain valid over every Z_d.
Exact mathematical passage
```
\Z_d
```

zIntroduction passage · 109 passages

Search aliases: z

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Introduction passage · Read passage · c_visual.tex, line 656
guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s) (∑_t B(t)z_T^t). Each factor is traced on its own variable block. The union trace is then their product. This is an ordinary factorization of a finit…
Exact mathematical passage
```
\tau_{z_S}(A)=\sum_s A(s)z_S^s
```

z^{c(s)}Introduction passage · 1 passage

Search aliases: z^c(s)

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Introduction passage · Read passage · g_main.tex, line 485
For strictly positive integer costs, replace each generator s by the monomial z^c(s)ρ(s). Every coefficient then counts finitely many words of a given total cost, and the same formal inverse is defined. Zero-cost cycles require separate treatment: they can create infinitely many words of cost zero, so the ordinary counting series need not exist. The finite atlases with free gates use shortest-path search with zero-cost edges instead. Multivariate variables can track several resources when the gr…
Exact mathematical passage
```
z^{c(s)}\rho(s)
```

\Z_8^{2^n-1}Introduction passage · 1 passage

Search aliases: Z_8^2^n-1 Z

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Introduction passage · Read passage · g_main.tex, line 3977
Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives ambient Clifford conjugation must therefore be invariant under the relevant symplectic action or must explicitly charge the frame change.
Exact mathematical passage
```
a\in\Z_8^{2^n-1}
```

\Z_dIntroduction passage · 8 passages

Search aliases: Z_d Z

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Introduction passage · Read passage · g_main.tex, line 3278
Connected higher-dimensional qudit families. The snapshot target A(u;s_1,…,s_v) from the core chapter is defined over any coefficient ring for which the displayed additions make sense. Over Z_d, it becomes a modular SUM circuit. Its explicit ambient word uses exactly two reverse-directed SUM gates and has length u+v+2. The interval-tree proof of exact restricted length is most transparent over fields, but the displayed upper-bound and resource-amortization mechanism remain valid over every Z_d.
Exact mathematical passage
```
\Z_d
```

Z_sIntroduction passage · 1 passage

Search aliases: Z_s

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Introduction passage · Read passage · g_main.tex, line 3313
Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t, Z_s · ↦ Z_s, · Z_t · ↦ Z_s^-aZ_t. Thus the modular linear action embeds into a symplectic transformation. The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency changes both X propagation and the contragredient Z propagation, then restores them.
Exact mathematical passage
```
X_s&\mapsto X_sX_t^a,
&
X_t&\mapsto X_t,\\
Z_s&\mapsto Z_s,
&
Z_t&\mapsto Z_s^{-a}Z_t.
```

z_SIntroduction passage · 1 passage

Search aliases: z_S

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Introduction passage · Read passage · c_visual.tex, line 656
guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s) (∑_t B(t)z_T^t). Each factor is traced on its own variable block. The union trace is then their product. This is an ordinary factorization of a finit…
Exact mathematical passage
```
\tau_{z_S}(A)=\sum_s A(s)z_S^s
```

Z_s^{-a}Introduction passage · 1 passage

Search aliases: Z_s^-a

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Introduction passage · Read passage · g_main.tex, line 3313
Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t, Z_s · ↦ Z_s, · Z_t · ↦ Z_s^-aZ_t. Thus the modular linear action embeds into a symplectic transformation. The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency changes both X propagation and the contragredient Z propagation, then restores them.
Exact mathematical passage
```
X_s&\mapsto X_sX_t^a,
&
X_t&\mapsto X_t,\\
Z_s&\mapsto Z_s,
&
Z_t&\mapsto Z_s^{-a}Z_t.
```

z_S^sIntroduction passage · 2 passages

Search aliases: z_S^s

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Introduction passage · Read passage · c_visual.tex, line 656
guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s) (∑_t B(t)z_T^t). Each factor is traced on its own variable block. The union trace is then their product. This is an ordinary factorization of a finit…
Exact mathematical passage
```
\tau_{z_S}(A)=\sum_s A(s)z_S^s
```

Z_tIntroduction passage · 1 passage

Search aliases: Z_t

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Introduction passage · Read passage · g_main.tex, line 3313
Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t, Z_s · ↦ Z_s, · Z_t · ↦ Z_s^-aZ_t. Thus the modular linear action embeds into a symplectic transformation. The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency changes both X propagation and the contragredient Z propagation, then restores them.
Exact mathematical passage
```
X_s&\mapsto X_sX_t^a,
&
X_t&\mapsto X_t,\\
Z_s&\mapsto Z_s,
&
Z_t&\mapsto Z_s^{-a}Z_t.
```

z_T^tIntroduction passage · 1 passage

Search aliases: z_T^t

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Introduction passage · Read passage · c_visual.tex, line 659
guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s) (∑_t B(t)z_T^t). Each factor is traced on its own variable block. The union trace is then their product. This is an ordinary factorization of a finit…
Exact mathematical passage
```
 \sum_{s,t} A(s)B(t)z_S^s z_T^t
 =\left(\sum_s A(s)z_S^s\right)
  \left(\sum_t B(t)z_T^t\right).
```

\rhoIntroduction passage · 36 passages

Search aliases: ρ rho

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Introduction passage · Read passage · g_main.tex, line 485
For strictly positive integer costs, replace each generator s by the monomial z^c(s)ρ(s). Every coefficient then counts finitely many words of a given total cost, and the same formal inverse is defined. Zero-cost cycles require separate treatment: they can create infinitely many words of cost zero, so the ordinary counting series need not exist. The finite atlases with free gates use shortest-path search with zero-cost edges instead. Multivariate variables can track several resources when the gr…
Exact mathematical passage
```
z^{c(s)}\rho(s)
```

\tau_wIntroduction passage · 5 passages

Search aliases: τ_w tau

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Introduction passage · Read passage · c_visual.tex, line 657
guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s) (∑_t B(t)z_T^t). Each factor is traced on its own variable block. The union trace is then their product. This is an ordinary factorization of a finit…
Exact mathematical passage
```
\tau_w
```

\tau_{z_S}Introduction passage · 1 passage

Search aliases: τ_z_S tau

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Introduction passage · Read passage · c_visual.tex, line 656
guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s) (∑_t B(t)z_T^t). Each factor is traced on its own variable block. The union trace is then their product. This is an ordinary factorization of a finit…
Exact mathematical passage
```
\tau_{z_S}(A)=\sum_s A(s)z_S^s
```

\daggerIntroduction passage · 56 passages

Search aliases: † dagger

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Introduction passage · Read passage · q_main.tex, line 51
Here D is a diagonal zero-or-one matrix, F is unitary, and † denotes conjugate transpose. Consequently P is an orthogonal projector. The transformation F is reversible. The selection P need not be. A complete decomposition consists of both Px and (I-P)x, whose sum recovers x.
Exact mathematical passage
```
\dagger
```

\ell_TIntroduction passage · 3 passages

Search aliases: ℓ_T ell

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Introduction passage · Read passage · g_main.tex, line 3978
Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives ambient Clifford conjugation must therefore be invariant under the relevant symplectic action or must explicitly charge the frame change.
Exact mathematical passage
```
\ell_T(P)=
\min_{a:\,P_a=P}
\operatorname{wt}(\operatorname{odd}(a)).
```

\mapstoIntroduction passage · 20 passages

Search aliases: ↦ mapsto

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Introduction passage · Read passage · c_visual.tex, line 424
guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. Its restored nonlinear cost is 2m-3.
Exact mathematical passage
```
x\mapsto L(x)
```

\prodIntroduction passage · 24 passages

Search aliases: ∏ prod

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Introduction passage · Read passage · c_visual.tex, line 420
guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. Its restored nonlinear cost is 2m-3.
Exact mathematical passage
```
 L_j(x)=\sum_{v:v_j=1}x_v,\qquad
 P_m(x)=\prod_{j=1}^m L_j(x).
```

\prod_{j=1}^mIntroduction passage · 4 passages

Search aliases: ∏_j=1^m prod

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Introduction passage · Read passage · c_visual.tex, line 420
guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. Its restored nonlinear cost is 2m-3.
Exact mathematical passage
```
 L_j(x)=\sum_{v:v_j=1}x_v,\qquad
 P_m(x)=\prod_{j=1}^m L_j(x).
```

\sum_sIntroduction passage · 2 passages

Search aliases: ∑_s sum

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Introduction passage · Read passage · c_visual.tex, line 656
guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s) (∑_t B(t)z_T^t). Each factor is traced on its own variable block. The union trace is then their product. This is an ordinary factorization of a finit…
Exact mathematical passage
```
\tau_{z_S}(A)=\sum_s A(s)z_S^s
```

\sum_{s,t}Introduction passage · 1 passage

Search aliases: ∑_s,t sum

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Introduction passage · Read passage · c_visual.tex, line 659
guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s) (∑_t B(t)z_T^t). Each factor is traced on its own variable block. The union trace is then their product. This is an ordinary factorization of a finit…
Exact mathematical passage
```
 \sum_{s,t} A(s)B(t)z_S^s z_T^t
 =\left(\sum_s A(s)z_S^s\right)
  \left(\sum_t B(t)z_T^t\right).
```

\sum_tIntroduction passage · 3 passages

Search aliases: ∑_t sum

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Introduction passage · Read passage · c_visual.tex, line 659
guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s) (∑_t B(t)z_T^t). Each factor is traced on its own variable block. The union trace is then their product. This is an ordinary factorization of a finit…
Exact mathematical passage
```
 \sum_{s,t} A(s)B(t)z_S^s z_T^t
 =\left(\sum_s A(s)z_S^s\right)
  \left(\sum_t B(t)z_T^t\right).
```

\sum_{v:v_j=1}Introduction passage · 2 passages

Search aliases: ∑_v:v_j=1 sum

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Introduction passage · Read passage · c_visual.tex, line 420
guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. Its restored nonlinear cost is 2m-3.
Exact mathematical passage
```
 L_j(x)=\sum_{v:v_j=1}x_v,\qquad
 P_m(x)=\prod_{j=1}^m L_j(x).
```

\sum_{\{v,v+a\}\subseteq V}Introduction passage · 1 passage

Search aliases: ∑_{v,v+a}⊆ V sum subseteq

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Introduction passage · Read passage · c_main.tex, line 442
Extracting the quadratic grade. [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The map v↦ v+a is a fixed-point-free involution of V, so the displayed terms form n/2 disjoint…
Exact mathematical passage
```
 Q_m=\sum_{\{v,v+a\}\subseteq V}x_vx_{v+a},
```

\setminusIntroduction passage · 6 passages

Search aliases: ∖ setminus

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Introduction passage · Read passage · c_visual.tex, line 416
guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. Its restored nonlinear cost is 2m-3.
Exact mathematical passage
```
 V=\F_2^m\setminus\{0,a\},\qquad n=2^m-2.
```

\neIntroduction passage · 37 passages

Search aliases: ≠ ne

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Introduction passage · Read passage · g_main.tex, line 1055
The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be chosen so that ellExc00(A_n)-ellExc11(A_n) ≥ lfloorfracn-52rfloor. Thus a rank-one excursion across one adjacent flag layer can create a line…
Exact mathematical passage
```
X_{ij}\ne0
```

\oplusIntroduction passage · 11 passages

Search aliases: ⊕ oplus

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Introduction passage · Read passage · g_appendix.tex, line 181
Tensor and direct-sum conventions. For linear maps, A⊕ B denotes a block-diagonal direct sum. The Kronecker product A⊗ B uses lexicographic ordering. In quantum chapters, U⊗ V denotes the physical tensor product on disjoint systems. These operations have different dimension and cost normalizations and are not interchanged silently.
Exact mathematical passage
```
A\oplus B
```

\mathbf1Use in source · 9 passages

Search aliases: 1 mathbf

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Use in source · Read passage · q_main.tex, line 265
1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d).
Exact mathematical passage
```
\mathbf1_{\gcd(a,q)=1}=\sum_{d\mid\gcd(a,q)}\mu(d).
```

\mathbf1_{12\mid n}Use in source · 1 passage

Search aliases: 1_12mid n mathbf mid

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Use in source · Read passage · q_main.tex, line 715
c(n)=21_2mid n-41_4mid n -61_6mid n+121_12mid n.
Exact mathematical passage
```
c(n)=2\mathbf1_{2\mid n}-4\mathbf1_{4\mid n}
-6\mathbf1_{6\mid n}+12\mathbf1_{12\mid n}.
```

\mathbf1_{2\mid n}Use in source · 1 passage

Search aliases: 1_2mid n mathbf mid

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Use in source · Read passage · q_main.tex, line 715
c(n)=21_2mid n-41_4mid n -61_6mid n+121_12mid n.
Exact mathematical passage
```
c(n)=2\mathbf1_{2\mid n}-4\mathbf1_{4\mid n}
-6\mathbf1_{6\mid n}+12\mathbf1_{12\mid n}.
```

\mathbf1_{4\mid n}Use in source · 1 passage

Search aliases: 1_4mid n mathbf mid

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Use in source · Read passage · q_main.tex, line 715
c(n)=21_2mid n-41_4mid n -61_6mid n+121_12mid n.
Exact mathematical passage
```
c(n)=2\mathbf1_{2\mid n}-4\mathbf1_{4\mid n}
-6\mathbf1_{6\mid n}+12\mathbf1_{12\mid n}.
```

\mathbf1_{6\mid n}Use in source · 1 passage

Search aliases: 1_6mid n mathbf mid

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Use in source · Read passage · q_main.tex, line 715
c(n)=21_2mid n-41_4mid n -61_6mid n+121_12mid n.
Exact mathematical passage
```
c(n)=2\mathbf1_{2\mid n}-4\mathbf1_{4\mid n}
-6\mathbf1_{6\mid n}+12\mathbf1_{12\mid n}.
```

\mathbf1_{a\in S_d}Use in source · 1 passage

Search aliases: 1_a∈ S_d mathbf in

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Use in source · Read passage · q_main.tex, line 342
The set S_d contains the labels of characters of exact order d. Put D_d=diag(1_a∈ S_d) and
Exact mathematical passage
```
D_d=\operatorname{diag}(\mathbf1_{a\in S_d})
```

\mathbf1_{a\in U_{12}}Use in source · 2 passages

Search aliases: 1_a∈ U_12 mathbf in

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Use in source · Read passage · q_main.tex, line 752
With D=diag(1_a∈ U_12) and F=F_12,
Exact mathematical passage
```
D=\operatorname{diag}(\mathbf1_{a\in U_{12}})
```

\mathbf1_{\gcd(a,q)=1}Use in source · 2 passages

Search aliases: 1_gcd(a,q)=1 mathbf gcd

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Use in source · Read passage · q_main.tex, line 265
1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d).
Exact mathematical passage
```
\mathbf1_{\gcd(a,q)=1}=\sum_{d\mid\gcd(a,q)}\mu(d).
```

\mathbf1_{\gcd(k,q)=1}Use in source · 1 passage

Search aliases: 1_gcd(k,q)=1 mathbf gcd

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Use in source · Read passage · q_visual.tex, line 222
integer waveform changes with the modulus, while the same arithmetic identity supplies it. fig19_other_arithmetic_signals1.0Four exact Ramanujan signals constructed by the divisor formula. Their periods and amplitudes differ. These examples illustrate the general formulas; their finite verification does not prove those formulas universally. For general q, character orthogonality gives widehatc_q(k)=q1_gcd(k,q)=1, ∑_n=0^q-1|c_q(n)|^2=qφ(q). The associated primitive-order projector has rank φ(q). …
Exact mathematical passage
```
 \widehat{c_q}(k)=q\mathbf1_{\gcd(k,q)=1},\qquad
 \sum_{n=0}^{q-1}|c_q(n)|^2=q\varphi(q).
```

\mathbf1_{k\in U_{12}}Use in source · 1 passage

Search aliases: 1_k∈ U_12 mathbf in

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Use in source · Read passage · q_visual.tex, line 81
s the selector leadThe Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform widehat x(k)=∑_n=0^11x(n)e^-2π ikn/12. fig04_exact_spectrum1.0The coefficient is exactly 12 at k=1,5,7,11 and exactly zero elsewhere. This is an algebraic identity, not a spectral thresholding decision. widehatc_12(k)=∑_a∈ U_12∑_n=0^11ζ^(a-k)n =12 1_k∈ U_12. The inner geometric sum is twelve when a=k modulo twelve and zero otherwise. The…
Exact mathematical passage
```
 \widehat{c_{12}}(k)=\sum_{a\in U_{12}}\sum_{n=0}^{11}\zeta^{(a-k)n}
 =12\,\mathbf1_{k\in U_{12}}.
```

\mathbf1_{k\in U_q}Use in source · 1 passage

Search aliases: 1_k∈ U_q mathbf in

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Use in source · Read passage · q_main.tex, line 298
widehatc_q(k)=q 1_k∈ U_q, k∈ I_q.
Exact mathematical passage
```
\boxed{\widehat{c_q}(k)=q\,\mathbf1_{k\in U_q},\qquad k\in I_q.}
```

\mathrm{2q-depth}Use in source · 1 passage

Search aliases: 2q-depth mathrm

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Use in source · Read passage · g_main.tex, line 3662
Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its total count is large. Conversely, a two-gate count saving can increase depth under locality constraints.
Exact mathematical passage
```
\Delta_{\mathrm{depth}}(U),
\qquad
\Delta_{\mathrm{2q-depth}}(U),
\qquad
\Delta_{\mathrm{space-time}}(U).
```

\mathcal AUse in source · 1 passage

Search aliases: A mathcal

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Use in source · Read passage · s_main.tex, line 1987
Metric collapse and existence collapse. The simulation theorems compare costs of two realizations that both exist. A broader restricted-realization framework also allows the restricted object to be absent. If R(x)=∅, A(x)≠∅, then enlarging the realization category changes feasibility rather than merely runtime.
Exact mathematical passage
```
\mathcal R(x)=\varnothing,
\qquad
\mathcal A(x)\neq\varnothing,
```

a'Use in source · 3 passages

Search aliases: a'

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Use in source · Read passage · q_main.tex, line 149
Proof. The equation (ζ_q^a)^t=1 holds exactly when q divides at. Write g=gcd(a,q), q=gq', and a=ga', with gcd(a',q')=1. The divisibility condition is q'mid a't, which is equivalent to q'mid t. Its smallest positive solution is t=q'. This includes a=0: then g=q and the order is one. square
Exact mathematical passage
```
a=ga'
```

a^{-1}Use in source · 3 passages

Search aliases: a^-1

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Use in source · Read passage · g_main.tex, line 1302
If the unique gate is R_t← R_t+aR_s, then aR_s=w. Choose a coordinate j for which w_j is a unit. Since a(R_s)_j=w_j, the factor a is a unit in the commutative ring R. Hence R_s=a^-1w.
Exact mathematical passage
```
R_s=a^{-1}w
```

A^{-T}Use in source · 3 passages

Search aliases: A^-T

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Use in source · Read passage · g_appendix.tex, line 164
Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).
Exact mathematical passage
```
\iota(A)=\diag(A,A^{-T})\in\Sp(2n,\R).
```

A^{[0]}Use in source · 1 passage

Search aliases: A^[0]

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Use in source · Read passage · g_main.tex, line 2699
Let A^[0]=A and A^[r+1]= Z(A^[r]). If A has size m, restricted length a, and ambient length b, then ellDown(A^[r]) = 2^r a+r m2^r-1, and ellAll(A^[r]) ≤ 2^r b+r m2^r-1. Thus DeltaDef(A^[r])≥2^rDeltaDef(A). For the canonical atom C, the first values are @rrrrr@ r · dimension · restricted exact · ambient upper bound · certified gap
Exact mathematical passage
```
A^{[0]}=A
```

A^{[r+1]}Use in source · 1 passage

Search aliases: A^[r+1]

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Use in source · Read passage · g_main.tex, line 2699
Let A^[0]=A and A^[r+1]= Z(A^[r]). If A has size m, restricted length a, and ambient length b, then ellDown(A^[r]) = 2^r a+r m2^r-1, and ellAll(A^[r]) ≤ 2^r b+r m2^r-1. Thus DeltaDef(A^[r])≥2^rDeltaDef(A). For the canonical atom C, the first values are @rrrrr@ r · dimension · restricted exact · ambient upper bound · certified gap
Exact mathematical passage
```
A^{[r+1]}=\mathcal Z(A^{[r]})
```

A^{[r]}Use in source · 4 passages

Search aliases: A^[r]

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Use in source · Read passage · g_main.tex, line 2699
Let A^[0]=A and A^[r+1]= Z(A^[r]). If A has size m, restricted length a, and ambient length b, then ellDown(A^[r]) = 2^r a+r m2^r-1, and ellAll(A^[r]) ≤ 2^r b+r m2^r-1. Thus DeltaDef(A^[r])≥2^rDeltaDef(A). For the canonical atom C, the first values are @rrrrr@ r · dimension · restricted exact · ambient upper bound · certified gap
Exact mathematical passage
```
A^{[r+1]}=\mathcal Z(A^{[r]})
```

a^kUse in source · 1 passage

Search aliases: a^k

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Use in source · Read passage · g_main.tex, line 854
The Chevalley conjugation formula has the schematic form x_α(a)x_β(b)x_α(a)^-1 = x_β(b) ∏_k>0 x_β+kα(c_k a^k b). A forbidden root α can therefore create several allowed root effects before being removed.
Exact mathematical passage
```
x_\alpha(a)x_\beta(b)x_\alpha(a)^{-1}
=
x_\beta(b)
\prod_{k>0}
x_{\beta+k\alpha}(c_k a^k b).
```

A^\ellUse in source · 3 passages

Search aliases: A^ℓ ell

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Use in source · Read passage · q_main.tex, line 414
E_a=frac1N∑_ℓ=0^N-1ζ_N^aℓA^ℓ
Exact mathematical passage
```
E_a=\frac1N\sum_{\ell=0}^{N-1}\zeta_N^{a\ell}A^\ell
```

A^{\otimes r}Use in source · 10 passages

Search aliases: A^⊗ r otimes

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Use in source · Read passage · g_main.tex, line 1097
[Tensor-amplified snapshot family] Let A=A(u;s_1,…,s_v) have dimension m=u+v+2. For N=m^r, ellDown(A^⊗ r) =r m^r-1(u+2v), while ellAll(A^⊗ r) ≤ r m^r. Consequently, ellDown(A^⊗ r)-ellAll(A^⊗ r) ≥ fracv-2mlog_2mNlog_2N.
Exact mathematical passage
```
\ellDown(A^{\otimes r})
=r m^{r-1}(u+2v),
```

A^{\otimes s}Use in source · 1 passage

Search aliases: A^⊗ s otimes

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Use in source · Read passage · g_main.tex, line 2886
The tensor inequality also defines a regularized ambient invariant. Let A∈U_m(F) and put b_r(A) = fracellAll(A^⊗ r)m^r-1. Applying [source reference] to A^⊗ r⊗ A^⊗ s gives b_r+s(A)≤ b_r(A)+b_s(A).
Exact mathematical passage
```
A^{\otimes r}\otimes A^{\otimes s}
```

A_1Use in source · 2 passages

Search aliases: A_1

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Use in source · Read passage · g_main.tex, line 2601
[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).
Exact mathematical passage
```
\ellDown\bigl(\operatorname{diag}(A_1,\ldots,A_k)\bigr)
=
\sum_{i=1}^k\ellDown(A_i).
```

a_1Use in source · 3 passages

Search aliases: a_1

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Use in source · Read passage · g_main.tex, line 1170
Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_1(a_1).
Exact mathematical passage
```
w=(\tau_{t_m,s_m}(a_m),\ldots,\tau_{t_1,s_1}(a_1))
```

A_2Use in source · 1 passage

Search aliases: A_2

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Use in source · Read passage · g_main.tex, line 5403
Suggested sequence: A_2→ B_2=C_2→ G_2→ A_r. Low-rank nonsimply laced systems are attractive because longer root strings may create more than one useful effect per consumer.
Exact mathematical passage
```
A_2\to B_2=C_2\to G_2\to A_r.
```

A_{21}Use in source · 4 passages

Search aliases: A_21

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Use in source · Read passage · g_main.tex, line 2813
Within the snapshot family, the best coefficient is obtained by the twenty-one-dimensional seed A_21=A(9;0,1,…,9), for which ellDown(A_21)=29, ellAll(A_21)≤21, DeltaDef(A_21)≥8.
Exact mathematical passage
```
A_{21}=A(9;0,1,\ldots,9),
```

A_{21}^{\otimes j}Use in source · 1 passage

Search aliases: A_21^⊗ j otimes

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Use in source · Read passage · g_main.tex, line 2950
[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^r+120 ≤ frac2120n. Also r≥log_21n-1. Therefore S≥ nlog_21n-frac4120n. Multiplying by 8/21 yields D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n.
Exact mathematical passage
```
A_{21}^{\otimes j}
```

A_{21}^{\otimes0}Use in source · 1 passage

Search aliases: A_21^⊗0 otimes

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Use in source · Read passage · g_main.tex, line 2950
[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^r+120 ≤ frac2120n. Also r≥log_21n-1. Therefore S≥ nlog_21n-frac4120n. Multiplying by 8/21 yields D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n.
Exact mathematical passage
```
A_{21}^{\otimes0}=I_1
```

A_{2^n}Use in source · 1 passage

Search aliases: A_2^n

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Use in source · Read passage · g_appendix.tex, line 160
tring x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric group.
Exact mathematical passage
```
A_{2^n}
```

A_5Use in source · 1 passage

Search aliases: A_5

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Use in source · Read passage · g_main.tex, line 870
Type texorpdfstringAA and beyond. Row transvections are root-group elements of type A. The established unitriangular atom occurs in A_5. In nonsimply laced systems, root strings can have length greater than two. This raises the possibility that one ambient root can create multiple restricted effects per consumer.
Exact mathematical passage
```
A_5
```

A_iUse in source · 6 passages

Search aliases: A_i

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Use in source · Read passage · c_main.tex, line 104
erage] Let a restored oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with target z, correctness of the full circuit gives R(x,z)=P(x,z+f(x)). When f(x)=0, this says B(x)=A(x). When f(x)=1, it says B(x)=P…
Exact mathematical passage
```
H_i=A_i+B_i
```

a_{i,S}Use in source · 1 passage

Search aliases: a_i,S

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Use in source · Read passage · g_main.tex, line 3712
Algebraic normal form. Every Boolean coordinate function has a unique algebraic normal form f_i(x) = ⊕_S⊆[n]a_i,S∏_j∈ Sx_j. Affine maps have degree at most one. A Toffoli gate targeting t with controls a,b performs x_t⟼ x_t+x_ax_b, which can raise the degree of later coordinates. The final affine target requires every higher-degree coefficient to cancel.
Exact mathematical passage
```
f_i(x)
=
\bigoplus_{S\subseteq[n]}a_{i,S}\prod_{j\in S}x_j.
```

a_i^\daggerUse in source · 1 passage

Search aliases: a_i^† dagger

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Use in source · Read passage · s_main.tex, line 1324
Gate updates. A number-preserving one-body unitary acts on mode creation operators by a_i^† ⟼ ∑_j U_jia_j^†. A generic dense mode transformation can induce many occupation-basis transitions, but for fixed N the basis dimension remains polynomial. Local hopping and phase gates are sparse. Few-body interactions remain polynomial to apply for fixed N, although the exponent can grow with N.
Exact mathematical passage
```
a_i^\dagger
\longmapsto
\sum_j U_{ji}a_j^\dagger.
```

A_{ij}Use in source · 1 passage

Search aliases: A_ij

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Use in source · Read passage · g_main.tex, line 830
Pattern groups and architecture constraints. For a poset P, define the pattern group U_P(K) = {I+A:A_ij=0 unless j<_P i}. When the allowed positions are transitively closed, U_P is a group. Compare the intrinsic metric generated by P-allowed transvections with the metric induced by all transvections.
Exact mathematical passage
```
U_P(K)
=
\{I+A:A_{ij}=0\text{ unless }j<_P i\}.
```

a_j^\daggerUse in source · 1 passage

Search aliases: a_j^† dagger

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Use in source · Read passage · s_main.tex, line 1324
Gate updates. A number-preserving one-body unitary acts on mode creation operators by a_i^† ⟼ ∑_j U_jia_j^†. A generic dense mode transformation can induce many occupation-basis transitions, but for fixed N the basis dimension remains polynomial. Local hopping and phase gates are sparse. Few-body interactions remain polynomial to apply for fixed N, although the exponent can grow with N.
Exact mathematical passage
```
a_i^\dagger
\longmapsto
\sum_j U_{ji}a_j^\dagger.
```

A_kUse in source · 2 passages

Search aliases: A_k

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Use in source · Read passage · g_main.tex, line 2601
[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).
Exact mathematical passage
```
\ellDown\bigl(\operatorname{diag}(A_1,\ldots,A_k)\bigr)
=
\sum_{i=1}^k\ellDown(A_i).
```

a_LUse in source · 1 passage

Search aliases: a_L

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Use in source · Read passage · g_main.tex, line 2410
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\Phi_\sigma:\Aff^L\longrightarrow\SL_n,
\qquad
(a_1,\ldots,a_L)
\longmapsto
(I+a_LE_{t_L,s_L})\cdots(I+a_1E_{t_1,s_1}).
```

a_mUse in source · 2 passages

Search aliases: a_m

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Use in source · Read passage · g_main.tex, line 1170
Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_1(a_1).
Exact mathematical passage
```
w=(\tau_{t_m,s_m}(a_m),\ldots,\tau_{t_1,s_1}(a_1))
```

a_nUse in source · 3 passages

Search aliases: a_n

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Use in source · Read passage · q_main.tex, line 530
Let L≥1 and let f(z)=∑_n≥0a_nz^n be a formal power series over a field containing a primitive Lth root ω and 1/L. For 0≤ r<L, define
Exact mathematical passage
```
f(z)=\sum_{n\ge0}a_nz^n
```

A_rUse in source · 1 passage

Search aliases: A_r

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Use in source · Read passage · g_main.tex, line 5403
Suggested sequence: A_2→ B_2=C_2→ G_2→ A_r. Low-rank nonsimply laced systems are attractive because longer root strings may create more than one useful effect per consumer.
Exact mathematical passage
```
A_2\to B_2=C_2\to G_2\to A_r.
```

a_rUse in source · 3 passages

Search aliases: a_r

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Use in source · Read passage · g_main.tex, line 5207
Template XI: stable capacity. If a_r+s≤ m^s a_r+m^r a_s, then b_r=fraca_rm^r-1 is subadditive and lim_r→∞fraca_rr m^r-1 exists by Fekete's lemma. This defines a regularized ambient cost and exposes possible tensor activation.
Exact mathematical passage
```
a_{r+s}\le m^s a_r+m^r a_s,
```

a_{r+Lm}Use in source · 1 passage

Search aliases: a_r+Lm

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Use in source · Read passage · q_main.tex, line 534
E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).
Exact mathematical passage
```
E_rf(t)=\sum_{m\ge0}a_{r+Lm}t^m,
\qquad
R_rf(z)=z^rE_rf(z^L).
```

a_{r+s}Use in source · 1 passage

Search aliases: a_r+s

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Use in source · Read passage · g_main.tex, line 5207
Template XI: stable capacity. If a_r+s≤ m^s a_r+m^r a_s, then b_r=fraca_rm^r-1 is subadditive and lim_r→∞fraca_rr m^r-1 exists by Fekete's lemma. This defines a regularized ambient cost and exposes possible tensor activation.
Exact mathematical passage
```
a_{r+s}\le m^s a_r+m^r a_s,
```

A_SUse in source · 12 passages

Search aliases: A_S

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Use in source · Read passage · g_main.tex, line 457
Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.
Exact mathematical passage
```
A_S=\sum_{s\in S}\rho(s).
```

a_sUse in source · 1 passage

Search aliases: a_s

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Use in source · Read passage · g_main.tex, line 5207
Template XI: stable capacity. If a_r+s≤ m^s a_r+m^r a_s, then b_r=fraca_rm^r-1 is subadditive and lim_r→∞fraca_rr m^r-1 exists by Fekete's lemma. This defines a regularized ambient cost and exposes possible tensor activation.
Exact mathematical passage
```
a_{r+s}\le m^s a_r+m^r a_s,
```

A_S^kUse in source · 5 passages

Search aliases: A_S^k

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Use in source · Read passage · g_main.tex, line 461
Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.
Exact mathematical passage
```
\langle\delta_g,A_S^k\delta_e\rangle
```

A_tUse in source · 2 passages

Search aliases: A_t

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Use in source · Read passage · s_main.tex, line 394
Adaptive packet enlargement. A practical simulator need not fix one packet forever. It can maintain an active set A_t of packets, estimate transfers P_α G_tP_β, and add a packet when its predicted mass exceeds a tolerance. The proof obligation becomes a global accounting identity: total error ≤ discarded transfer + numerical error + measurement approximation. The key research problem is to retain polynomial active size while certifying the tail.
Exact mathematical passage
```
A_t
```

A_\lambdaUse in source · 1 passage

Search aliases: A_λ lambda

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Use in source · Read passage · s_main.tex, line 995
Decomposition For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.
Exact mathematical passage
```
A
=
\bigoplus_\lambda
A_\lambda\otimes I_{\mathcal V_\lambda}.
```

\AffUse in source · 5 passages

Search aliases: Aff

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Use in source · Read passage · c_main.tex, line 148
For a positive input v, define instead β_v(f)=min{deg h:h∈ I_f, h(v)=1}, β(f)=max_v:f(v)=1β_v(f). Set β(0)=0. Thus β(f) is the degree needed before safe polynomials collectively detect every positive input. Finally, put β^*(f)=max_ℓ∈Aff_nβ(f+ℓ), where Aff_n is the set of affine Boolean functions on the logical inputs.
Exact mathematical passage
```
 \beta^*(f)=\max_{\ell\in\Aff_n}\beta(f+\ell),
```

\Aff^LUse in source · 1 passage

Search aliases: Aff^L Aff

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Use in source · Read passage · g_main.tex, line 2410
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\Phi_\sigma:\Aff^L\longrightarrow\SL_n,
\qquad
(a_1,\ldots,a_L)
\longmapsto
(I+a_LE_{t_L,s_L})\cdots(I+a_1E_{t_1,s_1}).
```

\Aff^NUse in source · 1 passage

Search aliases: Aff^N Aff

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Use in source · Read passage · g_main.tex, line 2490
After base change to an algebraic closure overline K, [source reference] gives dimoverline R^*_n,≤ N-1≤ N-1<N=dimU_n. Because the word maps are defined over K, this proper closed set descends to a proper closed subset of U_n over K. Its complement is a nonempty Zariski-open set Ω_n. Since K is infinite and U_n≅Aff^N, the set Ω_n(K) is nonempty and Zariski dense. For any A∈Ω_n(K), one has ellAll(A)≥ N. The inequalities ellAll(A)≤ellDown(A)≤ N force equality. Replacing each image by its Zariski cl…
Exact mathematical passage
```
\U_n\cong\Aff^N
```

\Aff_nUse in source · 3 passages

Search aliases: Aff_n Aff

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Use in source · Read passage · c_main.tex, line 148
For a positive input v, define instead β_v(f)=min{deg h:h∈ I_f, h(v)=1}, β(f)=max_v:f(v)=1β_v(f). Set β(0)=0. Thus β(f) is the degree needed before safe polynomials collectively detect every positive input. Finally, put β^*(f)=max_ℓ∈Aff_nβ(f+ℓ), where Aff_n is the set of affine Boolean functions on the logical inputs.
Exact mathematical passage
```
 \beta^*(f)=\max_{\ell\in\Aff_n}\beta(f+\ell),
```

\operatorname{AGL}Use in source · 6 passages

Search aliases: AGL operatorname

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Use in source · Read passage · g_appendix.tex, line 160
Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric group.
Exact mathematical passage
```
\operatorname{AGL}(n,2)
```

\mathrm{AND}Use in source · 1 passage

Search aliases: AND mathrm

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Use in source · Read passage · c_lean.tex, line 177
A degree-one complete safe cover for AND would include a detector of its sole accepting input. Safety forces that detector to be AND itself, contradicting the corner-parity lemma. The degree-two monomial is a valid detector. The source proves β(AND_2)=β^*(AND_2)=2 and therefore every restored implementation of AND_2 uses at least one Toffoli.
Exact mathematical passage
```
\beta(\mathrm{AND}_2)=\beta^*(\mathrm{AND}_2)=2
```

\mathrm{AND}_2Use in source · 1 passage

Search aliases: AND_2 mathrm

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Use in source · Read passage · c_lean.tex, line 177
A degree-one complete safe cover for AND would include a detector of its sole accepting input. Safety forces that detector to be AND itself, contradicting the corner-parity lemma. The degree-two monomial is a valid detector. The source proves β(AND_2)=β^*(AND_2)=2 and therefore every restored implementation of AND_2 uses at least one Toffoli.
Exact mathematical passage
```
\beta(\mathrm{AND}_2)=\beta^*(\mathrm{AND}_2)=2
```

\BUse in source · 9 passages

Search aliases: B

Link to this entry ·

Use in source · Read passage · c_main.tex, line 4
Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies alor b=a+b+ab. A statement h≤ f is pointwise and means that h has no positive value at a negative input of f.
Exact mathematical passage
```
f:\B^n\to\B
```

\mathcal BUse in source · 3 passages

Search aliases: B mathcal

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Use in source · Read passage · c_main.tex, line 495
The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
 \mathcal B=K^W,
```

\mathrm BUse in source · 1 passage

Search aliases: B mathrm

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Use in source · Read passage · s_main.tex, line 1449
Bosonic and fermionic analogues. For N fermions in m modes, D_ F(m,N)=binom mN. For N bosons in m modes, D_ B(m,N)=binomm+N-1N. The same binomial analysis gives:
Exact mathematical passage
```
D_{\mathrm B}(m,N)=\binom{m+N-1}{N}.
```

b^{-1}Use in source · 5 passages

Search aliases: b^-1

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1642
Let B_4={g∈GL_6(F_2):ellAll(g)≤4}. Exact breadth-first search gives |B_4|=110626. Group B_4 by right cosets of U_6(F_2). If a,b lie in the same right coset, then b^-1a∈U_6(F_2), and ellAll(b^-1a)≤ellAll(b)+ellAll(a)≤8. The implementation uses a canonical right-coset key obtained by reducing matrix columns from right to left. Right multiplication by a lower-unitriangular matrix changes column j only by a combination of columns with larger index, so this reduction is constant on right cosets and s…
Exact mathematical passage
```
b^{-1}a\in\U_6(\F_2),
```

\B^mUse in source · 4 passages

Search aliases: B^m B

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Use in source · Read passage · c_main.tex, line 494
The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
W=\B^m
```

\B^nUse in source · 2 passages

Search aliases: B^n B

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Use in source · Read passage · c_main.tex, line 4
Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies alor b=a+b+ab. A statement h≤ f is pointwise and means that h has no positive value at a negative input of f.
Exact mathematical passage
```
f:\B^n\to\B
```

B_0Use in source · 2 passages

Search aliases: B_0

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Use in source · Read passage · g_main.tex, line 4567
Define Q_R=fracI+Rn, Q_C=fracI+Cn, Q_S=fracI+Sn, B_0=fracJn^2. Then the five joint projectors are B_0, Q_R-B_0, Q_C-B_0, Q_S-B_0, I-Q_R-Q_C-Q_S+2B_0. Their multiplicities are 1, n-1, n-1, n-1, (n-1)(n-2). The Latin-square graph R+C+S has universal spectrum 3(n-1)^1, (n-3)^3(n-1), (-3)^(n-1)(n-2). At n=6 this becomes 15^1,qquad3^15, (-3)^20. Thus the graph spectrum cannot distinguish the twelve main classes of order six.
Exact mathematical passage
```
Q_R=\frac{I+R}{n},
\qquad
Q_C=\frac{I+C}{n},
\qquad
Q_S=\frac{I+S}{n},
\qquad
B_0=\frac{J}{n^2}.
```

B_0^*Use in source · 1 passage

Search aliases: B_0^*

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Use in source · Read passage · s_main.tex, line 458
[Exact dynamic-packet theorem] Suppose P_0ketψ_0=ketψ_0 and [source reference] holds for every gate. Put z_0=B_0^*ketψ_0, z_t=M_tz_t-1. Then G_t⋯ G_1ketψ_0=B_tz_t for every t.
Exact mathematical passage
```
z_0=B_0^*\ket{\psi_0},
\qquad
z_t=M_tz_{t-1}.
```

b_1Use in source · 1 passage

Search aliases: b_1

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Use in source · Read passage · s_main.tex, line 207
The packet access model. For each input length, suppose there is a packet projector P_n and a basis B_n= {ketb_1,…,ketb_D_n} of P_nHh_n. The simulation interface has four routines.
Exact mathematical passage
```
\B_n=
\{\ket{b_1},\ldots,\ket{b_{D_n}}\}
```

B_2Use in source · 2 passages

Search aliases: B_2

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Use in source · Read passage · g_main.tex, line 5403
Suggested sequence: A_2→ B_2=C_2→ G_2→ A_r. Low-rank nonsimply laced systems are attractive because longer root strings may create more than one useful effect per consumer.
Exact mathematical passage
```
A_2\to B_2=C_2\to G_2\to A_r.
```

B_4Use in source · 5 passages

Search aliases: B_4

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Use in source · Read passage · g_main.tex, line 1634
Let B_4={g∈GL_6(F_2):ellAll(g)≤4}. Exact breadth-first search gives |B_4|=110626. Group B_4 by right cosets of U_6(F_2). If a,b lie in the same right coset, then b^-1a∈U_6(F_2), and ellAll(b^-1a)≤ellAll(b)+ellAll(a)≤8. The implementation uses a canonical right-coset key obtained by reducing matrix columns from right to left. Right multiplication by a lower-unitriangular matrix changes column j only by a combination of columns with larger index, so this reduction is constant on right cosets and s…
Exact mathematical passage
```
B_4=\{g\in\GL_6(\F_2):\ellAll(g)\le4\}.
```

b_{D_n}Use in source · 1 passage

Search aliases: b_D_n

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Use in source · Read passage · s_main.tex, line 207
The packet access model. For each input length, suppose there is a packet projector P_n and a basis B_n= {ketb_1,…,ketb_D_n} of P_nHh_n. The simulation interface has four routines.
Exact mathematical passage
```
\B_n=
\{\ket{b_1},\ldots,\ket{b_{D_n}}\}
```

B_iUse in source · 2 passages

Search aliases: B_i

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Use in source · Read passage · c_main.tex, line 104
erage] Let a restored oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with target z, correctness of the full circuit gives R(x,z)=P(x,z+f(x)). When f(x)=0, this says B(x)=A(x). When f(x)=1, it says B(x)=P…
Exact mathematical passage
```
H_i=A_i+B_i
```

\B_nUse in source · 2 passages

Search aliases: B_n B

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Use in source · Read passage · s_main.tex, line 207
The packet access model. For each input length, suppose there is a packet projector P_n and a basis B_n= {ketb_1,…,ketb_D_n} of P_nHh_n. The simulation interface has four routines.
Exact mathematical passage
```
\B_n=
\{\ket{b_1},\ldots,\ket{b_{D_n}}\}
```

b_rUse in source · 4 passages

Search aliases: b_r

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Use in source · Read passage · g_main.tex, line 2881
The tensor inequality also defines a regularized ambient invariant. Let A∈U_m(F) and put b_r(A) = fracellAll(A^⊗ r)m^r-1. Applying [source reference] to A^⊗ r⊗ A^⊗ s gives b_r+s(A)≤ b_r(A)+b_s(A).
Exact mathematical passage
```
b_r(A)
=
\frac{\ellAll(A^{\otimes r})}{m^{r-1}}.
```

b_{r+s}Use in source · 1 passage

Search aliases: b_r+s

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Use in source · Read passage · g_main.tex, line 2887
The tensor inequality also defines a regularized ambient invariant. Let A∈U_m(F) and put b_r(A) = fracellAll(A^⊗ r)m^r-1. Applying [source reference] to A^⊗ r⊗ A^⊗ s gives b_r+s(A)≤ b_r(A)+b_s(A).
Exact mathematical passage
```
b_{r+s}(A)\le b_r(A)+b_s(A).
```

B_r^{(M)}Use in source · 2 passages

Search aliases: B_r^(M)

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Use in source · Read passage · s_main.tex, line 136
Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis and charge sectors in quantum theory.
Exact mathematical passage
```
B_r^{(M)}=
\sum_{k\equiv r\pmod M}P_k.
```

b_sUse in source · 1 passage

Search aliases: b_s

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Use in source · Read passage · g_main.tex, line 2887
The tensor inequality also defines a regularized ambient invariant. Let A∈U_m(F) and put b_r(A) = fracellAll(A^⊗ r)m^r-1. Applying [source reference] to A^⊗ r⊗ A^⊗ s gives b_r+s(A)≤ b_r(A)+b_s(A).
Exact mathematical passage
```
b_{r+s}(A)\le b_r(A)+b_s(A).
```

B_tUse in source · 4 passages

Search aliases: B_t

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Use in source · Read passage · s_main.tex, line 442
Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.
Exact mathematical passage
```
B_t:\C^{D_t}\longrightarrow\Hh,
\qquad
B_tB_t^*=P_t,
\qquad
B_t^*B_t=I_{D_t},
```

B_{t-1}Use in source · 3 passages

Search aliases: B_t-1

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Use in source · Read passage · s_main.tex, line 450
Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.
Exact mathematical passage
```
M_t=B_t^*G_tB_{t-1}.
\label{s:eq:dynamic-coordinate-update}
```

B_t^*Use in source · 3 passages

Search aliases: B_t^*

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Use in source · Read passage · s_main.tex, line 442
Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.
Exact mathematical passage
```
B_t:\C^{D_t}\longrightarrow\Hh,
\qquad
B_tB_t^*=P_t,
\qquad
B_t^*B_t=I_{D_t},
```

b_xUse in source · 4 passages

Search aliases: b_x

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Use in source · Read passage · g_main.tex, line 1508
Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y;…
Exact mathematical passage
```
b_x
```

B_\lambdaUse in source · 5 passages

Search aliases: B_λ lambda

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Use in source · Read passage · g_main.tex, line 4546
Spherical packets. Let T_1,…,T_m be commuting normal operators on a finite-dimensional Hilbert space. Their unique coarsest joint spectral resolution is T_j=∑_λt_j(λ)B_λ, ∑_λ B_λ=I. The tuple λ=(t_1(λ),…,t_m(λ)) is the spherical label. The abstract spectral packet records these labels and the ranks of B_λ.
Exact mathematical passage
```
T_j=\sum_{\lambda}t_j(\lambda)B_\lambda,
\qquad
\sum_\lambda B_\lambda=I.
```

\backslashUse in source · 2 passages

Search aliases: backslash

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Use in source · Read passage · g_main.tex, line 1253
Set G=Egrp_n(R), H=U_n(R). Consider the right Schreier graph on the left cosets Hbackslash G: a generator s∈ S_n^*(R) gives an edge Hg⟶ Hgs. A word representing h∈ H traces a based loop at the coset H. Its label product is the holonomy h.
Exact mathematical passage
```
H\backslash G
```

\bigveeUse in source · 2 passages

Search aliases: bigvee

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Use in source · Read passage · c_main.tex, line 105
d oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with target z, correctness of the full circuit gives R(x,z)=P(x,z+f(x)). When f(x)=0, this says B(x)=A(x). When f(x)=1, it says B(x)=P(x,1), and the discr…
Exact mathematical passage
```
\label{c:eq:cover}
 H_i\le f\quad\hbox{for every }i,\qquad
 \bigvee_iH_i=f.
```

\bigvee_iUse in source · 2 passages

Search aliases: bigvee_i bigvee

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Use in source · Read passage · c_main.tex, line 105
d oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with target z, correctness of the full circuit gives R(x,z)=P(x,z+f(x)). When f(x)=0, this says B(x)=A(x). When f(x)=1, it says B(x)=P(x,1), and the discr…
Exact mathematical passage
```
\label{c:eq:cover}
 H_i\le f\quad\hbox{for every }i,\qquad
 \bigvee_iH_i=f.
```

\bmodUse in source · 5 passages

Search aliases: bmod

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Use in source · Read passage · g_appendix.tex, line 51
Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.
Exact mathematical passage
```
\SUM^{(a)}_{s\to t}
\ket{x_s,x_t}
=
\ket{x_s,x_t+a x_s\bmod d}.
```

\mathrm{Bool}Use in source · 2 passages

Search aliases: Bool mathrm

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Use in source · Read passage · c_lean.tex, line 59
pageheadModel and definitionsTHE ASSUMPTIONS ARE PART OF THE THEOREM A general register state is a Boolean function W→Bool on any wire type with decidable equality. C01 does not assume that W is finite. The canonical degree model has a finite wire set with n input wires, a ancillas, and one target. A NOT complements a bit. A CNOT adds one control to a distinct target. A Toffoli adds the product of two distinct controls to a third distinct wire. Gate constructors carry the required inequalities.
Exact mathematical passage
```
W\to\mathrm{Bool}
```

\braUse in source · 8 passages

Search aliases: bra

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Use in source · Read passage · s_appendix.tex, line 40
Hypercube quotient coupling. Let ket k = binom nk^-1/2 ∑_|x|=kket x. Every weight-k string has n-k neighbors of weight k+1, while every weight-(k+1) string has k+1 neighbors of weight k. Counting the bipartite edges yields brak+1Aket k = fracbinom nk(n-k) sqrtbinom nkbinom nk+1 = sqrt(k+1)(n-k), proving [source reference].
Exact mathematical passage
```
\bra{k+1}A\ket k
=
\frac{\binom nk(n-k)}
{\sqrt{\binom nk\binom n{k+1}}}
=
\sqrt{(k+1)(n-k)},
```

\CUse in source · 5 passages

Search aliases: C

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Use in source · Read passage · s_main.tex, line 442
Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.
Exact mathematical passage
```
B_t:\C^{D_t}\longrightarrow\Hh,
\qquad
B_tB_t^*=P_t,
\qquad
B_t^*B_t=I_{D_t},
```

\mathcal CUse in source · 8 passages

Search aliases: C mathcal

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Use in source · Read passage · c_main.tex, line 95
Exact cut comparison and safe discrepancies. claimC01 [Cut-comparison identity and complete coverage] Let a restored oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with target z, correctness of the full …
Exact mathematical passage
```
\mathcal C=Q\circ P
```

\mathfrak CUse in source · 1 passage

Search aliases: C mathfrak

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Use in source · Read passage · s_main.tex, line 33
The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:
Exact mathematical passage
```
\mathfrak C_n=
\bigl(
N_{\mathrm{packets}},
D_{\mathrm{internal}},
S_{\mathrm{transfer}},
L_{\mathrm{leakage}}
\bigr).
```

\C^2Use in source · 1 passage

Search aliases: C^2 C

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Use in source · Read passage · s_main.tex, line 736
Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.
Exact mathematical passage
```
(\C^2)^{\otimes n}
=
\bigoplus_{k=0}^n\Hh_{n,k}
```

\C^dUse in source · 3 passages

Search aliases: C^d C

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Use in source · Read passage · s_main.tex, line 961
Symmetric qudits. For local dimension d, the symmetric subspace has occupation-number basis (n_1,…,n_d), n_1+⋯+n_d=n, and dimension dimSym^n(C^d) = binomn+d-1d-1. For fixed d, this is polynomial in n.
Exact mathematical passage
```
\dim\Sym^n(\C^d)
=
\binom{n+d-1}{d-1}.
\label{s:eq:symmetric-qudit-dim}
```

\C^{D_t}Use in source · 1 passage

Search aliases: C^D_t C

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Use in source · Read passage · s_main.tex, line 442
Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.
Exact mathematical passage
```
B_t:\C^{D_t}\longrightarrow\Hh,
\qquad
B_tB_t^*=P_t,
\qquad
B_t^*B_t=I_{D_t},
```

\mathbb C^NUse in source · 3 passages

Search aliases: C^N mathbb

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Use in source · Read passage · q_main.tex, line 469
For every x∈ C^N,
Exact mathematical passage
```
x\in\mathbb C^N
```

C^\daggerUse in source · 5 passages

Search aliases: C^† dagger

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Use in source · Read passage · g_appendix.tex, line 175
Pauli rotations. For a Hermitian Pauli P, R_P(θ)=e^-iθ P/2. Clifford conjugation gives C R_P(θ) C^†=R_CPC^†(θ). The sign of CPC^† can be absorbed by changing θ to -θ.
Exact mathematical passage
```
C R_P(\theta) C^\dagger=R_{CPC^\dagger}(\theta).
```

C^{\oplus k}Use in source · 3 passages

Search aliases: C^⊕ k oplus

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Use in source · Read passage · g_main.tex, line 2638
[Block-sum amplification] For every field and every n≥6, there is an A_n∈U_n(F) with ellDown(A_n)-ellAll(A_n) ≥ lfloorfrac n6rfloor. For n=6k, the target C^⊕ k satisfies ellDown(C^⊕ k)=7k, ellAll(C^⊕ k)≤6k.
Exact mathematical passage
```
C^{\oplus k}
```

c_{(n-m)\bmod12}Use in source · 1 passage

Search aliases: c_(n-m)bmod12 bmod

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Use in source · Read passage · q_lean.tex, line 123
The cyclic matrix has entry c_(n-m)bmod12/12. The original layer proves its symmetry, idempotence, preservation of c, rejection of the chosen perturbation, and exact recovery from the displayed contaminated signal. The exact energy contract is 150=48+102.
Exact mathematical passage
```
c_{(n-m)\bmod12}/12
```

c_0Use in source · 2 passages

Search aliases: c_0

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Use in source · Read passage · c_visual.tex, line 335
Any affine polynomial is a coefficient covector c=(c_0,c_a,c_b). Safety means that it pairs to zero with each negative row. The displayed dependence therefore forces its value at (1,1) to be zero. No safe affine packet detects the positive point.
Exact mathematical passage
```
c=(c_0,c_a,c_b)
```

c_{12}Use in source · 16 passages

Search aliases: c_12

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Use in source · Read passage · q_visual.tex, line 4
twelve will become an integer signal, an exact rational filter, and a normalized quantum state. Every bridge will be written down. The guide follows one object through three coordinate systems. The starting condition is coprimality. The signal realization adds its selected waves. The quantum realization normalizes the amplitude vector and distinguishes coherent transformation from measurement. U_12={1,5,7,11}, c_12(n)=∑_a∈ U_12e^2π ian/12. We will build a projector that keeps exactly those modes…
Exact mathematical passage
```
 U_{12}=\{1,5,7,11\},\qquad
 c_{12}(n)=\sum_{a\in U_{12}}e^{2\pi ian/12}.
```

C_2Use in source · 2 passages

Search aliases: C_2

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Use in source · Read passage · g_main.tex, line 5403
Suggested sequence: A_2→ B_2=C_2→ G_2→ A_r. Low-rank nonsimply laced systems are attractive because longer root strings may create more than one useful effect per consumer.
Exact mathematical passage
```
A_2\to B_2=C_2\to G_2\to A_r.
```

c_2Use in source · 1 passage

Search aliases: c_2

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Use in source · Read passage · q_main.tex, line 771
c_0 · =(4,0,-4,0), c_1 · =(0,-2,0,2), c_2 · =(2,0,-2,0).
Exact mathematical passage
```
\begin{aligned}
c_0&=(4,0,-4,0),\\
c_1&=(0,-2,0,2),\\
c_2&=(2,0,-2,0).
\end{aligned}
```

c_aUse in source · 2 passages

Search aliases: c_a

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Use in source · Read passage · c_visual.tex, line 335
Any affine polynomial is a coefficient covector c=(c_0,c_a,c_b). Safety means that it pairs to zero with each negative row. The displayed dependence therefore forces its value at (1,1) to be zero. No safe affine packet detects the positive point.
Exact mathematical passage
```
c=(c_0,c_a,c_b)
```

\mathcal C_{\mathrm{amb}}Use in source · 5 passages

Search aliases: C_amb mathcal mathrm

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Use in source · Read passage · g_main.tex, line 188
Metric and existence defects in one category. Let C_res⊆ C_amb be categories with weighted generating morphisms. For a morphism f in the restricted target class, the same definition Δ(f)=c_res(f)-c_amb(f) includes group-word metrics, adaptive quantum channels, tensor-network realizations, and orthogonality designs. This categorical form is essential when the ambient model permits measurement, branching, postselection, or entangled objects that do not lie in a common group.
Exact mathematical passage
```
\mathcal C_{\mathrm{res}}\subseteq\mathcal C_{\mathrm{amb}}
```

c_{\mathrm{amb}}Use in source · 5 passages

Search aliases: c_amb mathrm

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Use in source · Read passage · g_main.tex, line 189
Metric and existence defects in one category. Let C_res⊆ C_amb be categories with weighted generating morphisms. For a morphism f in the restricted target class, the same definition Δ(f)=c_res(f)-c_amb(f) includes group-word metrics, adaptive quantum channels, tensor-network realizations, and orthogonality designs. This categorical form is essential when the ambient model permits measurement, branching, postselection, or entangled objects that do not lie in a common group.
Exact mathematical passage
```
\Delta(f)=c_{\mathrm{res}}(f)-c_{\mathrm{amb}}(f)
```

c_bUse in source · 1 passage

Search aliases: c_b

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Use in source · Read passage · c_visual.tex, line 335
Any affine polynomial is a coefficient covector c=(c_0,c_a,c_b). Safety means that it pairs to zero with each negative row. The displayed dependence therefore forces its value at (1,1) to be zero. No safe affine packet detects the positive point.
Exact mathematical passage
```
c=(c_0,c_a,c_b)
```

c_{\mathrm{close}}Use in source · 2 passages

Search aliases: c_close mathrm

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Use in source · Read passage · g_main.tex, line 5076
Template II: amortized consumers. Suppose a catalyst costs c_open+c_close and each of m consumers saves at least s_i restricted cost. Then Δ(U) ≥ ∑_i=1^m s_i -c_open-c_close-r, where r is repair or interference cost. A threshold occurs when the right side first becomes positive.
Exact mathematical passage
```
c_{\mathrm{open}}+c_{\mathrm{close}}
```

C_{\mathrm{compile}}Use in source · 1 passage

Search aliases: C_compile mathrm

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Use in source · Read passage · s_main.tex, line 227
State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectorization, and basis compilation determine actual runtime. The asymptotic distinction is exact when D is polynomial and 2^n is exponential.
Exact mathematical passage
```
C_{\mathrm{packet}}=O(dnD)+C_{\mathrm{compile}}.
```

c_dUse in source · 8 passages

Search aliases: c_d

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Use in source · Read passage · q_main.tex, line 357
(P_d^(N))_nm=frac1N c_d(n-m).
Exact mathematical passage
```
\boxed{(P_d^{(N)})_{nm}=\frac1N c_d(n-m).}
```

C_{\mathrm{err}}Use in source · 1 passage

Search aliases: C_err mathrm

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Use in source · Read passage · g_main.tex, line 4450
For stochastic error rates p_g, a first-order weighted cost is C_err(W)=∑_g∈ W-log(1-p_g). This additive approximation can be inserted directly into Dijkstra search. It does not capture coherent or correlated errors, but it produces a declared and reproducible metric.
Exact mathematical passage
```
C_{\mathrm{err}}(W)=\sum_{g\in W}-\log(1-p_g).
```

C_{\mathrm{full}}Use in source · 1 passage

Search aliases: C_full mathrm

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Use in source · Read passage · s_main.tex, line 231
State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectorization, and basis compilation determine actual runtime. The asymptotic distinction is exact when D is polynomial and 2^n is exponential.
Exact mathematical passage
```
C_{\mathrm{full}}=O(dn2^n).
```

c_iUse in source · 1 passage

Search aliases: c_i

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Use in source · Read passage · c_main.tex, line 373
A cubic on three variables has the form x_1x_2x_3+q(x)+ℓ(x), with q quadratic and ℓ affine. Translating x_i↦ x_i+c_i adjusts independently the coefficients of x_jx_k, so all quadratic terms can be cancelled. The resulting function is a three-variable monomial plus an affine function. Affine input changes and affine output additions cost no Toffolis. Corollary~[source reference] gives both the lower bound three and a matching restored construction.
Exact mathematical passage
```
x_i\mapsto x_i+c_i
```

c_jUse in source · 3 passages

Search aliases: c_j

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Use in source · Read passage · g_main.tex, line 3148
For an invertible binary matrix A, let c_j be column j. Starting at j=n-1 and moving left, reduce c_j by the already canonicalized columns c_j+1,…,c_n-1, using their pivot coordinates. The resulting ordered reduced columns form the key κ(A).
Exact mathematical passage
```
c_j
```

c_{j+1}Use in source · 1 passage

Search aliases: c_j+1

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Use in source · Read passage · g_main.tex, line 3148
For an invertible binary matrix A, let c_j be column j. Starting at j=n-1 and moving left, reduce c_j by the already canonicalized columns c_j+1,…,c_n-1, using their pivot coordinates. The resulting ordered reduced columns form the key κ(A).
Exact mathematical passage
```
c_{j+1},\ldots,c_{n-1}
```

c_j^\daggerUse in source · 1 passage

Search aliases: c_j^† dagger

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Use in source · Read passage · g_main.tex, line 4188
Fermionic Gaussian and matchgate circuits. Let c_j,c_j^† be fermionic creation and annihilation operators. Quadratic Hamiltonians generate Gaussian unitaries, and their action is determined by an orthogonal transformation on Majoranas. Number-conserving Gaussian operations correspond to a unitary action on modes, while general Gaussian operations allow particle-hole mixing.
Exact mathematical passage
```
c_j,c_j^\dagger
```

c_kUse in source · 1 passage

Search aliases: c_k

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Use in source · Read passage · g_main.tex, line 854
The Chevalley conjugation formula has the schematic form x_α(a)x_β(b)x_α(a)^-1 = x_β(b) ∏_k>0 x_β+kα(c_k a^k b). A forbidden root α can therefore create several allowed root effects before being removed.
Exact mathematical passage
```
x_\alpha(a)x_\beta(b)x_\alpha(a)^{-1}
=
x_\beta(b)
\prod_{k>0}
x_{\beta+k\alpha}(c_k a^k b).
```

\mathfrak C_nUse in source · 1 passage

Search aliases: C_n mathfrak

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Use in source · Read passage · s_main.tex, line 33
The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:
Exact mathematical passage
```
\mathfrak C_n=
\bigl(
N_{\mathrm{packets}},
D_{\mathrm{internal}},
S_{\mathrm{transfer}},
L_{\mathrm{leakage}}
\bigr).
```

c_{n-1}Use in source · 1 passage

Search aliases: c_n-1

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Use in source · Read passage · g_main.tex, line 3148
For an invertible binary matrix A, let c_j be column j. Starting at j=n-1 and moving left, reduce c_j by the already canonicalized columns c_j+1,…,c_n-1, using their pivot coordinates. The resulting ordered reduced columns form the key κ(A).
Exact mathematical passage
```
c_{j+1},\ldots,c_{n-1}
```

c_{\mathrm{open}}Use in source · 2 passages

Search aliases: c_open mathrm

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Use in source · Read passage · g_main.tex, line 5076
Template II: amortized consumers. Suppose a catalyst costs c_open+c_close and each of m consumers saves at least s_i restricted cost. Then Δ(U) ≥ ∑_i=1^m s_i -c_open-c_close-r, where r is repair or interference cost. A threshold occurs when the right side first becomes positive.
Exact mathematical passage
```
c_{\mathrm{open}}+c_{\mathrm{close}}
```

C_{\mathrm{packet}}Use in source · 1 passage

Search aliases: C_packet mathrm

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Use in source · Read passage · s_main.tex, line 227
State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectorization, and basis compilation determine actual runtime. The asymptotic distinction is exact when D is polynomial and 2^n is exponential.
Exact mathematical passage
```
C_{\mathrm{packet}}=O(dnD)+C_{\mathrm{compile}}.
```

c_qUse in source · 12 passages

Search aliases: c_q

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Use in source · Read passage · q_main.tex, line 229
c_q(n)=∑_a∈ U_qζ_q^an, n∈ Z.
Exact mathematical passage
```
c_q(n)=\sum_{a\in U_q}\zeta_q^{an},
\qquad n\in\mathbb Z.
```

\mathcal C_{\mathrm{res}}Use in source · 6 passages

Search aliases: C_res mathcal mathrm

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Use in source · Read passage · g_main.tex, line 188
Metric and existence defects in one category. Let C_res⊆ C_amb be categories with weighted generating morphisms. For a morphism f in the restricted target class, the same definition Δ(f)=c_res(f)-c_amb(f) includes group-word metrics, adaptive quantum channels, tensor-network realizations, and orthogonality designs. This categorical form is essential when the ambient model permits measurement, branching, postselection, or entangled objects that do not lie in a common group.
Exact mathematical passage
```
\mathcal C_{\mathrm{res}}\subseteq\mathcal C_{\mathrm{amb}}
```

c_{\mathrm{res}}Use in source · 4 passages

Search aliases: c_res mathrm

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Use in source · Read passage · g_main.tex, line 189
Metric and existence defects in one category. Let C_res⊆ C_amb be categories with weighted generating morphisms. For a morphism f in the restricted target class, the same definition Δ(f)=c_res(f)-c_amb(f) includes group-word metrics, adaptive quantum channels, tensor-network realizations, and orthogonality designs. This categorical form is essential when the ambient model permits measurement, branching, postselection, or entangled objects that do not lie in a common group.
Exact mathematical passage
```
\Delta(f)=c_{\mathrm{res}}(f)-c_{\mathrm{amb}}(f)
```

c_{\rm amb}Use in source · 2 passages

Search aliases: c_rm amb rm

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Use in source · Read passage · s_main.tex, line 2007
Unified terminology. For a target x, let c_rm res(x) and c_rm amb(x) be minimum costs, with +∞ assigned when a realization does not exist. Then Δ(x) = c_rm res(x)-c_rm amb(x) includes both finite metric defects and infinite existence defects. Symmetry collapse concerns the special case where the ambient quantum description has a cheaper classical realization because its reachable representation is compressed.
Exact mathematical passage
```
c_{\rm amb}(x)
```

C_{\rm apply}Use in source · 3 passages

Search aliases: C_rm apply rm

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Use in source · Read passage · s_main.tex, line 828
Repeated application versus one shot. Let C_rm compile be the cost of constructing the packet basis and restricted gate plans, and let C_rm apply be one packet application. Reusing the compiled plan R times costs C_rm packet(R) = C_rm compile+RC_rm apply. If the full path costs RC_rm full, the packet method breaks even when R> fracC_rm compile C_rm full-C_rm apply. This is why time evolution, parameter sweeps, variational loops, and repeated likelihood evaluations are natural applications.
Exact mathematical passage
```
C_{\rm apply}
```

C_{\rm compile}Use in source · 3 passages

Search aliases: C_rm compile rm

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Use in source · Read passage · s_main.tex, line 828
Repeated application versus one shot. Let C_rm compile be the cost of constructing the packet basis and restricted gate plans, and let C_rm apply be one packet application. Reusing the compiled plan R times costs C_rm packet(R) = C_rm compile+RC_rm apply. If the full path costs RC_rm full, the packet method breaks even when R> fracC_rm compile C_rm full-C_rm apply. This is why time evolution, parameter sweeps, variational loops, and repeated likelihood evaluations are natural applications.
Exact mathematical passage
```
C_{\rm compile}
```

C_{\rm full}Use in source · 2 passages

Search aliases: C_rm full rm

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Use in source · Read passage · s_main.tex, line 834
Repeated application versus one shot. Let C_rm compile be the cost of constructing the packet basis and restricted gate plans, and let C_rm apply be one packet application. Reusing the compiled plan R times costs C_rm packet(R) = C_rm compile+RC_rm apply. If the full path costs RC_rm full, the packet method breaks even when R> fracC_rm compile C_rm full-C_rm apply. This is why time evolution, parameter sweeps, variational loops, and repeated likelihood evaluations are natural applications.
Exact mathematical passage
```
RC_{\rm full}
```

C_{\rm packet}Use in source · 1 passage

Search aliases: C_rm packet rm

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Use in source · Read passage · s_main.tex, line 829
Repeated application versus one shot. Let C_rm compile be the cost of constructing the packet basis and restricted gate plans, and let C_rm apply be one packet application. Reusing the compiled plan R times costs C_rm packet(R) = C_rm compile+RC_rm apply. If the full path costs RC_rm full, the packet method breaks even when R> fracC_rm compile C_rm full-C_rm apply. This is why time evolution, parameter sweeps, variational loops, and repeated likelihood evaluations are natural applications.
Exact mathematical passage
```
C_{\rm packet}(R)
=
C_{\rm compile}+RC_{\rm apply}.
```

c_{\rm res}Use in source · 2 passages

Search aliases: c_rm res rm

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Use in source · Read passage · s_main.tex, line 2007
Unified terminology. For a target x, let c_rm res(x) and c_rm amb(x) be minimum costs, with +∞ assigned when a realization does not exist. Then Δ(x) = c_rm res(x)-c_rm amb(x) includes both finite metric defects and infinite existence defects. Symmetry collapse concerns the special case where the ambient quantum description has a cheaper classical realization because its reachable representation is compressed.
Exact mathematical passage
```
c_{\rm res}(x)
```

c_uUse in source · 4 passages

Search aliases: c_u

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Use in source · Read passage · c_main.tex, line 525
ltiplicative map into a K-algebra. Suppose Φ(δ_v)≠0 for every v∈ W. Then dim_K A≥2^m. No unital hypothesis on Φ is required. Write e_v=Φ(δ_v). Multiplicativity makes these nonzero orthogonal idempotents. If ∑_v∈ Wc_ve_v=0, multiply the equality by e_u. Orthogonality annihilates every term except c_ue_u, which must be zero. Since K is a field and e_u≠0, c_u=0. This holds for every u, so the 2^m vectors e_v are linearly independent.
Exact mathematical passage
```
c_ue_u
```

c_vUse in source · 2 passages

Search aliases: c_v

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Use in source · Read passage · c_main.tex, line 522
a faithful multiplicative summary] Let Φ: B→ A be a K-linear multiplicative map into a K-algebra. Suppose Φ(δ_v)≠0 for every v∈ W. Then dim_K A≥2^m. No unital hypothesis on Φ is required. Write e_v=Φ(δ_v). Multiplicativity makes these nonzero orthogonal idempotents. If ∑_v∈ Wc_ve_v=0, multiply the equality by e_u. Orthogonality annihilates every term except c_ue_u, which must be zero. Since K is a field and e_u≠0, c_u=0. This holds for every u, so the 2^m vectors e_v are linearly independent.
Exact mathematical passage
```
 \sum_{v\in W}c_ve_v=0,
```

C_xUse in source · 4 passages

Search aliases: C_x

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Use in source · Read passage · s_main.tex, line 199
Circuit families. A quantum decision procedure is a polynomial-time uniform family of circuits {C_x}, where the input string x determines a circuit on n=n(|x|) qubits with at most poly(|x|) gates. Let C_x=G_m⋯ G_1. A classical simulation theorem must describe an algorithm receiving the same succinct description of x and the circuit. It is not enough to assume that an exponentially large matrix or a complete orbit list has already been supplied.
Exact mathematical passage
```
\{C_x\}
```

c_yUse in source · 5 passages

Search aliases: c_y

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Use in source · Read passage · g_main.tex, line 1508
Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y;…
Exact mathematical passage
```
c_y
```

\mathrm{cat}Use in source · 1 passage

Search aliases: cat mathrm

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Use in source · Read passage · g_main.tex, line 353
Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.
Exact mathematical passage
```
w_{\mathrm{cat}}=
\max_t
\#\{j:t\in I_j\}.
```

\mathrm{clean}Use in source · 1 passage

Search aliases: clean mathrm

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Use in source · Read passage · g_main.tex, line 4375
Ancillas define a family of defects Δ_a,clean(U), Δ_a,dirty(U), where a is the number of ancillary qubits or modes. Stable ancilla capacity asks whether a fixed number of reusable ancillas can yield an unbounded saving as the data size grows.
Exact mathematical passage
```
\Delta_{a,\mathrm{clean}}(U),
\qquad
\Delta_{a,\mathrm{dirty}}(U),
```

\mathrm{close}Use in source · 3 passages

Search aliases: close mathrm

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Use in source · Read passage · g_main.tex, line 343
Catalytic interval geometry. Associate to every matched opening and closing operation an interval I_j=[t_j^open,t_j^close]. The family of intervals can be disjoint, nested, laminar, or crossing.
Exact mathematical passage
```
I_j=[t_j^{\mathrm{open}},t_j^{\mathrm{close}}].
```

\operatorname{code}Use in source · 1 passage

Search aliases: code operatorname

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Use in source · Read passage · g_main.tex, line 3095
A lower-unitriangular binary matrix is encoded by reading the strict lower entries row by row: (1,0), (2,0),(2,1), …, (5,0),…,(5,4). The resulting fifteen-bit integer is denoted by code(A). A row is displayed as the integer whose binary expansion is its support vector. [source reference] lists all defects. Every row has ellDown=7 and ellAll=6.
Exact mathematical passage
```
\operatorname{code}(A)
```

\codimUse in source · 5 passages

Search aliases: codim

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Use in source · Read passage · g_main.tex, line 582
Defect codimension. If every target has restricted cost at most N, then defect at least d forces ambient cost at most N-d. The dimension-per-gate theorem gives codim_Voverline{U:Δ(U)≥ d} ≥ d when each ambient gate carries one parameter.
Exact mathematical passage
```
\codim_V\overline{\{U:\Delta(U)\ge d\}}
\ge d
```

\codim_{(\U_n)_{\overline K}}Use in source · 1 passage

Search aliases: codim_(U_n)_overline K codim U overline

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Use in source · Read passage · g_main.tex, line 1077
-rigidity-intro Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline K overline D_n,≥ d(overline K)≥ d.
Exact mathematical passage
```
\codim_{(\U_n)_{\overline K}}
\overline{\mathcal D_{n,\ge d}(\overline K)}\ge d.
```

\codim_{\U_n}Use in source · 1 passage

Search aliases: codim_U_n codim U

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Use in source · Read passage · g_main.tex, line 2512
[Defect--codimension bound] Let K be algebraically closed. For every 1≤ d≤ N, dimoverline D_n,≥ d≤ N-d. Equivalently, codim_U_noverline D_n,≥ d≥ d. More finely, on the stratum where ellDown(A)=L and DeltaDef(A)≥ d, the closure has dimension at most L-d.
Exact mathematical passage
```
\codim_{\U_n}\overline{\mathcal D_{n,\ge d}}\ge d.
```

\codim_VUse in source · 1 passage

Search aliases: codim_V codim

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Use in source · Read passage · g_main.tex, line 582
Defect codimension. If every target has restricted cost at most N, then defect at least d forces ambient cost at most N-d. The dimension-per-gate theorem gives codim_Voverline{U:Δ(U)≥ d} ≥ d when each ambient gate carries one parameter.
Exact mathematical passage
```
\codim_V\overline{\{U:\Delta(U)\ge d\}}
\ge d
```

\mathrm{compile}Use in source · 1 passage

Search aliases: compile mathrm

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Use in source · Read passage · s_main.tex, line 227
State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectorization, and basis compilation determine actual runtime. The asymptotic distinction is exact when D is polynomial and 2^n is exponential.
Exact mathematical passage
```
C_{\mathrm{packet}}=O(dnD)+C_{\mathrm{compile}}.
```

\mathrm{conn}Use in source · 1 passage

Search aliases: conn mathrm

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Use in source · Read passage · g_main.tex, line 4262
Architecture and routing. On a line or grid, Gaussian and matchgate circuits may be dominated by routing. Let G_conn be the hardware graph. A mode permutation π can be treated as a catalyst if it is opened, many interactions are executed in the reordered frame, and π^-1 restores the logical labeling. The benefit is routing saved across consumers - 2 cost(π). This is the same amortization law as the linear snapshot family, now applied to mode geometry.
Exact mathematical passage
```
G_{\mathrm{conn}}
```

\mathrm{controls}Use in source · 3 passages

Search aliases: controls mathrm

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Use in source · Read passage · c_lean.tex, line 94
Suppose the next gate toggles target u by the product of a control list of length q. A product that omits u is unchanged. If u occurs, idempotence removes all its repetitions. With R the remaining list, the updated product is (s_u∏_i∈ Rs_i) x (∏_j∈controlss_j∏_i∈ Rs_i). The two old-state products have certified degrees at most t+1+|R| and t+q+|R|. Since |R|+1≤ |L|, the invariant's budget rises by at most max{q-1,0}: zero for NOT and CNOT, one for Toffoli.
Exact mathematical passage
```
\left(s_u\prod_{i\in R}s_i\right)
\x
\left(\prod_{j\in\mathrm{controls}}s_j\prod_{i\in R}s_i\right).
```

\cosUse in source · 9 passages

Search aliases: cos

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Use in source · Read passage · q_lean.tex, line 103
The rational signal and its filter. The first layer uses literal rational lists. The chosen perturbation is r_n=3cos(π n/2)+2, whose displayed rational entries can be certified without numerical trigonometric evaluation. Its sum with c is the test signal.
Exact mathematical passage
```
r_n=3\cos(\pi n/2)+2
```

\costUse in source · 2 passages

Search aliases: cost

Link to this entry ·

Use in source · Read passage · g_main.tex, line 4303
For a target channel Φ in the restricted category, define c_res(Φ) = inf{cost(W):W realizes Φ in C_res} and similarly c_amb(Φ). The value +∞ is allowed. A finite positive difference is a metric defect; an infinite difference is an existence defect.
Exact mathematical passage
```
c_{\mathrm{res}}(\Phi)
=
\inf\{\cost(W):W\text{ realizes }\Phi\text{ in }\mathcal C_{\mathrm{res}}\}
```

\operatorname{cost}Use in source · 3 passages

Search aliases: cost operatorname

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2336
A hardware or fault model may penalize an upper-directed interaction more heavily than a lower-directed one. Fix λ>0. If w has d(w) lower-directed gates and u(w) upper-directed gates, put cost_λ(w)=d(w)+λ u(w), and define ellWeightλ(A) = inf{cost_λ(w):[w]=A}. At λ=1 this is the ordinary ambient length, while large λ increasingly penalizes flag excursions.
Exact mathematical passage
```
\operatorname{cost}_{\lambda}(w)=d(w)+\lambda u(w),
```

\operatorname{cost}_{\lambda}Use in source · 2 passages

Search aliases: cost_λ operatorname lambda

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Use in source · Read passage · g_main.tex, line 2336
A hardware or fault model may penalize an upper-directed interaction more heavily than a lower-directed one. Fix λ>0. If w has d(w) lower-directed gates and u(w) upper-directed gates, put cost_λ(w)=d(w)+λ u(w), and define ellWeightλ(A) = inf{cost_λ(w):[w]=A}. At λ=1 this is the ordinary ambient length, while large λ increasingly penalizes flag excursions.
Exact mathematical passage
```
\operatorname{cost}_{\lambda}(w)=d(w)+\lambda u(w),
```

\CX_{0\to1}Use in source · 9 passages

Search aliases: CX_0→1 CX to

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Use in source · Read passage · g_appendix.tex, line 117
1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.
Exact mathematical passage
```
\CX_{0\to1},\ S_1,\ \CX_{1\to0},\ \CX_{0\to1},
```

\CX_{0\to2}Use in source · 4 passages

Search aliases: CX_0→2 CX to

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 140
0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.
Exact mathematical passage
```
\CX_{0\to2},\ \CZ_{1,2},\ \CX_{2\to0},\ \CX_{0\to2},
\ \CX_{0\to1},\ \CX_{1\to0},\ \CX_{0\to1},
```

\CX_{1\to0}Use in source · 9 passages

Search aliases: CX_1→0 CX to

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 117
1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.
Exact mathematical passage
```
\CX_{0\to1},\ S_1,\ \CX_{1\to0},\ \CX_{0\to1},
```

\CX_{1\to2}Use in source · 2 passages

Search aliases: CX_1→2 CX to

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 145
0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.
Exact mathematical passage
```
H_1,\ \CX_{1\to2},\ \CX_{2\to1},\ \CX_{2\to0},\ \CX_{0\to2},\ H_0.
```

\CX_{2\to0}Use in source · 4 passages

Search aliases: CX_2→0 CX to

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Use in source · Read passage · g_appendix.tex, line 140
0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.
Exact mathematical passage
```
\CX_{0\to2},\ \CZ_{1,2},\ \CX_{2\to0},\ \CX_{0\to2},
\ \CX_{0\to1},\ \CX_{1\to0},\ \CX_{0\to1},
```

\CX_{2\to1}Use in source · 2 passages

Search aliases: CX_2→1 CX to

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Use in source · Read passage · g_appendix.tex, line 145
0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.
Exact mathematical passage
```
H_1,\ \CX_{1\to2},\ \CX_{2\to1},\ \CX_{2\to0},\ \CX_{0\to2},\ H_0.
```

\CX_{c\to t}Use in source · 1 passage

Search aliases: CX_c→ t CX to

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Use in source · Read passage · g_appendix.tex, line 154
Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric group.
Exact mathematical passage
```
\CX_{c\to t}:x_t\leftarrow x_t+x_c,
```

\CZ_{1,2}Use in source · 1 passage

Search aliases: CZ_1,2 CZ

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Use in source · Read passage · g_appendix.tex, line 140
0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.
Exact mathematical passage
```
\CX_{0\to2},\ \CZ_{1,2},\ \CX_{2\to0},\ \CX_{0\to2},
\ \CX_{0\to1},\ \CX_{1\to0},\ \CX_{0\to1},
```

\CZ_{12}Use in source · 1 passage

Search aliases: CZ_12 CZ

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Use in source · Read passage · g_main.tex, line 3488
0 · 1 · 0 · 1 · 0 · 0 . A restricted word of length seven is CX_0→2, CZ_12, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1. An ambient word of length six is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.
Exact mathematical passage
```
\CX_{0\to2},
\CZ_{12},
\CX_{2\to0},
\CX_{0\to2},
\CX_{0\to1},
\CX_{1\to0},
\CX_{0\to1}.
```

\mathcal DUse in source · 6 passages

Search aliases: D mathcal

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Use in source · Read passage · g_main.tex, line 1072
[Generic rigidity and defect codimension] Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline K overline D_n,≥ d(overline K)≥ d.
Exact mathematical passage
```
\mathcal D_{n,\ge d}(\overline K)
=\{A\in\U_n(\overline K):\DeltaDef(A)\ge d\},
```

D^2Use in source · 1 passage

Search aliases: D^2

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Use in source · Read passage · q_visual.tex, line 94
With the positive-sign unitary matrix F, the negative-sign DFT satisfies F^† x=widehat x/sqrt12. fig05_transported_projector1.0Left: the frequency selector D. Right: its transported sample-basis matrix P. The right colorbar displays its five possible rational entries. The two panels use different scales. P=FDF^†, P_nm=fracc_12(n-m)12. Because D^2=D and F^† F=I, we obtain P^2=P. Because D^†=D, we also obtain P^†=P. This is an orthogonal projector of rank four.
Exact mathematical passage
```
D^2=D
```

d^2Use in source · 1 passage

Search aliases: d^2

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Use in source · Read passage · s_main.tex, line 1166
umber of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).
Exact mathematical passage
```
\sum_\lambda u_\lambda^2
=O_d\left(n^{d^2-1}\right).
\label{s:eq:commutant-parameter-bound}
```

D^\daggerUse in source · 1 passage

Search aliases: D^† dagger

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Use in source · Read passage · q_visual.tex, line 94
-sign DFT satisfies F^† x=widehat x/sqrt12. fig05_transported_projector1.0Left: the frequency selector D. Right: its transported sample-basis matrix P. The right colorbar displays its five possible rational entries. The two panels use different scales. P=FDF^†, P_nm=fracc_12(n-m)12. Because D^2=D and F^† F=I, we obtain P^2=P. Because D^†=D, we also obtain P^†=P. This is an orthogonal projector of rank four.
Exact mathematical passage
```
D^\dagger=D
```

D_{\mathrm B}Use in source · 1 passage

Search aliases: D_ B mathrm

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Use in source · Read passage · s_main.tex, line 1449
Bosonic and fermionic analogues. For N fermions in m modes, D_ F(m,N)=binom mN. For N bosons in m modes, D_ B(m,N)=binomm+N-1N. The same binomial analysis gives:
Exact mathematical passage
```
D_{\mathrm B}(m,N)=\binom{m+N-1}{N}.
```

D_{\mathrm F}Use in source · 1 passage

Search aliases: D_ F mathrm

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Use in source · Read passage · s_main.tex, line 1445
Bosonic and fermionic analogues. For N fermions in m modes, D_ F(m,N)=binom mN. For N bosons in m modes, D_ B(m,N)=binomm+N-1N. The same binomial analysis gives:
Exact mathematical passage
```
D_{\mathrm F}(m,N)=\binom mN.
```

d_*Use in source · 1 passage

Search aliases: d_*

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Use in source · Read passage · g_main.tex, line 3167
[Radius-four right-coset meet in the middle] Compute the exact radius-four ball B_4 in GL_6(F_2). Bucket its elements by κ. For each bucket and every ordered pair (b,a) in that bucket, compute u=b^-1a∈U_6 and update d_*(u)← min(d_*(u),d(b)+d(a)).
Exact mathematical passage
```
d_*(u)\leftarrow
\min\bigl(d_*(u),d(b)+d(a)\bigr).
```

d_0Use in source · 2 passages

Search aliases: d_0

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Use in source · Read passage · g_main.tex, line 3589
The four exceptional matrices admit one common lower-bound proof. Work in the declared symplectic quotient and entangling metric, with H_i and S_i free and SWAP excluded as a primitive. Put J_2=0 · 1 1 · 0, D=diag(d_0,d_1), d_0,d_1∈F_2, and define M_D=J_2 · 0
Exact mathematical passage
```
J_2=\begin{pmatrix}0&1\\1&0\end{pmatrix},
\qquad D=\operatorname{diag}(d_0,d_1),
\qquad d_0,d_1\in\F_2,
```

d_1Use in source · 2 passages

Search aliases: d_1

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Use in source · Read passage · g_main.tex, line 3589
The four exceptional matrices admit one common lower-bound proof. Work in the declared symplectic quotient and entangling metric, with H_i and S_i free and SWAP excluded as a primitive. Put J_2=0 · 1 1 · 0, D=diag(d_0,d_1), d_0,d_1∈F_2, and define M_D=J_2 · 0
Exact mathematical passage
```
J_2=\begin{pmatrix}0&1\\1&0\end{pmatrix},
\qquad D=\operatorname{diag}(d_0,d_1),
\qquad d_0,d_1\in\F_2,
```

d_AUse in source · 3 passages

Search aliases: d_A

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Use in source · Read passage · g_main.tex, line 4865
[Breadth-first defect census] Given restricted generators S_R, ambient generators S_A, and identity e: item compute d_R on ⟨ S_R⟩; item compute d_A on ⟨ S_A⟩; item for every restricted state x, compare d_R(x) and d_A(x); item record the pair (d_R(x),d_A(x)) and reconstruct witnesses for strict inequalities.
Exact mathematical passage
```
d_A
```

D_{aa}Use in source · 1 passage

Search aliases: D_aa

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Use in source · Read passage · q_visual.tex, line 89
A diagonal selector becomes a rational filter. addcontentslinetocsectionA diagonal selector becomes a rational filter leadThe filter is diagonal in one basis and a rational convolution in the other. Let F_na=ζ^na/sqrt12 and D_aa=1_a∈ U_12. With the positive-sign unitary matrix F, the negative-sign DFT satisfies F^† x=widehat x/sqrt12. fig05_transported_projector1.0Left: the frequency selector D. Right: its transported sample-basis matrix P. The right colorbar displays its five possible rational …
Exact mathematical passage
```
D_{aa}=\mathbf1_{a\in U_{12}}
```

D_dUse in source · 6 passages

Search aliases: D_d

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Use in source · Read passage · q_main.tex, line 342
The set S_d contains the labels of characters of exact order d. Put D_d=diag(1_a∈ S_d) and
Exact mathematical passage
```
D_d=\operatorname{diag}(\mathbf1_{a\in S_d})
```

D_eUse in source · 1 passage

Search aliases: D_e

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Use in source · Read passage · q_main.tex, line 392
QA04 proves rationality. Each D_d is a self-adjoint diagonal projector. The sets S_d partition I_N, because every character has exactly one order dividing N. Hence D_dD_e=δ_deD_d and ∑_dmid ND_d=Id_N. Conjugating these identities by the unitary F_N proves the projector identities. The rank is unchanged under multiplication by invertible matrices, so it equals the number of selected diagonal entries, |S_d|=φ(d). The diagonal kernel entries are all c_d(0)/N=φ(d)/N, whose sum is φ(d). square
Exact mathematical passage
```
D_dD_e=\delta_{de}D_d
```

D_iUse in source · 3 passages

Search aliases: D_i

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Use in source · Read passage · c_lean.tex, line 298
Cut transport · Inverse suffix meets the appropriate forward target branch; H_i=fD_i gives safety and pointwise completeness
Exact mathematical passage
```
H_i=fD_i
```

D_{\mathrm{internal}}Use in source · 2 passages

Search aliases: D_internal mathrm

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Use in source · Read passage · s_main.tex, line 33
The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:
Exact mathematical passage
```
\mathfrak C_n=
\bigl(
N_{\mathrm{packets}},
D_{\mathrm{internal}},
S_{\mathrm{transfer}},
L_{\mathrm{leakage}}
\bigr).
```

d_jUse in source · 5 passages

Search aliases: d_j

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Use in source · Read passage · g_main.tex, line 2945
[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^r+120 ≤ frac2120n. Also r≥log_21n-1. Therefore S≥ nlog_21n-frac4120n. Multiplying by 8/21 yields D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n.
Exact mathematical passage
```
n=\sum_{j=0}^{r}d_jm^j,
\qquad
0\le d_j<m.
```

D_{n,k}Use in source · 4 passages

Search aliases: D_n,k

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Use in source · Read passage · s_main.tex, line 1350
Exact sector size is a complexity phase variable. The statement ``fixed particle number is polynomial'' has a sharp extension. Let the particle number depend on the system size: k=k(n). The explicit-amplitude simulation cost of one fermionic or qubit weight packet is governed by D_n,k=binom nk. For 1≤ k≤ n, the standard bounds (frac nk)^k ≤ binom nk ≤ (fracenk)^k show that the growth changes continuously between polynomial and exponential regimes.
Exact mathematical passage
```
D_{n,k}=\binom nk.
```

\mathcal D_{n,\ge d}Use in source · 3 passages

Search aliases: D_n,≥ d mathcal ge

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Use in source · Read passage · g_main.tex, line 1072
[Generic rigidity and defect codimension] Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline K overline D_n,≥ d(overline K)≥ d.
Exact mathematical passage
```
\mathcal D_{n,\ge d}(\overline K)
=\{A\in\U_n(\overline K):\DeltaDef(A)\ge d\},
```

D_n^{(1)}Use in source · 1 passage

Search aliases: D_n^(1)

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Use in source · Read passage · g_main.tex, line 367
Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.
Exact mathematical passage
```
D_n^{(1)}=\Omega(n)
```

D_n^{(w)}Use in source · 1 passage

Search aliases: D_n^(w)

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Use in source · Read passage · g_main.tex, line 359
Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.
Exact mathematical passage
```
D_n^{(w)}
=
\max_{U}
\left(
\ellRes(U)-\ellAmb^{(w)}(U)
\right),
```

D_n^kUse in source · 4 passages

Search aliases: D_n^k

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Use in source · Read passage · s_main.tex, line 905
The symmetric subspace. Let the symmetric group S_n act by permuting the n tensor factors. The fully symmetric qubit subspace is spanned by the Dicke states ketD_n^k = binom nk^-1/2 ∑_|x|=kket x, 0≤ k≤ n. Its dimension is n+1.
Exact mathematical passage
```
\ket{D_n^k}
=
\binom nk^{-1/2}
\sum_{|x|=k}\ket x,
\qquad
0\le k\le n.
```

D_n^{k+1}Use in source · 1 passage

Search aliases: D_n^k+1

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Use in source · Read passage · s_main.tex, line 941
uad a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1, J_-ketD_n^k = sqrtk(n-k+1)ketD_n^k-1. Thus collective rotations and polynomial functions of collectiv…
Exact mathematical passage
```
J_+\ket{D_n^k}
=
\sqrt{(n-k)(k+1)}\ket{D_n^{k+1}},
```

D_n^{k-1}Use in source · 1 passage

Search aliases: D_n^k-1

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Use in source · Read passage · s_main.tex, line 946
mmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1, J_-ketD_n^k = sqrtk(n-k+1)ketD_n^k-1. Thus collective rotations and polynomial functions of collective spin admit banded or structured updates on n+1…
Exact mathematical passage
```
J_-\ket{D_n^k}
=
\sqrt{k(n-k+1)}\ket{D_n^{k-1}}.
```

d_RUse in source · 3 passages

Search aliases: d_R

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Use in source · Read passage · g_main.tex, line 4864
[Breadth-first defect census] Given restricted generators S_R, ambient generators S_A, and identity e: item compute d_R on ⟨ S_R⟩; item compute d_A on ⟨ S_A⟩; item for every restricted state x, compare d_R(x) and d_A(x); item record the pair (d_R(x),d_A(x)) and reconstruct witnesses for strict inequalities.
Exact mathematical passage
```
d_R
```

D_{\rm boson}Use in source · 1 passage

Search aliases: D_rm boson rm

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Use in source · Read passage · s_main.tex, line 1290
Bosonic occupation space. For N indistinguishable bosons in m modes, a Fock basis state is an occupation vector (n_1,…,n_m), n_1+⋯+n_m=N. The dimension is D_rm boson(m,N) = binomN+m-1N. For fixed N, this is O(m^N).
Exact mathematical passage
```
D_{\rm boson}(m,N)
=
\binom{N+m-1}{N}.
\label{s:eq:boson-dim}
```

D_{\rm fermion}Use in source · 1 passage

Search aliases: D_rm fermion rm

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Use in source · Read passage · s_main.tex, line 1300
Fermionic occupation space. For N fermions in m modes, each mode is occupied at most once and D_rm fermion(m,N) = binom mN. For fixed N, this is also polynomial in m.
Exact mathematical passage
```
D_{\rm fermion}(m,N)
=
\binom mN.
\label{s:eq:fermion-dim}
```

D_{\rm internal}Use in source · 1 passage

Search aliases: D_rm internal rm

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Use in source · Read passage · s_main.tex, line 2273
The core research invariant is the complete classical representation record ( discovery cost, N_rm packets, D_rm internal, S_rm transfer, L_rm leakage, measurement cost ). A future compiler that computes and certifies this record can decide when the full state vector is unnecessary, when a hybrid representation is preferable, and when symmetry alone is insufficient.
Exact mathematical passage
```
\bigl(
\text{discovery cost},
N_{\rm packets},
D_{\rm internal},
S_{\rm transfer},
L_{\rm leakage},
\text{measurement cost}
\bigr).
```

D_{\rm packet}Use in source · 4 passages

Search aliases: D_rm packet rm

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Use in source · Read passage · s_main.tex, line 890
Entanglement inside a packet. A small number of symmetry labels does not imply low entanglement. The half-filled packet can support volume-law entanglement. Tensor-network bond dimension and packet dimension are independent resources. A robust simulator should therefore track both (D_rm packet,χ_rm internal), where χ_rm internal measures the cost of representing the state inside the packet.
Exact mathematical passage
```
(D_{\rm packet},\chi_{\rm internal}),
```

D_SUse in source · 3 passages

Search aliases: D_S

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Use in source · Read passage · q_main.tex, line 460
For any set S⊆ I_N, let D_S be its diagonal indicator and P_S=F_ND_SF_N^†. This construction does not require S to be an order class. Rationality is special, but orthogonal projection is general.
Exact mathematical passage
```
D_S
```

d_SUse in source · 3 passages

Search aliases: d_S

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Use in source · Read passage · g_main.tex, line 468
[Endpoint Green valuation] The word distance is d_S(e,g) = ord_z=0 ⟨ δ_g,(I-zA_S)^-1δ_e ⟩. Expand the resolvent as a formal power series: (I-zA_S)^-1=∑_k≥0z^kA_S^k. The first nonzero endpoint coefficient occurs at the shortest word length.
Exact mathematical passage
```
d_S(e,g)
=
\ord_{z=0}
\left\langle
\delta_g,(I-zA_S)^{-1}\delta_e
\right\rangle.
```

D_tUse in source · 4 passages

Search aliases: D_t

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Use in source · Read passage · s_main.tex, line 442
Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.
Exact mathematical passage
```
B_t:\C^{D_t}\longrightarrow\Hh,
\qquad
B_tB_t^*=P_t,
\qquad
B_t^*B_t=I_{D_t},
```

d_tUse in source · 3 passages

Search aliases: d_t

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Use in source · Read passage · s_main.tex, line 679
Fix the projectors chosen during the particular run. Put d_t=(I-P_t)G_tϕ_t-1. Subtraction gives the exact recurrence ψ_t-ϕ_t=G_t(ψ_t-1-ϕ_t-1)+d_t. Since ψ_0-ϕ_0=0, iteration yields ψ_m-ϕ_m=∑_t=1^mG_m⋯ G_t+1d_t. Every unitary product preserves norm, so the triangle inequality proves the state estimate. Also ψ_m =1 and ϕ_m ≤1 because each projector is contractive. Expanding the difference of the two quadratic forms and using M ≤1 gives the factor two. This proof applies to the chosen run even when…
Exact mathematical passage
```
d_t=(I-P_t)G_t\phi_{t-1}
```

D_uUse in source · 4 passages

Search aliases: D_u

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Use in source · Read passage · g_main.tex, line 3726
A candidate lower-bound invariant must do more than track maximum degree. One Toffoli creates degree two and a second can remove it. Useful refinements include: item the dimension of the quadratic monomial span, item directional derivatives D_u f(x)=f(x+u)+f(x), item the rank of the bilinear polarization, item the number of outputs consuming a nonlinear workspace term, item restoration obligations for every nonlinear carrier.
Exact mathematical passage
```
D_u f(x)=f(x+u)+f(x)
```

d_\chiUse in source · 2 passages

Search aliases: d_χ chi

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Use in source · Read passage · s_main.tex, line 157
Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).
Exact mathematical passage
```
d_\chi
```

D_{\le K}Use in source · 2 passages

Search aliases: D_≤ K le

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1432
A bounded-excitation hierarchy. The union of the first K weight sectors has dimension D_≤ K(n) = ∑_k=0^Kbinom nk. For fixed K, D_≤ K(n)=O(n^K). If K=O(log n), the same union is quasipolynomial. This refines [source reference]: bounded symmetry breaking preserves polynomial simulation, while logarithmically growing excitation gives a controlled quasipolynomial regime.
Exact mathematical passage
```
D_{\le K}(n)
=
\sum_{k=0}^{K}\binom nk.
```

\operatorname{decode}Use in source · 2 passages

Search aliases: decode operatorname

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1809
Let R_n be a class of classical descriptions, and let decode: R_n→ P_nHh_n associate a quantum state with each description.
Exact mathematical passage
```
\operatorname{decode}:\mathcal R_n\to P_n\Hh_n
```

\degUse in source · 15 passages

Search aliases: deg

Link to this entry ·

Use in source · Read passage · c_lean.tex, line 129
pageheadFrom the cut to a numerical lower boundEXISTENCE, BALANCE, AND COMPLETE COVERAGE Write c=pmathbin++q. At this cut, Block 01 defines H_i(x)=run(p)(x,0^a,0)_ix run(q^-1)(x,0^a,0)_i and proves H_i≤ f, f(x)=1iff∃ i H_i(x)=1. Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives deg(H_i)≤max{TC(p),TC(q)}+1.
Exact mathematical passage
```
\deg(H_i)\le\max\{\TC(p),\TC(q)\}+1.
```

\mathrm{depth}Use in source · 1 passage

Search aliases: depth mathrm

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Use in source · Read passage · g_main.tex, line 3662
Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its total count is large. Conversely, a two-gate count saving can increase depth under locality constraints.
Exact mathematical passage
```
\Delta_{\mathrm{depth}}(U),
\qquad
\Delta_{\mathrm{2q-depth}}(U),
\qquad
\Delta_{\mathrm{space-time}}(U).
```

\diagUse in source · 1 passage

Search aliases: diag

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Use in source · Read passage · g_appendix.tex, line 164
Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).
Exact mathematical passage
```
\iota(A)=\diag(A,A^{-T})\in\Sp(2n,\R).
```

\mathrm{diag}Use in source · 3 passages

Search aliases: diag mathrm

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Use in source · Read passage · q_main.tex, line 1383
ρ_diag=diag(c(n)^2/48)_n=0^11.
Exact mathematical passage
```
\rho_{\mathrm{diag}}=\operatorname{diag}\big(c(n)^2/48\big)_{n=0}^{11}.
```

\operatorname{diag}Use in source · 7 passages

Search aliases: diag operatorname

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Use in source · Read passage · g_main.tex, line 2601
[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).
Exact mathematical passage
```
\ellDown\bigl(\operatorname{diag}(A_1,\ldots,A_k)\bigr)
=
\sum_{i=1}^k\ellDown(A_i).
```

\operatorname{diam}Use in source · 2 passages

Search aliases: diam operatorname

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Use in source · Read passage · g_appendix.tex, line 193
Clifford census summary. @ccccrrrr@ Qubits · Metric · SWAP · Restricted size · Ambient size · diam_R · diam_A · Defects
Exact mathematical passage
```
\operatorname{diam}_R
```

\operatorname{diam}_AUse in source · 1 passage

Search aliases: diam_A operatorname

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Use in source · Read passage · g_appendix.tex, line 193
Clifford census summary. @ccccrrrr@ Qubits · Metric · SWAP · Restricted size · Ambient size · diam_R · diam_A · Defects
Exact mathematical passage
```
\operatorname{diam}_A
```

\operatorname{diam}_RUse in source · 1 passage

Search aliases: diam_R operatorname

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Use in source · Read passage · g_appendix.tex, line 193
Clifford census summary. @ccccrrrr@ Qubits · Metric · SWAP · Restricted size · Ambient size · diam_R · diam_A · Defects
Exact mathematical passage
```
\operatorname{diam}_R
```

\dimUse in source · 15 passages

Search aliases: dim

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Use in source · Read passage · c_main.tex, line 409
ratic-cost For even n, let Q(x)=x_1x_2+x_3x_4+⋯+x_n-1x_n. Then M(Q)=KTrest(Q)=n/2. The polar bilinear form of Q is the full-rank alternating form made of n/2 disjoint 2×2 blocks. If Q is affine on an affine subspace with direction space U, its second differences vanish on U, so U is totally isotropic. Nondegeneracy gives dim U≤ n/2.
Exact mathematical passage
```
\dim U\le n/2
```

\dim_KUse in source · 4 passages

Search aliases: dim_K dim

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Use in source · Read passage · c_main.tex, line 515
[Dimension of a faithful multiplicative summary] Let Φ: B→ A be a K-linear multiplicative map into a K-algebra. Suppose Φ(δ_v)≠0 for every v∈ W. Then dim_K A≥2^m. No unital hypothesis on Φ is required. Write e_v=Φ(δ_v). Multiplicativity makes these nonzero orthogonal idempotents. If ∑_v∈ Wc_ve_v=0, multiply the equality by e_u. Orthogonality annihilates every term except c_ue_u, which must be zero. Since K is a field and e_u≠0, c_u=0. This holds for every u, so the 2^m vectors e_v are linearly i…
Exact mathematical passage
```
 \dim_K A\ge2^m.
```

\mathrm{dirty}Use in source · 1 passage

Search aliases: dirty mathrm

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Use in source · Read passage · g_main.tex, line 4375
Ancillas define a family of defects Δ_a,clean(U), Δ_a,dirty(U), where a is the number of ancillary qubits or modes. Stable ancilla capacity asks whether a fixed number of reusable ancillas can yield an unbounded saving as the data size grows.
Exact mathematical passage
```
\Delta_{a,\mathrm{clean}}(U),
\qquad
\Delta_{a,\mathrm{dirty}}(U),
```

EUse in source · 17 passages

Search aliases: E

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Use in source · Read passage · g_appendix.tex, line 8
Row-transvection convention. Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection τ_t,s(a)=I+aE_t,s performs R_t← R_t+aR_s. In the binary CNOT interpretation, s is the control and t is the target.
Exact mathematical passage
```
\tau_{t,s}(a)=I+aE_{t,s}
```

e^{-2\pi ikn/12}Use in source · 1 passage

Search aliases: e^-2π ikn/12 pi

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Use in source · Read passage · q_visual.tex, line 77
The spectrum recovers the selector. addcontentslinetocsectionThe spectrum recovers the selector leadThe Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform widehat x(k)=∑_n=0^11x(n)e^-2π ikn/12. fig04_exact_spectrum1.0The coefficient is exactly 12 at k=1,5,7,11 and exactly zero elsewhere. This is an algebraic identity, not a spectral thresholding decision. widehatc_12(k)=∑_a∈ U_12∑_n=0^11ζ^(a-k)n =12 1_k∈ U_12.…
Exact mathematical passage
```
 \widehat x(k)=\sum_{n=0}^{11}x(n)e^{-2\pi ikn/12}.
```

e^{-2\pi ikr/12}Use in source · 1 passage

Search aliases: e^-2π ikr/12 pi

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Use in source · Read passage · q_visual.tex, line 211
Phase factors connect the lane spectra. addcontentslinetocsectionPhase factors connect the lane spectra leadEach local Fourier transform contributes to every appropriate global frequency. With four-point negative-sign DFTs in the lanes, widehat x(k)=∑_r=0^2e^-2π ikr/12 widehatE_rx(kbmod4). fig18_polyphase_alias_assembly1.0Left: magnitudes of the three local transforms. Right: the recovered global spectrum. Assembly uses the complex coefficients and phase factors; adding the displayed magnitudes …
Exact mathematical passage
```
 \widehat x(k)=\sum_{r=0}^{2}e^{-2\pi ikr/12}
 \widehat{E_rx}(k\bmod4).
```

e^{-2\pi i\ell/3}Use in source · 1 passage

Search aliases: e^-2π iℓ/3 pi ell

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Use in source · Read passage · q_main.tex, line 1023
2(4+2e^-2π iℓ/3+2e^-4π iℓ/3).
Exact mathematical passage
```
2\left(4+2e^{-2\pi i\ell/3}+2e^{-4\pi i\ell/3}\right).
```

e^{-4\pi i\ell/3}Use in source · 1 passage

Search aliases: e^-4π iℓ/3 pi ell

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Use in source · Read passage · q_main.tex, line 1023
2(4+2e^-2π iℓ/3+2e^-4π iℓ/3).
Exact mathematical passage
```
2\left(4+2e^{-2\pi i\ell/3}+2e^{-4\pi i\ell/3}\right).
```

e^{-i(t-s)H}Use in source · 2 passages

Search aliases: e^-i(t-s)H

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Use in source · Read passage · s_main.tex, line 343
Define F(s)=e^-i(t-s)HPe^-isHP. Then F(0)=e^-itHP and F(t)=Pe^-itHP. Differentiation gives F'(s)=ie^-i(t-s)H[H,P]e^-isHP. Therefore (I-P)e^-itHP =-(I-P)∫_0^tF'(s) ds. The unitary factors and projectors have norm at most one, so the integral is bounded by |t| [H,P] .
Exact mathematical passage
```
F(s)=e^{-i(t-s)H}Pe^{-isH}P.
```

e^{-isH}Use in source · 2 passages

Search aliases: e^-isH

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Use in source · Read passage · s_main.tex, line 343
Define F(s)=e^-i(t-s)HPe^-isHP. Then F(0)=e^-itHP and F(t)=Pe^-itHP. Differentiation gives F'(s)=ie^-i(t-s)H[H,P]e^-isHP. Therefore (I-P)e^-itHP =-(I-P)∫_0^tF'(s) ds. The unitary factors and projectors have norm at most one, so the integral is bounded by |t| [H,P] .
Exact mathematical passage
```
F(s)=e^{-i(t-s)H}Pe^{-isH}P.
```

e^{-itA}Use in source · 1 passage

Search aliases: e^-itA

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Use in source · Read passage · s_main.tex, line 1272
Fourier packets. On a cycle, translation characters diagonalize the adjacency operator. The continuous-time walk ketψ(t)=e^-itAket0 is a sum of Fourier-character packets. This is the same discrete Fourier transform that appears as a signal-processing coordinate change, a group-character decomposition, and a quantum basis transformation.
Exact mathematical passage
```
\ket{\psi(t)}=e^{-itA}\ket0
```

e^{-itH}Use in source · 6 passages

Search aliases: e^-itH

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Use in source · Read passage · s_main.tex, line 323
Hamiltonian leakage. Exact conservation is not the only useful regime. Let P be an orthogonal projector and let U(t)=e^-itH for a bounded self-adjoint Hamiltonian H. The leakage operator from the packet is L_P(t)=(I-P)e^-itHP.
Exact mathematical passage
```
U(t)=e^{-itH}
```

e^{-i\theta P/2}Use in source · 2 passages

Search aliases: e^-iθ P/2 theta

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Use in source · Read passage · g_appendix.tex, line 171
Pauli rotations. For a Hermitian Pauli P, R_P(θ)=e^-iθ P/2. Clifford conjugation gives C R_P(θ) C^†=R_CPC^†(θ). The sign of CPC^† can be absorbed by changing θ to -θ.
Exact mathematical passage
```
R_P(\theta)=e^{-i\theta P/2}.
```

e^{-i\phi}Use in source · 1 passage

Search aliases: e^-iϕ phi

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Use in source · Read passage · s_main.tex, line 748
Number-preserving gates. Single-qubit R_Z gates commute with widehat N. A two-qubit exchange rotation acts in the ordered basis ket00,ket01,ket10,ket11 as U_XY(θ,ϕ) = 1 · 0 · 0 · 0 0 · cosθ · -ie^-iϕsinθ · 0 0 · -ie^iϕsinθ · cosθ · 0 0 · 0 · 0 · 1 . It rotates ket01 and ket10 while fixing ket00 and ket11, so it preserves exact Hamming weight.
Exact mathematical passage
```
U_{XY}(\theta,\phi)
=
\begin{pmatrix}
1&0&0&0\\
0&\cos\theta&-ie^{-i\phi}\sin\theta&0\\
0&-ie^{i\phi}\sin\theta&\cos\theta&0\\
0&0&0&1
\end{pmatrix}.
```

e^{-\varepsilon X}Use in source · 1 passage

Search aliases: e^-ε X varepsilon

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Use in source · Read passage · g_main.tex, line 383
Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.
Exact mathematical passage
```
e^{\varepsilon X}e^{\varepsilon Y}e^{-\varepsilon X}e^{-\varepsilon Y}
=
e^{\varepsilon^2[X,Y]+O(\varepsilon^3)}.
```

e^{-\varepsilon Y}Use in source · 1 passage

Search aliases: e^-ε Y varepsilon

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Use in source · Read passage · g_main.tex, line 383
Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.
Exact mathematical passage
```
e^{\varepsilon X}e^{\varepsilon Y}e^{-\varepsilon X}e^{-\varepsilon Y}
=
e^{\varepsilon^2[X,Y]+O(\varepsilon^3)}.
```

e^{2\pi i/12}Use in source · 1 passage

Search aliases: e^2π i/12 pi

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Use in source · Read passage · q_visual.tex, line 21
Coprimality selects primitive roots. addcontentslinetocsectionCoprimality selects primitive roots leadA selected residue is a label for a wave whose phase has exact order twelve. Set ζ=e^2π i/12. The phase ζ^a first returns to 1 after 12/gcd(a,12) powers. Thus the coprime labels 1,5,7,11 select exactly the primitive twelfth roots. fig01_primitive_roots0.91The circle is the complex unit circle. Teal points are the four primitive roots; pale points are the other eight roots. Labels are residues a,…
Exact mathematical passage
```
\zeta=e^{2\pi i/12}
```

e^{2\pi i/M}Use in source · 1 passage

Search aliases: e^2π i/M pi

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Use in source · Read passage · s_main.tex, line 141
Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis and charge sectors in quantum theory.
Exact mathematical passage
```
V_M=\exp\left(\frac{2\pi i}{M}Q\right),
\qquad
\omega_M=e^{2\pi i/M}.
```

e^{2\pi i/q}Use in source · 1 passage

Search aliases: e^2π i/q pi

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Use in source · Read passage · q_main.tex, line 109
ζ_q=e^2π i/q.
Exact mathematical passage
```
\zeta_q=e^{2\pi i/q}.
```

e^{2\pi ian/12}Use in source · 1 passage

Search aliases: e^2π ian/12 pi

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Use in source · Read passage · q_visual.tex, line 4
twelve will become an integer signal, an exact rational filter, and a normalized quantum state. Every bridge will be written down. The guide follows one object through three coordinate systems. The starting condition is coprimality. The signal realization adds its selected waves. The quantum realization normalizes the amplitude vector and distinguishes coherent transformation from measurement. U_12={1,5,7,11}, c_12(n)=∑_a∈ U_12e^2π ian/12. We will build a projector that keeps exactly those modes…
Exact mathematical passage
```
 U_{12}=\{1,5,7,11\},\qquad
 c_{12}(n)=\sum_{a\in U_{12}}e^{2\pi ian/12}.
```

e^{2\pi ibn/(q/d)}Use in source · 2 passages

Search aliases: e^2π ibn/(q/d) pi

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Use in source · Read passage · q_main.tex, line 272
c_q(n) · =∑_dmid qμ(d) ∑_b=0^q/d-1e^2π ibn/(q/d) · =∑_substackdmid q (q/d)mid nμ(d)frac qd.
Exact mathematical passage
```
\begin{aligned}
c_q(n)
&=\sum_{d\mid q}\mu(d)
  \sum_{b=0}^{q/d-1}e^{2\pi ibn/(q/d)}\\
&=\sum_{\substack{d\mid q\\(q/d)\mid n}}\mu(d)\frac qd.
\end{aligned}
```

e^{2\pi i\ell Q/M}Use in source · 1 passage

Search aliases: e^2π iℓ Q/M pi ell

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Use in source · Read passage · s_appendix.tex, line 6
Residue projector formula. Let Qketψ_k=kketψ_k. Then frac1M∑_ℓ=0^M-1 ω_M^-rℓe^2π iℓ Q/Mketψ_k = ( frac1M∑_ℓ=0^M-1 ω_M^(k-r)ℓ )ketψ_k. The bracket equals one when k≡ rpmod M and zero otherwise. This proves [source reference] and the orthogonality of the residue packets.
Exact mathematical passage
```
\frac1M\sum_{\ell=0}^{M-1}
\omega_M^{-r\ell}e^{2\pi i\ell Q/M}\ket{\psi_k}
=
\left(
\frac1M\sum_{\ell=0}^{M-1}
\omega_M^{(k-r)\ell}
\right)\ket{\psi_k}.
```

e^{i\theta}Use in source · 8 passages

Search aliases: e^iθ theta

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Use in source · Read passage · q_main.tex, line 1316
χ_θ=v_++e^iθv_-.
Exact mathematical passage
```
\chi_\theta=v_++e^{i\theta}v_-.
```

e^{i\pi/4}Use in source · 1 passage

Search aliases: e^iπ/4 pi

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Use in source · Read passage · g_main.tex, line 3901
Let ω=e^iπ/4. A phase-polynomial circuit on n qubits has the form [source reference] Uketx=ω^P(x)ketAx+b, where A∈GL(n,2), b∈F_2^n, and P:F_2^n→Z_8 is represented as a sum of binary linear forms with coefficients in Z_8. The diagonal case has A=I and b=0.
Exact mathematical passage
```
\omega=e^{i\pi/4}
```

e^{i\phi}Use in source · 1 passage

Search aliases: e^iϕ phi

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Use in source · Read passage · s_main.tex, line 748
Number-preserving gates. Single-qubit R_Z gates commute with widehat N. A two-qubit exchange rotation acts in the ordered basis ket00,ket01,ket10,ket11 as U_XY(θ,ϕ) = 1 · 0 · 0 · 0 0 · cosθ · -ie^-iϕsinθ · 0 0 · -ie^iϕsinθ · cosθ · 0 0 · 0 · 0 · 1 . It rotates ket01 and ket10 while fixing ket00 and ket11, so it preserves exact Hamming weight.
Exact mathematical passage
```
U_{XY}(\theta,\phi)
=
\begin{pmatrix}
1&0&0&0\\
0&\cos\theta&-ie^{-i\phi}\sin\theta&0\\
0&-ie^{i\phi}\sin\theta&\cos\theta&0\\
0&0&0&1
\end{pmatrix}.
```

e^{\varepsilon X}Use in source · 1 passage

Search aliases: e^ε X varepsilon

Link to this entry ·

Use in source · Read passage · g_main.tex, line 383
Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.
Exact mathematical passage
```
e^{\varepsilon X}e^{\varepsilon Y}e^{-\varepsilon X}e^{-\varepsilon Y}
=
e^{\varepsilon^2[X,Y]+O(\varepsilon^3)}.
```

e^{\varepsilon Y}Use in source · 1 passage

Search aliases: e^ε Y varepsilon

Link to this entry ·

Use in source · Read passage · g_main.tex, line 383
Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.
Exact mathematical passage
```
e^{\varepsilon X}e^{\varepsilon Y}e^{-\varepsilon X}e^{-\varepsilon Y}
=
e^{\varepsilon^2[X,Y]+O(\varepsilon^3)}.
```

e^{\varepsilon^2[X,Y]+O(\varepsilon^3)}Use in source · 1 passage

Search aliases: e^ε^2[X,Y]+O(ε^3) varepsilon varepsilon

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Use in source · Read passage · g_main.tex, line 383
Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.
Exact mathematical passage
```
e^{\varepsilon X}e^{\varepsilon Y}e^{-\varepsilon X}e^{-\varepsilon Y}
=
e^{\varepsilon^2[X,Y]+O(\varepsilon^3)}.
```

e^{\pi i/6}Use in source · 1 passage

Search aliases: e^π i/6 pi

Link to this entry ·

Use in source · Read passage · q_main.tex, line 680
U_12={1,5,7,11}, φ(12)=4, ζ=e^π i/6=fracsqrt32+frac i2.
Exact mathematical passage
```
U_{12}=\{1,5,7,11\},
\qquad \varphi(12)=4,
\qquad \zeta=e^{\pi i/6}=\frac{\sqrt3}{2}+\frac i2.
```

e_0Use in source · 14 passages

Search aliases: e_0

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Use in source · Read passage · g_main.tex, line 963
The six-gate ambient circuit uses row 1 as a dirty workspace. It first adds row 2 into row 1, exports the carrier e_1+e_2, changes the carrier to e_0+e_1+e_2, exports it twice, and finally removes the temporary e_2 component. No extra wire is introduced. The workspace is one of the target rows and is restored to its prescribed value.
Exact mathematical passage
```
e_0+e_1+e_2
```

e_1Use in source · 14 passages

Search aliases: e_1

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Use in source · Read passage · g_main.tex, line 963
The six-gate ambient circuit uses row 1 as a dirty workspace. It first adds row 2 into row 1, exports the carrier e_1+e_2, changes the carrier to e_0+e_1+e_2, exports it twice, and finally removes the temporary e_2 component. No extra wire is introduced. The workspace is one of the target rows and is restored to its prescribed value.
Exact mathematical passage
```
e_1+e_2
```

e_2Use in source · 17 passages

Search aliases: e_2

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Use in source · Read passage · g_main.tex, line 963
The six-gate ambient circuit uses row 1 as a dirty workspace. It first adds row 2 into row 1, exports the carrier e_1+e_2, changes the carrier to e_0+e_1+e_2, exports it twice, and finally removes the temporary e_2 component. No extra wire is introduced. The workspace is one of the target rows and is restored to its prescribed value.
Exact mathematical passage
```
e_1+e_2
```

e_{2n-1}Use in source · 1 passage

Search aliases: e_2n-1

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Use in source · Read passage · g_appendix.tex, line 101
The parabolic subgroup used in the census is generated by S_i, CZ_ij, and CX_i→ j. It stabilizes the Z Lagrangian L_Z=Span(e_n,…,e_2n-1).
Exact mathematical passage
```
L_Z=\Span(e_n,\ldots,e_{2n-1}).
```

e_3Use in source · 1 passage

Search aliases: e_3

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Use in source · Read passage · g_main.tex, line 1318
Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.
Exact mathematical passage
```
R_0&=e_0,
& R_1&=e_1+e_0,
& R_2&=e_2,\\
R_3&=e_3+v,
& R_4&=e_4+w,
& R_5&=e_5+w.
```

e_4Use in source · 1 passage

Search aliases: e_4

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Use in source · Read passage · g_main.tex, line 1318
Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.
Exact mathematical passage
```
R_0&=e_0,
& R_1&=e_1+e_0,
& R_2&=e_2,\\
R_3&=e_3+v,
& R_4&=e_4+w,
& R_5&=e_5+w.
```

e_5Use in source · 1 passage

Search aliases: e_5

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Use in source · Read passage · g_main.tex, line 1318
Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.
Exact mathematical passage
```
R_0&=e_0,
& R_1&=e_1+e_0,
& R_2&=e_2,\\
R_3&=e_3+v,
& R_4&=e_4+w,
& R_5&=e_5+w.
```

E_aUse in source · 2 passages

Search aliases: E_a

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Use in source · Read passage · q_main.tex, line 414
E_a=frac1N∑_ℓ=0^N-1ζ_N^aℓA^ℓ
Exact mathematical passage
```
E_a=\frac1N\sum_{\ell=0}^{N-1}\zeta_N^{a\ell}A^\ell
```

e_iUse in source · 5 passages

Search aliases: e_i

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Use in source · Read passage · g_appendix.tex, line 79
Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.
Exact mathematical passage
```
H_i:e_i\leftrightarrow e_{n+i}.
```

e_{i_r}Use in source · 2 passages

Search aliases: e_i_r

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Use in source · Read passage · g_main.tex, line 781
C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
Q_{>r}e_{i_r}e_{j_r}^{T}P_{<r}
```

E_{ij}Use in source · 1 passage

Search aliases: E_ij

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Use in source · Read passage · g_main.tex, line 4133
Under a Gaussian gate alphabet in which each embedded elementary transvection ι(I+aE_ij) is one primitive and direction is the only restriction, the universal six-row catalytic atom embeds into a twelve-quadrature Gaussian atom with restricted length seven and ambient length six.
Exact mathematical passage
```
\iota(I+aE_{ij})
```

e_{j_r}^{T}Use in source · 2 passages

Search aliases: e_j_r^T

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Use in source · Read passage · g_main.tex, line 781
C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
Q_{>r}e_{i_r}e_{j_r}^{T}P_{<r}
```

e_nUse in source · 1 passage

Search aliases: e_n

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Use in source · Read passage · g_appendix.tex, line 101
The parabolic subgroup used in the census is generated by S_i, CZ_ij, and CX_i→ j. It stabilizes the Z Lagrangian L_Z=Span(e_n,…,e_2n-1).
Exact mathematical passage
```
L_Z=\Span(e_n,\ldots,e_{2n-1}).
```

e_{n+i}Use in source · 1 passage

Search aliases: e_n+i

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Use in source · Read passage · g_appendix.tex, line 79
Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.
Exact mathematical passage
```
H_i:e_i\leftrightarrow e_{n+i}.
```

e_pUse in source · 5 passages

Search aliases: e_p

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Use in source · Read passage · g_main.tex, line 1519
inF^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y; [R_p← R_p-ab_xR_x]_x∈ X; R_q← R_…
Exact mathematical passage
```
R_p=e_p-aB,
\qquad
R_q=e_q+B,
\qquad
R_y=e_y+c_ye_p+ac_ye_q
\quad(y\in Y).
```

e_qUse in source · 8 passages

Search aliases: e_q

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Use in source · Read passage · g_main.tex, line 1519
inF^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y; [R_p← R_p-ab_xR_x]_x∈ X; R_q← R_…
Exact mathematical passage
```
R_p=e_p-aB,
\qquad
R_q=e_q+B,
\qquad
R_y=e_y+c_ye_p+ac_ye_q
\quad(y\in Y).
```

E_rUse in source · 2 passages

Search aliases: E_r

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Use in source · Read passage · q_main.tex, line 534
E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).
Exact mathematical passage
```
E_rf(t)=\sum_{m\ge0}a_{r+Lm}t^m,
\qquad
R_rf(z)=z^rE_rf(z^L).
```

e_rUse in source · 5 passages

Search aliases: e_r

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Use in source · Read passage · g_main.tex, line 1425
Suppose instead that an output row, say row r∈{3,4,5}, is targeted twice. The other two outputs and row 1 are targeted once. We claim that row r cannot lie in W between its two targetings. To cancel its unique quotient pivot e_r at the first targeting, the source would already have to contain an e_r component. Before row r is first targeted, the only way another row can acquire such a component is to be targeted using row r as source. That other row is targeted only once and would then retain an…
Exact mathematical passage
```
e_r
```

e_sUse in source · 2 passages

Search aliases: e_s

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Use in source · Read passage · q_visual.tex, line 117
Projection and unitary change are different operations. addcontentslinetocsectionProjection and unitary change are different operations leadA basis change preserves the whole vector. Projection keeps only one orthogonal component. Let e_s=c_12/sqrt48 and e_r=(I-P)x/sqrt102. They are orthonormal, so the normalized input has the exact two-dimensional representation fracxsqrt150=sqrtfrac825e_s+sqrtfrac1725e_r. fig20_projection_geometry0.97This plane is the actual span of the selected signal and its…
Exact mathematical passage
```
e_s=c_{12}/\sqrt{48}
```

e_tUse in source · 2 passages

Search aliases: e_t

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Use in source · Read passage · g_main.tex, line 1293
[One-target source condition] Let W≤ R^n. Suppose row t starts at e_t, is targeted exactly once, and finishes at e_t+w, w∈ W, where at least one coordinate of w is a unit of R. At the moment of its unique targeting, the gate coefficient is a unit and the source row is a unit multiple of w.
Exact mathematical passage
```
e_t
```

E_{t,s}Use in source · 6 passages

Search aliases: E_t,s

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Use in source · Read passage · g_appendix.tex, line 8
Row-transvection convention. Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection τ_t,s(a)=I+aE_t,s performs R_t← R_t+aR_s. In the binary CNOT interpretation, s is the control and t is the target.
Exact mathematical passage
```
\tau_{t,s}(a)=I+aE_{t,s}
```

E_{t_1,s_1}Use in source · 1 passage

Search aliases: E_t_1,s_1

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Use in source · Read passage · g_main.tex, line 2410
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\Phi_\sigma:\Aff^L\longrightarrow\SL_n,
\qquad
(a_1,\ldots,a_L)
\longmapsto
(I+a_LE_{t_L,s_L})\cdots(I+a_1E_{t_1,s_1}).
```

e_{t_j}Use in source · 1 passage

Search aliases: e_t_j

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Use in source · Read passage · g_main.tex, line 2021
[ 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.
Exact mathematical passage
```
\label{g:eq:snapshot-output}
R_{t_j}=e_{t_j}+e_q+w_{s_j}
\qquad(1\le j\le v).
```

E_{t_L,s_L}Use in source · 1 passage

Search aliases: E_t_L,s_L

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Use in source · Read passage · g_main.tex, line 2410
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\Phi_\sigma:\Aff^L\longrightarrow\SL_n,
\qquad
(a_1,\ldots,a_L)
\longmapsto
(I+a_LE_{t_L,s_L})\cdots(I+a_1E_{t_1,s_1}).
```

e_xUse in source · 1 passage

Search aliases: e_x

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Use in source · Read passage · g_main.tex, line 1518
Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y;…
Exact mathematical passage
```
B=\sum_{x\in X}b_xe_x
```

e_{x_i}Use in source · 2 passages

Search aliases: e_x_i

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Use in source · Read passage · g_main.tex, line 2005
Fix integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.
Exact mathematical passage
```
\label{g:eq:prefix-state}
w_s=e_p+\sum_{i=1}^{s}e_{x_i}.
```

e_yUse in source · 1 passage

Search aliases: e_y

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Use in source · Read passage · g_main.tex, line 1519
inF^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y; [R_p← R_p-ab_xR_x]_x∈ X; R_q← R_…
Exact mathematical passage
```
R_p=e_p-aB,
\qquad
R_q=e_q+B,
\qquad
R_y=e_y+c_ye_p+ac_ye_q
\quad(y\in Y).
```

\EgrpUse in source · 5 passages

Search aliases: Egrp

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Use in source · Read passage · g_main.tex, line 933
rected when s<t. The lower-directed gates generate U_n(R). We compare ellDown(A) =min{number of lower-directed transvections producing A} with ellAll(A) =min{number of arbitrary row transvections producing A}, A∈U_n(R). For a field F, the first metric is intrinsic to the unipotent radical and the second is induced from the elementary group Egrp_n(F)=SL_n(F).
Exact mathematical passage
```
\Egrp_n(\F)=\SL_n(\F)
```

\Egrp_nUse in source · 5 passages

Search aliases: Egrp_n Egrp

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Use in source · Read passage · g_main.tex, line 933
rected when s<t. The lower-directed gates generate U_n(R). We compare ellDown(A) =min{number of lower-directed transvections producing A} with ellAll(A) =min{number of arbitrary row transvections producing A}, A∈U_n(R). For a field F, the first metric is intrinsic to the unipotent radical and the second is induced from the elementary group Egrp_n(F)=SL_n(F).
Exact mathematical passage
```
\Egrp_n(\F)=\SL_n(\F)
```

\ellAmb^{(w)}Use in source · 1 passage

Search aliases: ellAmb^(w) ellAmb

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Use in source · Read passage · g_main.tex, line 359
Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.
Exact mathematical passage
```
D_n^{(w)}
=
\max_{U}
\left(
\ellRes(U)-\ellAmb^{(w)}(U)
\right),
```

\ellWeightUse in source · 5 passages

Search aliases: ellWeight

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Use in source · Read passage · g_main.tex, line 1047
en gates and weighted penalties] For every field and every n≥7, there is a connected target A_n∈U_n(F) such that ellBudget0(A_n)-ellBudget2(A_n) ≥ lfloorfracn-52rfloor, where ellBudgetq permits at most q upper-directed gates. More generally, if lower gates have cost 1 and upper gates have cost λ>0, then ellDown(A_n)-ellWeightλ(A_n) ≥ lfloorfracn-12rfloor-2λ. Thus no fixed upper-gate budget of two, and no fixed finite upper-gate penalty, suppresses catalytic escape asymptotically.
Exact mathematical passage
```
\ellDown(A_n)-\ellWeight{\lambda}(A_n)
\ge
\left\lfloor\frac{n-1}{2}\right\rfloor-2\lambda.
```

\mathrm{encode}Use in source · 3 passages

Search aliases: encode mathrm

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Use in source · Read passage · c_lean.tex, line 211
pageheadReading the formal model literallyQUANTIFIERS AND ENCODING CARRY THE HYPOTHESES The generic cut theory uses a logical-input type X, a wire type W with decidable equality, and a boundary structure. Its essential fields are an encoding encode:X→Bool→(W→Bool), a target wire, and the proof encode(x,b)(target)=b. This target equation separates the two target branches for each fixed input. It excludes an encoding that silently ignores its supplied target bit. The source proves codeBoundary.tar…
Exact mathematical passage
```
\mathrm{encode}:X\to\mathrm{Bool}\to(W\to\mathrm{Bool})
```

\mathrm{entangled}Use in source · 1 passage

Search aliases: entangled mathrm

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Use in source · Read passage · g_main.tex, line 4698
This is naturally expressed as Real_product(P_6)=∅, Real_entangled(P_6)≠∅. It is an existence defect, not a gate-count defect.
Exact mathematical passage
```
\operatorname{Real}_{\mathrm{product}}(P_6)=\varnothing,
\qquad
\operatorname{Real}_{\mathrm{entangled}}(P_6)\ne\varnothing.
```

\mathrm{err}Use in source · 1 passage

Search aliases: err mathrm

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Use in source · Read passage · g_main.tex, line 4450
For stochastic error rates p_g, a first-order weighted cost is C_err(W)=∑_g∈ W-log(1-p_g). This additive approximation can be inserted directly into Dijkstra search. It does not capture coherent or correlated errors, but it produces a declared and reproducible metric.
Exact mathematical passage
```
C_{\mathrm{err}}(W)=\sum_{g\in W}-\log(1-p_g).
```

\mathrm{exist}Use in source · 1 passage

Search aliases: exist mathrm

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Use in source · Read passage · g_main.tex, line 148
Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.
Exact mathematical passage
```
\tau_{\mathrm{metric}}&=\min\{n:\exists U,\ 0<\Delta(U)<\infty\},\\
\tau_{\mathrm{exist}}&=\min\{n:\exists U,\ \ellRes(U)=\infty,\ \ellAmb(U)<\infty\},\\
\tau_{q}&=\min\{n:\exists U,\ \ell_{\le q}(U)<\ell_{\le0}(U)\},
```

\ExpUse in source · 2 passages

Search aliases: Exp

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Use in source · Read passage · g_main.tex, line 2552
[Finite-field defect density] For 1≤ d≤ N, Prob_A∈U_n(F_q)[DeltaDef(A)≥ d] ≤ min{1,M^N-dq^-d}. In particular, for fixed n, Prob[DeltaDef(A)>0]=O_n(q^-1), Exp[DeltaDef(A)]=O_n(q^-1) as q→∞.
Exact mathematical passage
```
\Prob[\DeltaDef(A)>0]=O_n(q^{-1}),
\qquad
\Exp[\DeltaDef(A)]=O_n(q^{-1})
```

\expUse in source · 3 passages

Search aliases: exp

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Use in source · Read passage · q_lean.tex, line 33
The typed mathematical target. The complex matrices act on vectors indexed by Fin(12). The Fourier convention is F_nk=frac1sqrt12exp (frac2π i nk12), n,k∈{0,…,11}. The adjoint F^* therefore carries the negative exponential sign. The selector D is diagonal with value one precisely at the residues U={1,5,7,11}. The transported operator is P=FDF^*.
Exact mathematical passage
```
 F_{nk}=\frac{1}{\sqrt{12}}\exp\!\left(\frac{2\pi i nk}{12}\right),
 \qquad n,k\in\{0,\ldots,11\}.
```

\mathbb FUse in source · 1 passage

Search aliases: F mathbb

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Use in source · Read passage · c_lean.tex, line 61
All arithmetic is over F_2, implemented using Boolean XOR and AND. Each gate is its own inverse. Circuit reversal, restoration, and every cut are inherited from the unchanged Block 01 source.
Exact mathematical passage
```
\mathbb F_2
```

\mathrm FUse in source · 1 passage

Search aliases: F mathrm

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Use in source · Read passage · s_main.tex, line 1445
Bosonic and fermionic analogues. For N fermions in m modes, D_ F(m,N)=binom mN. For N bosons in m modes, D_ B(m,N)=binomm+N-1N. The same binomial analysis gives:
Exact mathematical passage
```
D_{\mathrm F}(m,N)=\binom mN.
```

F'Use in source · 2 passages

Search aliases: F'

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Use in source · Read passage · s_main.tex, line 347
Define F(s)=e^-i(t-s)HPe^-isHP. Then F(0)=e^-itHP and F(t)=Pe^-itHP. Differentiation gives F'(s)=ie^-i(t-s)H[H,P]e^-isHP. Therefore (I-P)e^-itHP =-(I-P)∫_0^tF'(s) ds. The unitary factors and projectors have norm at most one, so the integral is bounded by |t| [H,P] .
Exact mathematical passage
```
F'(s)=ie^{-i(t-s)H}[H,P]e^{-isH}P.
```

f^{(0)}Use in source · 1 passage

Search aliases: f^(0)

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Use in source · Read passage · g_main.tex, line 3816
Higher-order Möbius coefficients. The ANF coefficients are the Möbius transform of the truth table. Projectors onto degree layers give an exact packet decomposition f=f^(0)+f^(1)+f^(2)+⋯. A restricted affine circuit remains in degrees zero and one. The ambient circuit can populate higher packets temporarily. A successful lower bound must charge the useful flow through those packets, not merely their instantaneous dimension.
Exact mathematical passage
```
f=f^{(0)}+f^{(1)}+f^{(2)}+\cdots.
```

f^{(1)}Use in source · 1 passage

Search aliases: f^(1)

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Use in source · Read passage · g_main.tex, line 3816
Higher-order Möbius coefficients. The ANF coefficients are the Möbius transform of the truth table. Projectors onto degree layers give an exact packet decomposition f=f^(0)+f^(1)+f^(2)+⋯. A restricted affine circuit remains in degrees zero and one. The ambient circuit can populate higher packets temporarily. A successful lower bound must charge the useful flow through those packets, not merely their instantaneous dimension.
Exact mathematical passage
```
f=f^{(0)}+f^{(1)}+f^{(2)}+\cdots.
```

f^{(2)}Use in source · 1 passage

Search aliases: f^(2)

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Use in source · Read passage · g_main.tex, line 3816
Higher-order Möbius coefficients. The ANF coefficients are the Möbius transform of the truth table. Projectors onto degree layers give an exact packet decomposition f=f^(0)+f^(1)+f^(2)+⋯. A restricted affine circuit remains in degrees zero and one. The ambient circuit can populate higher packets temporarily. A successful lower bound must charge the useful flow through those packets, not merely their instantaneous dimension.
Exact mathematical passage
```
f=f^{(0)}+f^{(1)}+f^{(2)}+\cdots.
```

F^*Use in source · 6 passages

Search aliases: F^*

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Use in source · Read passage · q_lean.tex, line 37
The typed mathematical target. The complex matrices act on vectors indexed by Fin(12). The Fourier convention is F_nk=frac1sqrt12exp (frac2π i nk12), n,k∈{0,…,11}. The adjoint F^* therefore carries the negative exponential sign. The selector D is diagonal with value one precisely at the residues U={1,5,7,11}. The transported operator is P=FDF^*.
Exact mathematical passage
```
F^*
```

f^{-1}Use in source · 2 passages

Search aliases: f^-1

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Use in source · Read passage · c_main.tex, line 211
Exact evaluation-span certificates. For an integer d≥0, form the evaluation vector v_d(x)=(x^S)_S⊆[n], |S|≤ d, and write U=f^-1(0).
Exact mathematical passage
```
U=f^{-1}(0)
```

\F^mUse in source · 2 passages

Search aliases: F^m F

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Use in source · Read passage · c_main.tex, line 415
A compact whole function. Let m≥2, let a=(1,…,1)∈F^m, and introduce one logical bit x_v for each v∈ V=F^m∖{0,a}. There are n=2^m-2 such bits. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^mL_j(x).
Exact mathematical passage
```
a=(1,\ldots,1)\in\F^m
```

\F^nUse in source · 1 passage

Search aliases: F^n F

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Use in source · Read passage · g_main.tex, line 1231
For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0⊂ V_1⊂⋯⊂ V_n-1. An upper-directed gate leaves this fiber of the flag map. A strict defect means tha…
Exact mathematical passage
```
\F^n
```

\F^\timesUse in source · 1 passage

Search aliases: F^× F times

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Use in source · Read passage · g_main.tex, line 1508
Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y;…
Exact mathematical passage
```
a\in\F^\times
```

F^\daggerUse in source · 28 passages

Search aliases: F^† dagger

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Use in source · Read passage · q_lean.tex, line 1031
The later source explicitly supplies FDF^†=P, both normalized state-action identities, the conventional exponential formula for F, identical initial probabilities, orthogonality, exact selected probability one versus zero, histogram nonreconstruction, and projector noninjectivity. It therefore closes the representation-transport obligations that the earlier package leaves outside its formal boundary.
Exact mathematical passage
```
FDF^\dagger=P
```

F_{12}Use in source · 2 passages

Search aliases: F_12

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Use in source · Read passage · q_main.tex, line 752
With D=diag(1_a∈ U_12) and F=F_12,
Exact mathematical passage
```
F=F_{12}
```

\F_{16}Use in source · 1 passage

Search aliases: F_16 F

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Use in source · Read passage · c_visual.tex, line 701
The field is F_16=F_2[θ]/(θ^4+θ+1). The checker stores field elements as four-bit coefficient vectors: θ is integer 2 and θ^2 is integer 4. These integers encode field elements; they are not ordinary integer multiplication. Exact field multiplication gives the reported mismatch.
Exact mathematical passage
```
\F_{16}=\F_2[\theta]/(\theta^4+\theta+1)
```

\F_2Use in source · 38 passages

Search aliases: F_2 F

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Use in source · Read passage · c_visual.tex, line 18
All polynomial arithmetic in the Boolean state model is over F_2, with x_i^2=x_i. Every function has a unique multilinear algebraic normal form. Its degree is the largest support size with a nonzero coefficient; the zero function has mathematical degree -1 in the Python helpers. A formal degree bound uses a natural number, so zero has a degree-zero certificate.
Exact mathematical passage
```
\F_2
```

\mathbb F_2Use in source · 1 passage

Search aliases: F_2 mathbb

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Use in source · Read passage · c_lean.tex, line 61
All arithmetic is over F_2, implemented using Boolean XOR and AND. Each gate is its own inverse. Circuit reversal, restoration, and every cut are inherited from the unchanged Block 01 source.
Exact mathematical passage
```
\mathbb F_2
```

\F_2^{2n}Use in source · 3 passages

Search aliases: F_2^2n F

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Use in source · Read passage · g_main.tex, line 3387
Binary symplectic representation. Modulo global phase and Pauli translations, an n-qubit Clifford operation acts on Pauli exponent vectors v=(x_0,…,x_n-1mid z_0,…,z_n-1)^T∈F_2^2n through a symplectic matrix M∈Sp(2n,2). The elementary gates used here act as follows: H_i · : (x_i,z_i)↦(z_i,x_i),
Exact mathematical passage
```
v=(x_0,\ldots,x_{n-1}\mid z_0,\ldots,z_{n-1})^T\in\F_2^{2n}
```

\F_2^3Use in source · 1 passage

Search aliases: F_2^3 F

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Use in source · Read passage · g_main.tex, line 1766
Over F_2, write x=e_0, p=e_1, q=e_2. The canonical ambient word uses the six nonzero vectors x, p, q, p+q, p+q+x, p+x inside Span(x,p,q)≅F_2^3. The only missing nonzero vector is x+q.
Exact mathematical passage
```
\Span(x,p,q)\cong\F_2^3
```

\F_2^nUse in source · 5 passages

Search aliases: F_2^n F

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Use in source · Read passage · g_appendix.tex, line 150
Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric group.
Exact mathematical passage
```
x\in\F_2^n
```

F_3Use in source · 3 passages

Search aliases: F_3

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Use in source · Read passage · c_visual.tex, line 234
guidepageOne safe packet need not cover every accepting point For k=3 consider F_3(a,b,z_1,z_2,z_3)=ab OR z_1z_2z_3. The packet ab is safe and has degree two. It detects every teal point below. It misses three additional positives, which require the other packet z_1z_2z_3.
Exact mathematical passage
```
 F_3(a,b,z_1,z_2,z_3)=ab\ \mathrm{OR}\ z_1z_2z_3.
```

\F_3Use in source · 13 passages

Search aliases: F_3 F

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Use in source · Read passage · g_main.tex, line 935
The equality ellDown(A)=ellAll(A) is tempting because the target and the obvious Gaussian-elimination paths are lower unitriangular. It is false. The failure first occurs at six wires over F_2 and F_3, and it is caused by a reusable temporary carrier rather than by a better ordering of the same lower gates.
Exact mathematical passage
```
\F_3
```

\F_5Use in source · 3 passages

Search aliases: F_5 F

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Use in source · Read passage · g_main.tex, line 1730
The n=5 ternary computation uses radius-three and radius-four balls, checks 38415205 right-coset pairs, and recovers all 59049 targets. It finds no defect. The field-independent [source reference] supplies a defect at n=6, proving [source reference]. Separate exact searches over F_3 and F_5 verify the distances of the canonical atom directly. The F_5 calculation is an audit of the atom, not a complete threshold computation below dimension six.
Exact mathematical passage
```
\F_5
```

f_aUse in source · 3 passages

Search aliases: f_a

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Use in source · Read passage · q_main.tex, line 411
Let A be the cyclic shift (Ax)(n)=x(n-1). A Fourier column f_a(n)=N^-1/2ζ_N^na satisfies Af_a=ζ_N^-af_a. Therefore
Exact mathematical passage
```
f_a(n)=N^{-1/2}\zeta_N^{na}
```

f_iUse in source · 1 passage

Search aliases: f_i

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Use in source · Read passage · g_main.tex, line 3712
Algebraic normal form. Every Boolean coordinate function has a unique algebraic normal form f_i(x) = ⊕_S⊆[n]a_i,S∏_j∈ Sx_j. Affine maps have degree at most one. A Toffoli gate targeting t with controls a,b performs x_t⟼ x_t+x_ax_b, which can raise the degree of later coordinates. The final affine target requires every higher-degree coefficient to cancel.
Exact mathematical passage
```
f_i(x)
=
\bigoplus_{S\subseteq[n]}a_{i,S}\prod_{j\in S}x_j.
```

F_kUse in source · 14 passages

Search aliases: F_k

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Use in source · Read passage · c_main.tex, line 274
Calibration families and exact small oracles. claimC03 and C04 An unbalanced disjunction. For k≥2 and n=k+2, define B=z_1⋯ z_k, F_k(a,b,z)=(ab)lor B=ab+B+abB, G_k=F_k+a. This is a deliberately easy family with two differently sized accepting regions. It separates the cost of finding one safe packet from the degree needed for complete coverage.
Exact mathematical passage
```
 B=z_1\cdots z_k,\qquad
 F_k(a,b,z)=(ab)\lor B=ab+B+abB,\qquad G_k=F_k+a.
```

F_NUse in source · 4 passages

Search aliases: F_N

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Use in source · Read passage · q_main.tex, line 344
P_d^(N)=F_ND_dF_N^†.
Exact mathematical passage
```
P_d^{(N)}=F_ND_dF_N^\dagger.
```

F_N^\daggerUse in source · 3 passages

Search aliases: F_N^† dagger

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Use in source · Read passage · q_main.tex, line 344
P_d^(N)=F_ND_dF_N^†.
Exact mathematical passage
```
P_d^{(N)}=F_ND_dF_N^\dagger.
```

F_{na}Use in source · 1 passage

Search aliases: F_na

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Use in source · Read passage · q_visual.tex, line 89
A diagonal selector becomes a rational filter. addcontentslinetocsectionA diagonal selector becomes a rational filter leadThe filter is diagonal in one basis and a rational convolution in the other. Let F_na=ζ^na/sqrt12 and D_aa=1_a∈ U_12. With the positive-sign unitary matrix F, the negative-sign DFT satisfies F^† x=widehat x/sqrt12. fig05_transported_projector1.0Left: the frequency selector D. Right: its transported sample-basis matrix P. The right colorbar displays its five possible rational …
Exact mathematical passage
```
F_{na}=\zeta^{na}/\sqrt{12}
```

F_{nk}Use in source · 1 passage

Search aliases: F_nk

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Use in source · Read passage · q_lean.tex, line 33
The typed mathematical target. The complex matrices act on vectors indexed by Fin(12). The Fourier convention is F_nk=frac1sqrt12exp (frac2π i nk12), n,k∈{0,…,11}. The adjoint F^* therefore carries the negative exponential sign. The selector D is diagonal with value one precisely at the residues U={1,5,7,11}. The transported operator is P=FDF^*.
Exact mathematical passage
```
 F_{nk}=\frac{1}{\sqrt{12}}\exp\!\left(\frac{2\pi i nk}{12}\right),
 \qquad n,k\in\{0,\ldots,11\}.
```

F_qUse in source · 5 passages

Search aliases: F_q

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Use in source · Read passage · q_main.tex, line 129
(F_q)_na=q^-1/2ζ_q^na.
Exact mathematical passage
```
(F_q)_{na}=q^{-1/2}\zeta_q^{na}.
```

\F_qUse in source · 9 passages

Search aliases: F_q F

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Use in source · Read passage · g_main.tex, line 595
Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estimate can be sharpened by orbit counting, stabilizers, and the geometry of word-map fibers.
Exact mathematical passage
```
\F_q
```

F_q^\daggerUse in source · 6 passages

Search aliases: F_q^† dagger

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Use in source · Read passage · q_main.tex, line 134
Thus F_q synthesizes a signal from normalized frequency coordinates, while F_q^† x=widehatx/sqrt q analyzes it. This convention explains all signs and all square-root factors below. The negative-sign transform and the positive-sign synthesis are adjoints of one another.
Exact mathematical passage
```
F_q^\dagger x=\widehat{x}/\sqrt q
```

f_\lambdaUse in source · 4 passages

Search aliases: f_λ lambda

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Use in source · Read passage · s_main.tex, line 1094
Exact commutant record. Let u_λ=dim U_λ, f_λ=dim V_λ. Schur's lemma gives the following normal form.
Exact mathematical passage
```
u_\lambda=\dim\mathcal U_\lambda,
\qquad
f_\lambda=\dim\mathcal V_\lambda.
```

\operatorname{Fin}Use in source · 3 passages

Search aliases: Fin operatorname

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Use in source · Read passage · q_lean.tex, line 31
The typed mathematical target. The complex matrices act on vectors indexed by Fin(12). The Fourier convention is F_nk=frac1sqrt12exp (frac2π i nk12), n,k∈{0,…,11}. The adjoint F^* therefore carries the negative exponential sign. The selector D is diagonal with value one precisely at the residues U={1,5,7,11}. The transported operator is P=FDF^*.
Exact mathematical passage
```
\operatorname{Fin}(12)
```

\mathrm{final}Use in source · 1 passage

Search aliases: final mathrm

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Use in source · Read passage · g_main.tex, line 791
endpmatrix. A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
P_{\mathrm{final}}=I_s,
\qquad
Q_{\mathrm{final}}=I_t.
```

\mathrm{full}Use in source · 1 passage

Search aliases: full mathrm

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Use in source · Read passage · s_main.tex, line 231
State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectorization, and basis compilation determine actual runtime. The asymptotic distinction is exact when D is polynomial and 2^n is exponential.
Exact mathematical passage
```
C_{\mathrm{full}}=O(dn2^n).
```

\mathcal GUse in source · 12 passages

Search aliases: G mathcal

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Use in source · Read passage · g_main.tex, line 493
Why eigenvalues alone are insufficient. If A_S=∑_λλΠ_λ, then G_S,g(z) = ∑_λ frac⟨δ_g,Π_λδ_e⟩1-λ z. The eigenvalues and multiplicities do not determine the endpoint coefficients ⟨δ_g,Π_λδ_e⟩. Two Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.
Exact mathematical passage
```
\mathcal G_{S,g}(z)
=
\sum_\lambda
\frac{\langle\delta_g,\Pi_\lambda\delta_e\rangle}{1-\lambda z}.
```

gUse in source · 78 passages

Search aliases: g

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Use in source · Read passage · c_lean.tex, line 70
β^*(f) · Least upper bound on β(fx g) over all degree-one g.
Exact mathematical passage
```
\beta(f\x g)
```

G'Use in source · 2 passages

Search aliases: G'

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Use in source · Read passage · g_main.tex, line 840
[Retraction inheritance] Suppose a smaller nested pair H≤ G embeds into H'≤ G' by homomorphisms preserving the inclusion, the ambient embedding does not increase gate cost, and there is a homomorphic retraction π:H'→ H onto the embedded copy. Require ℓ_H(π(s))≤ c(s) for every restricted generator s of H'. Then every defect in H≤ G gives a defect at least as large in H'≤ G'. The proof projects a restricted word in the larger system and embeds the shorter ambient word from the smaller system.
Exact mathematical passage
```
H'\le G'
```

g^{-1}Use in source · 14 passages

Search aliases: g^-1

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Use in source · Read passage · g_main.tex, line 270
The basic conjugation inequality. Let g∈ G and k∈ H satisfy gkg^-1∈ H. Then ℓ_G(gkg^-1) ≤ 2ℓ_G(g)+ℓ_H(k). Define the intrinsic conjugation stretch Stretch_g(k) = ℓ_H(gkg^-1)-ℓ_H(k).
Exact mathematical passage
```
gkg^{-1}\in H
```

G^LUse in source · 1 passage

Search aliases: G^L

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Use in source · Read passage · g_main.tex, line 2471
Every irreducible component of the Cartesian power X^× L has dimension at most rL. Multiplication G^L→ G is a morphism, so the closure of the image of each component has dimension at most rL. There are finitely many components, and intersecting the resulting finite union with V cannot increase dimension. Since the identity belongs to X, products of length at most L are already contained in X^(L).
Exact mathematical passage
```
G^L\to G
```

\mathcal G_{*,C}Use in source · 2 passages

Search aliases: G_*,C mathcal

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Use in source · Read passage · g_main.tex, line 1894
For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.
Exact mathematical passage
```
\mathcal G_{\downarrow,C}(z)
&=
543z^7+25080z^8+510703z^9+10824940z^{10}+\cdots,\\
\mathcal G_{*,C}(z)
&=
2z^6+776z^7+39454z^8+\cdots.
```

G_2Use in source · 5 passages

Search aliases: G_2

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Use in source · Read passage · g_main.tex, line 873
[Long-root catalytic gain] There exist gate alphabets in types B, C, or G_2 for which a clean one-catalyst multiplicative distortion exceeds the type-A snapshot limit of 3/2. No lower-bound certificate is presently supplied. The conjecture identifies a concrete reason to test nonsimply laced root systems rather than merely repeating type A computations.
Exact mathematical passage
```
G_2
```

G_{\mathrm{conn}}Use in source · 1 passage

Search aliases: G_conn mathrm

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Use in source · Read passage · g_main.tex, line 4262
Architecture and routing. On a line or grid, Gaussian and matchgate circuits may be dominated by routing. Let G_conn be the hardware graph. A mode permutation π can be treated as a catalyst if it is opened, many interactions are executed in the reordered frame, and π^-1 restores the logical labeling. The benefit is routing saved across consumers - 2 cost(π). This is the same amortization law as the linear snapshot family, now applied to mode geometry.
Exact mathematical passage
```
G_{\mathrm{conn}}
```

\mathcal G_{\downarrow,C}Use in source · 2 passages

Search aliases: G_downarrow,C mathcal downarrow

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Use in source · Read passage · g_main.tex, line 1894
For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.
Exact mathematical passage
```
\mathcal G_{\downarrow,C}(z)
&=
543z^7+25080z^8+510703z^9+10824940z^{10}+\cdots,\\
\mathcal G_{*,C}(z)
&=
2z^6+776z^7+39454z^8+\cdots.
```

G_{\mathrm{Gauss}}Use in source · 1 passage

Search aliases: G_Gauss mathrm

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Use in source · Read passage · g_main.tex, line 4236
Temporary non-Gaussianity. Suppose a non-Gaussian primitive g is introduced and later removed: U=g h_m⋯ h_1 g^-1, U∈ G_Gauss. If g h_i g^-1 generates several useful Gaussian effects, the excursion may be catalytic. However, many standard non-Gaussianity monotones vanish only on states rather than unitaries, and cancellation at the endpoint complicates direct lower bounds.
Exact mathematical passage
```
U=g h_m\cdots h_1 g^{-1},
\qquad U\in G_{\mathrm{Gauss}}.
```

G_jUse in source · 10 passages

Search aliases: G_j

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Use in source · Read passage · s_appendix.tex, line 18
Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].
Exact mathematical passage
```
V_j=PG_jP
```

G_kUse in source · 12 passages

Search aliases: G_k

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Use in source · Read passage · c_main.tex, line 274
Calibration families and exact small oracles. claimC03 and C04 An unbalanced disjunction. For k≥2 and n=k+2, define B=z_1⋯ z_k, F_k(a,b,z)=(ab)lor B=ab+B+abB, G_k=F_k+a. This is a deliberately easy family with two differently sized accepting regions. It separates the cost of finding one safe packet from the degree needed for complete coverage.
Exact mathematical passage
```
 B=z_1\cdots z_k,\qquad
 F_k(a,b,z)=(ab)\lor B=ab+B+abB,\qquad G_k=F_k+a.
```

G_{k+1}Use in source · 2 passages

Search aliases: G_k+1

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Use in source · Read passage · s_appendix.tex, line 19
Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].
Exact mathematical passage
```
G_m\cdots G_1P-V_m\cdots V_1P
=
\sum_{k=1}^m
G_m\cdots G_{k+1}(G_k-V_k)V_{k-1}\cdots V_1P.
```

\mathcal G_{S,g}Use in source · 9 passages

Search aliases: G_S,g mathcal

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Use in source · Read passage · g_main.tex, line 493
Why eigenvalues alone are insufficient. If A_S=∑_λλΠ_λ, then G_S,g(z) = ∑_λ frac⟨δ_g,Π_λδ_e⟩1-λ z. The eigenvalues and multiplicities do not determine the endpoint coefficients ⟨δ_g,Π_λδ_e⟩. Two Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.
Exact mathematical passage
```
\mathcal G_{S,g}(z)
=
\sum_\lambda
\frac{\langle\delta_g,\Pi_\lambda\delta_e\rangle}{1-\lambda z}.
```

\mathcal G_{S_A,g}Use in source · 1 passage

Search aliases: G_S_A,g mathcal

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Use in source · Read passage · g_main.tex, line 5104
[First-arrival template] If word multiplicities are nonnegative, then d_S(e,g)=ord_z=0 G_S,g(z). For restricted and ambient sets, Δ(g)=ord_0 G_S_R,g-ord_0 G_S_A,g.
Exact mathematical passage
```
\Delta(g)=\ord_0\mathcal G_{S_R,g}-\ord_0\mathcal G_{S_A,g}.
```

\mathcal G_{S_R,g}Use in source · 1 passage

Search aliases: G_S_R,g mathcal

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Use in source · Read passage · g_main.tex, line 5104
[First-arrival template] If word multiplicities are nonnegative, then d_S(e,g)=ord_z=0 G_S,g(z). For restricted and ambient sets, Δ(g)=ord_0 G_S_R,g-ord_0 G_S_A,g.
Exact mathematical passage
```
\Delta(g)=\ord_0\mathcal G_{S_R,g}-\ord_0\mathcal G_{S_A,g}.
```

g_tUse in source · 2 passages

Search aliases: g_t

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Use in source · Read passage · g_main.tex, line 4439
Depth, count, and space-time volume. Let g_t be the set of gates scheduled in layer t. Common scalar costs are ∑_t |g_t|, #{t:g_t≠∅}, ∑_t #{active qubits in layer t}. A catalyst may reduce count but increase depth, or vice versa. Space-time volume is often the more stable hardware quantity.
Exact mathematical passage
```
g_t
```

G_{t+1}Use in source · 1 passage

Search aliases: G_t+1

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Use in source · Read passage · s_main.tex, line 684
Fix the projectors chosen during the particular run. Put d_t=(I-P_t)G_tϕ_t-1. Subtraction gives the exact recurrence ψ_t-ϕ_t=G_t(ψ_t-1-ϕ_t-1)+d_t. Since ψ_0-ϕ_0=0, iteration yields ψ_m-ϕ_m=∑_t=1^mG_m⋯ G_t+1d_t. Every unitary product preserves norm, so the triangle inequality proves the state estimate. Also ψ_m =1 and ϕ_m ≤1 because each projector is contractive. Expanding the difference of the two quadratic forms and using M ≤1 gives the factor two. This proof applies to the chosen run even when…
Exact mathematical passage
```
\psi_m-\phi_m=\sum_{t=1}^mG_m\cdots G_{t+1}d_t.
```

G_{t;\beta\alpha}Use in source · 2 passages

Search aliases: G_t;βα beta alpha

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Use in source · Read passage · s_main.tex, line 540
Sparse packet-transfer graphs. Let I=∑_α∈ΛP_α be an orthogonal packet decomposition. Write the gate blocks as G_t;βα=P_β G_tP_α. Starting from an active packet set S_0, define recursively S_t = { β:∃α∈ S_t-1 with G_t;βα≠0 }.
Exact mathematical passage
```
G_{t;\beta\alpha}=P_\beta G_tP_\alpha.
```

\mathrm{Gauss}Use in source · 1 passage

Search aliases: Gauss mathrm

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Use in source · Read passage · g_main.tex, line 4236
Temporary non-Gaussianity. Suppose a non-Gaussian primitive g is introduced and later removed: U=g h_m⋯ h_1 g^-1, U∈ G_Gauss. If g h_i g^-1 generates several useful Gaussian effects, the excursion may be catalytic. However, many standard non-Gaussianity monotones vanish only on states rather than unitaries, and cancellation at the endpoint complicates direct lower bounds.
Exact mathematical passage
```
U=g h_m\cdots h_1 g^{-1},
\qquad U\in G_{\mathrm{Gauss}}.
```

\gcdUse in source · 16 passages

Search aliases: gcd

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Use in source · Read passage · q_main.tex, line 144
frac qgcd(a,q).
Exact mathematical passage
```
\frac q{\gcd(a,q)}.
```

\mathrm{GF}Use in source · 5 passages

Search aliases: GF mathrm

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Use in source · Read passage · c_main.tex, line 548
The finite scalar controls enumerate 256 linear maps from a two-point witness algebra to GF(16), 65,536 maps for four points, and 256 maps from an eight-point algebra to F. The respective counts of unital algebra homomorphisms are two, four, and eight, matching the evaluation classification. The all-size result rests on the proof above.
Exact mathematical passage
```
\mathrm{GF}(16)
```

\GLUse in source · 13 passages

Search aliases: GL

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Use in source · Read passage · g_appendix.tex, line 163
Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).
Exact mathematical passage
```
A\in\GL(n,\R)
```

\GL_6Use in source · 2 passages

Search aliases: GL_6 GL

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Use in source · Read passage · g_main.tex, line 1634
Let B_4={g∈GL_6(F_2):ellAll(g)≤4}. Exact breadth-first search gives |B_4|=110626. Group B_4 by right cosets of U_6(F_2). If a,b lie in the same right coset, then b^-1a∈U_6(F_2), and ellAll(b^-1a)≤ellAll(b)+ellAll(a)≤8. The implementation uses a canonical right-coset key obtained by reducing matrix columns from right to left. Right multiplication by a lower-unitriangular matrix changes column j only by a combination of columns with larger index, so this reduction is constant on right cosets and s…
Exact mathematical passage
```
B_4=\{g\in\GL_6(\F_2):\ellAll(g)\le4\}.
```

\GL_nUse in source · 5 passages

Search aliases: GL_n GL

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Use in source · Read passage · g_main.tex, line 910
A CNOT circuit on n labeled wires represents an element of GL_n(F_2)=SL_n(F_2), and minimizing its gate count is the word-metric problem for transvection generators. This formulation supports asymptotically efficient synthesis, architecture-aware elimination, computable lower bounds, and exact small-width optimization [source reference]. Recent work has also produced stronger explicit lower-bound families and established NP-completeness of exact all-to-all CNOT distance [source reference]. The p…
Exact mathematical passage
```
\GL_n(\F_2)=\SL_n(\F_2)
```

\mathcal HUse in source · 2 passages

Search aliases: H mathcal

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Use in source · Read passage · g_main.tex, line 4147
Bosonic gate hierarchies. Natural nested models include triangular shears ⊂ all Gaussian unitaries ⊂ Gaussian plus a non-Gaussian primitive ⊂ U( H). Typical primitive resources are: item SUM or controlled displacement gates, item beam splitters, item single-mode and two-mode squeezers, item Fourier transforms, item cubic phase gates, item photon counting or non-Gaussian state injection. Each inclusion supports a different defect. A target Gaussian transformation might be shortened by leaving a t…
Exact mathematical passage
```
\text{triangular shears}
\subset
\text{all Gaussian unitaries}
\subset
\text{Gaussian plus a non-Gaussian primitive}
\subset
U(\mathcal H).
```

H'Use in source · 4 passages

Search aliases: H'

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Use in source · Read passage · g_main.tex, line 840
[Retraction inheritance] Suppose a smaller nested pair H≤ G embeds into H'≤ G' by homomorphisms preserving the inclusion, the ambient embedding does not increase gate cost, and there is a homomorphic retraction π:H'→ H onto the embedded copy. Require ℓ_H(π(s))≤ c(s) for every restricted generator s of H'. Then every defect in H≤ G gives a defect at least as large in H'≤ G'. The proof projects a restricted word in the larger system and embeds the shorter ambient word from the smaller system.
Exact mathematical passage
```
H'\le G'
```

h^{-1}Use in source · 2 passages

Search aliases: h^-1

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Use in source · Read passage · g_main.tex, line 379
Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.
Exact mathematical passage
```
[g,h]=ghg^{-1}h^{-1}.
```

H_0Use in source · 6 passages

Search aliases: H_0

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Use in source · Read passage · g_appendix.tex, line 121
1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.
Exact mathematical passage
```
H_1,\ \CX_{1\to0},\ \CX_{0\to1},\ H_0,\ S_1,\ S_0,
```

H_1Use in source · 6 passages

Search aliases: H_1

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Use in source · Read passage · g_appendix.tex, line 121
1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.
Exact mathematical passage
```
H_1,\ \CX_{1\to0},\ \CX_{0\to1},\ H_0,\ S_1,\ S_0,
```

H_2Use in source · 5 passages

Search aliases: H_2

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Use in source · Read passage · s_main.tex, line 1390
t dimension is polynomial in n if r(n)=O(1); item the packet dimension is superpolynomial whenever r(n)→∞; item if r(n)=O(log n), then D_n,k=2^O((log n)^2), so the explicit packet simulation is quasipolynomial; item if r(n)=O(n^α) for a fixed 0<α<1, then D_n,k=2^O(n^αlog n), which is subexponential; item if k(n)/n→ p∈(0,1), then log_2 D_n,k = nH_2(k(n)/n)+O(log n) = nH_2(p)+o(n), where H_2 is binary entropy, and the packet remains exponential.
Exact mathematical passage
```
\log_2 D_{n,k}
=
nH_2(k(n)/n)+O(\log n)
=
nH_2(p)+o(n),
```

H_7Use in source · 3 passages

Search aliases: H_7

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Use in source · Read passage · g_main.tex, line 2090
[Exact binary cases in dimensions seven through nine] Over F_2, let H_7=A(2;0,1,2), H_8=A(3;0,1,2), H_9=A(3;0,1,2,3). Then (ellDown(H_7),ellAll(H_7))=(8,7), (ellDown(H_8),ellAll(H_8))=(9,8), and (ellDown(H_9),ellAll(H_9))=(11,9).
Exact mathematical passage
```
H_7=A(2;0,1,2),
\qquad
H_8=A(3;0,1,2),
\qquad
H_9=A(3;0,1,2,3).
```

H_8Use in source · 3 passages

Search aliases: H_8

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Use in source · Read passage · g_main.tex, line 2090
[Exact binary cases in dimensions seven through nine] Over F_2, let H_7=A(2;0,1,2), H_8=A(3;0,1,2), H_9=A(3;0,1,2,3). Then (ellDown(H_7),ellAll(H_7))=(8,7), (ellDown(H_8),ellAll(H_8))=(9,8), and (ellDown(H_9),ellAll(H_9))=(11,9).
Exact mathematical passage
```
H_7=A(2;0,1,2),
\qquad
H_8=A(3;0,1,2),
\qquad
H_9=A(3;0,1,2,3).
```

H_9Use in source · 3 passages

Search aliases: H_9

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Use in source · Read passage · g_main.tex, line 2090
[Exact binary cases in dimensions seven through nine] Over F_2, let H_7=A(2;0,1,2), H_8=A(3;0,1,2), H_9=A(3;0,1,2,3). Then (ellDown(H_7),ellAll(H_7))=(8,7), (ellDown(H_8),ellAll(H_8))=(9,8), and (ellDown(H_9),ellAll(H_9))=(11,9).
Exact mathematical passage
```
H_7=A(2;0,1,2),
\qquad
H_8=A(3;0,1,2),
\qquad
H_9=A(3;0,1,2,3).
```

h_{i_1j_1}Use in source · 1 passage

Search aliases: h_i_1j_1

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Use in source · Read passage · g_main.tex, line 4653
itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj_k), item the corresponding packets for row, column, and symbol roles.
Exact mathematical passage
```
\operatorname{tr}(h_{i_1j_1}\cdots h_{i_kj_k}),
```

h_{i_kj_k}Use in source · 1 passage

Search aliases: h_i_kj_k

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Use in source · Read passage · g_main.tex, line 4653
itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj_k), item the corresponding packets for row, column, and symbol roles.
Exact mathematical passage
```
\operatorname{tr}(h_{i_1j_1}\cdots h_{i_kj_k}),
```

h_{ij}Use in source · 4 passages

Search aliases: h_ij

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Use in source · Read passage · g_main.tex, line 4639
Holonomy packets. Let a family of local charts or rows be represented by permutations ρ_i. Define normalized transitions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj_k), item the corresponding packets …
Exact mathematical passage
```
h_{ij}=\rho_i\rho_j^{-1}
```

H_QUse in source · 2 passages

Search aliases: H_Q

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Use in source · Read passage · g_main.tex, line 4312
Symmetry-preserving circuits. Let Q be a conserved charge and let H_Q={U:[U,Q]=0}. Restricted circuits use gates commuting with Q. Ambient circuits may temporarily leave H_Q but must return to a charge-preserving endpoint.
Exact mathematical passage
```
H_Q=\{U:[U,Q]=0\}.
```

\mathcal H_qUse in source · 1 passage

Search aliases: H_q mathcal

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Use in source · Read passage · g_main.tex, line 4328
The Hilbert space decomposes as H=⊕_q H_q. A restricted circuit is block diagonal. An ambient catalyst may couple sectors, use the enlarged route, and decouple them at the endpoint.
Exact mathematical passage
```
\mathcal H=\bigoplus_q\mathcal H_q.
```

H_rUse in source · 2 passages

Search aliases: H_r

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Use in source · Read passage · g_main.tex, line 4643
Holonomy packets. Let a family of local charts or rows be represented by permutations ρ_i. Define normalized transitions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj_k), item the corresponding packets …
Exact mathematical passage
```
H_r(L)=\langle h_{ij}:i,j\rangle.
```

\hboxUse in source · 1 passage

Search aliases: hbox

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Use in source · Read passage · c_main.tex, line 105
d oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with target z, correctness of the full circuit gives R(x,z)=P(x,z+f(x)). When f(x)=0, this says B(x)=A(x). When f(x)=1, it says B(x)=P(x,1), and the discr…
Exact mathematical passage
```
\label{c:eq:cover}
 H_i\le f\quad\hbox{for every }i,\qquad
 \bigvee_iH_i=f.
```

\Hh_nUse in source · 2 passages

Search aliases: Hh_n Hh

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Use in source · Read passage · s_main.tex, line 211
The packet access model. For each input length, suppose there is a packet projector P_n and a basis B_n= {ketb_1,…,ketb_D_n} of P_nHh_n. The simulation interface has four routines.
Exact mathematical passage
```
P_n\Hh_n
```

\Hh_{n,d}Use in source · 2 passages

Search aliases: Hh_n,d Hh

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Use in source · Read passage · s_main.tex, line 1039
Let S_n act on Hh_n,d=(C^d)^⊗ n by permuting tensor factors. Schur--Weyl duality gives Hh_n,d ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where S_n acts trivially on the first tensor factor and irreducibly on V_λ.
Exact mathematical passage
```
\Hh_{n,d}=(\C^d)^{\otimes n}
```

\Hh_{n,k}Use in source · 4 passages

Search aliases: Hh_n,k Hh

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Use in source · Read passage · s_main.tex, line 579
[Bounded-excitation collapse] Consider n qubits and suppose the initial state and every circuit prefix are certified to lie in Hh_≤ K = ⊕_k=0^KHh_n,k for a fixed constant K. If initialization, the gate actions, and the measurement are computable in polynomial time in the occupation basis, then the circuit family lies in P, even when individual gates change excitation number.
Exact mathematical passage
```
\Hh_{\le K}
=
\bigoplus_{k=0}^{K}\Hh_{n,k}
```

\Hh_{\le K}Use in source · 2 passages

Search aliases: Hh_≤ K Hh le

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Use in source · Read passage · s_main.tex, line 579
[Bounded-excitation collapse] Consider n qubits and suppose the initial state and every circuit prefix are certified to lie in Hh_≤ K = ⊕_k=0^KHh_n,k for a fixed constant K. If initialization, the gate actions, and the measurement are computable in polynomial time in the occupation basis, then the circuit family lies in P, even when individual gates change excitation number.
Exact mathematical passage
```
\Hh_{\le K}
=
\bigoplus_{k=0}^{K}\Hh_{n,k}
```

\hlineUse in source · 3 passages

Search aliases: hline

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Use in source · Read passage · c_main.tex, line 377
There are 4+16+256=276 functions over the three input sizes. The distribution is c|rrr Optimum Toffolis · 0 · 1 · 3 Number of functions · 28 · 120 · 128. The source constructs and checks the actual circuits. Optimality is obtained from the written lower bounds and matching constructions, not from exhaustive enumeration of all reversible circuits.
Exact mathematical passage
```
 \begin{array}{c|rrr}
 \text{Optimum Toffolis}&0&1&3\\\hline
 \text{Number of functions}&28&120&128.
 \end{array}
```

i^2Use in source · 1 passage

Search aliases: i^2

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Use in source · Read passage · q_lean.tex, line 180
The exact phase arithmetic. The structure Cyclo12 represents (a,b,c,d)⟼ a+bsqrt3+i(c+dsqrt3), a,b,c,d∈ Q. Its multiplication explicitly incorporates (sqrt3)^2=3 and i^2=-1. It is a concrete arithmetic representation. The development does not need or install a global field instance to certify the finite example.
Exact mathematical passage
```
i^2=-1
```

I_{\mathcal V_\lambda}Use in source · 4 passages

Search aliases: I_ V_λ mathcal lambda

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Use in source · Read passage · s_main.tex, line 995
Decomposition For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.
Exact mathematical passage
```
A
=
\bigoplus_\lambda
A_\lambda\otimes I_{\mathcal V_\lambda}.
```

I_1Use in source · 1 passage

Search aliases: I_1

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Use in source · Read passage · g_main.tex, line 2950
[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^r+120 ≤ frac2120n. Also r≥log_21n-1. Therefore S≥ nlog_21n-frac4120n. Multiplying by 8/21 yields D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n.
Exact mathematical passage
```
A_{21}^{\otimes0}=I_1
```

i_1Use in source · 1 passage

Search aliases: i_1

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Use in source · Read passage · g_main.tex, line 4653
itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj_k), item the corresponding packets for row, column, and symbol roles.
Exact mathematical passage
```
\operatorname{tr}(h_{i_1j_1}\cdots h_{i_kj_k}),
```

I_2Use in source · 7 passages

Search aliases: I_2

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Use in source · Read passage · g_main.tex, line 3595
1 · 0, D=diag(d_0,d_1), d_0,d_1∈F_2, and define M_D=J_2 · 0 I_2+DJ_2 · J_2.
Exact mathematical passage
```
M_D=\begin{pmatrix}J_2&0\\I_2+DJ_2&J_2\end{pmatrix}.
```

I_4Use in source · 1 passage

Search aliases: I_4

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Use in source · Read passage · q_main.tex, line 934
This is a mathematical quantum-state construction. An actual realization would additionally require state preparation, a twelve-dimensional unitary implementation, coherence control, and sampling. A twelve-dimensional Fourier matrix is not the ordinary sixteen-dimensional Fourier transformation on four qubits. The unitary direct sum F_12⊕ I_4 supplies a mathematical sixteen-dimensional embedding, but it supplies no gate synthesis or resource bound by itself.
Exact mathematical passage
```
F_{12}\oplus I_4
```

I_AUse in source · 1 passage

Search aliases: I_A

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Use in source · Read passage · g_main.tex, line 5233
Template XIII: ancilla restoration. Let W act on data D and ancilla A. To certify a dirty catalyst, prove the operator identity W=U_D⊗ I_A on the full joint space, not only on one ancilla input. For a clean catalyst, prove W(ketψ⊗ket0) =(Uketψ)⊗ket0 for all ketψ. The distinction changes both the target equations and lower bounds.
Exact mathematical passage
```
W=U_D\otimes I_A
```

I_{D_t}Use in source · 1 passage

Search aliases: I_D_t

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Use in source · Read passage · s_main.tex, line 442
Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.
Exact mathematical passage
```
B_t:\C^{D_t}\longrightarrow\Hh,
\qquad
B_tB_t^*=P_t,
\qquad
B_t^*B_t=I_{D_t},
```

I_jUse in source · 2 passages

Search aliases: I_j

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Use in source · Read passage · g_main.tex, line 343
Catalytic interval geometry. Associate to every matched opening and closing operation an interval I_j=[t_j^open,t_j^close]. The family of intervals can be disjoint, nested, laminar, or crossing.
Exact mathematical passage
```
I_j=[t_j^{\mathrm{open}},t_j^{\mathrm{close}}].
```

i_kUse in source · 1 passage

Search aliases: i_k

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Use in source · Read passage · g_main.tex, line 4653
itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj_k), item the corresponding packets for row, column, and symbol roles.
Exact mathematical passage
```
\operatorname{tr}(h_{i_1j_1}\cdots h_{i_kj_k}),
```

I_MUse in source · 1 passage

Search aliases: I_M

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Use in source · Read passage · q_main.tex, line 611
Conversely, for ℓ∈ I_M,
Exact mathematical passage
```
\ell\in I_M
```

I_mUse in source · 2 passages

Search aliases: I_m

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Use in source · Read passage · g_main.tex, line 2770
For the matching lower construction, first realize I_m⊗ B using m copies of a circuit for B. Then realize A⊗ I_n. Each gate of a circuit for A lifts to n coordinatewise gates. Since (A⊗ I_n)(I_m⊗ B)=A⊗ B, the cost is the right side of [source reference]. The same construction with ambient circuits proves [source reference].
Exact mathematical passage
```
I_m\otimes B
```

I_nUse in source · 3 passages

Search aliases: I_n

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Use in source · Read passage · g_appendix.tex, line 65
Binary symplectic form. Pauli exponents are ordered as (x_0,…,x_n-1,z_0,…,z_n-1). The symplectic form is J= 0 · I_n I_n · 0 over F_2. A matrix M is symplectic when M^TJM=J.
Exact mathematical passage
```
J=
\begin{pmatrix}
0&I_n\\
I_n&0
\end{pmatrix}
```

I_{n-6}Use in source · 1 passage

Search aliases: I_n-6

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Use in source · Read passage · g_main.tex, line 1441
Embed C as C⊕ I_n-6. A six-gate ambient word acts on the first six rows. In a lower-directed circuit, rows with index at least six cannot be used as sources for targets among the first six rows, so the restricted lower bound of seven persists.
Exact mathematical passage
```
C\oplus I_{n-6}
```

I_{P_3}Use in source · 1 passage

Search aliases: I_P_3

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Use in source · Read passage · c_visual.tex, line 516
The three linear forms evaluate to L=(0,1,1), so P_3=L_1L_2L_3=0. The quadratic grade has pairs (1,6), (2,5), (3,4), so Q_3=x_1x_6+x_2x_5+x_3x_4=0+0+1=1. Thus Q_3∉ I_P_3: homogeneous extraction does not preserve the positive-support ideal. The whole polynomial is supported on its own accepting set, but its isolated homogeneous components need not be.
Exact mathematical passage
```
Q_3\notin I_{P_3}
```

I_rUse in source · 1 passage

Search aliases: I_r

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Use in source · Read passage · g_main.tex, line 2647
For n=6k+r, take C^⊕ k⊕ I_r and apply block additivity together with k copies of the ambient six-gate word.
Exact mathematical passage
```
C^{\oplus k}\oplus I_r
```

i_rUse in source · 2 passages

Search aliases: i_r

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Use in source · Read passage · g_main.tex, line 781
C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
Q_{>r}e_{i_r}e_{j_r}^{T}P_{<r}
```

I_sUse in source · 2 passages

Search aliases: I_s

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Use in source · Read passage · g_main.tex, line 773
Dynamic rank-one factorization. Consider a height-two block shear E(C)= I_s · 0 C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
E(C)=
\begin{pmatrix}
I_s&0\\
C&I_t
\end{pmatrix}.
```

I_tUse in source · 2 passages

Search aliases: I_t

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Use in source · Read passage · g_main.tex, line 773
Dynamic rank-one factorization. Consider a height-two block shear E(C)= I_s · 0 C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
E(C)=
\begin{pmatrix}
I_s&0\\
C&I_t
\end{pmatrix}.
```

\operatorname{id}Use in source · 2 passages

Search aliases: id operatorname

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Use in source · Read passage · g_main.tex, line 399
When defects are impossible. A useful positive theorem is a criterion guaranteeing geodesic convexity. [One-Lipschitz retraction criterion] Suppose there is a group homomorphism π:G→ H with π|_H=id_H such that every ambient generator s∈ S_G satisfies ℓ_H(π(s))≤ c(s). Then ℓ_H(h)≤ℓ_G(h) for every h∈ H, hence the two metrics agree on H. For an ambient word h=s_L⋯ s_1, multiplicativity gives h=π(h)=π(s_L)⋯π(s_1). Subadditivity and the generator bounds give ℓ_H(h)≤∑_j c(s_j). Taking the infimum over…
Exact mathematical passage
```
\pi|_H=\operatorname{id}_H
```

\operatorname{id}_HUse in source · 1 passage

Search aliases: id_H operatorname

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Use in source · Read passage · g_main.tex, line 399
When defects are impossible. A useful positive theorem is a criterion guaranteeing geodesic convexity. [One-Lipschitz retraction criterion] Suppose there is a group homomorphism π:G→ H with π|_H=id_H such that every ambient generator s∈ S_G satisfies ℓ_H(π(s))≤ c(s). Then ℓ_H(h)≤ℓ_G(h) for every h∈ H, hence the two metrics agree on H. For an ambient word h=s_L⋯ s_1, multiplicativity gives h=π(h)=π(s_L)⋯π(s_1). Subadditivity and the generator bounds give ℓ_H(h)≤∑_j c(s_j). Taking the infimum over…
Exact mathematical passage
```
\pi|_H=\operatorname{id}_H
```

\operatorname{Id}_NUse in source · 2 passages

Search aliases: Id_N operatorname

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Use in source · Read passage · q_main.tex, line 363

(P_d^(N))^† · =P_d^(N), P_d^(N)P_e^(N) · =δ_deP_d^(N), ∑_dmid NP_d^(N) · =Id_N, rankP_d^(N) · =trP_d^(N)=φ(d).

Exact mathematical passage


\begin{aligned}
(P_d^{(N)})^\dagger&=P_d^{(N)},\\
P_d^{(N)}P_e^{(N)}&=\delta_{de}P_d^{(N)},\\
\sum_{d\mid N}P_d^{(N)}&=\operatorname{Id}_N,\\
\operatorname{rank}P_d^{(N)}&=\operatorname{tr}P_d^{(N)}=\varphi(d).
\end{aligned}

\iffUse in source · 1 passage

Search aliases: iff

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Use in source · Read passage · c_lean.tex, line 125
pageheadFrom the cut to a numerical lower boundEXISTENCE, BALANCE, AND COMPLETE COVERAGE Write c=pmathbin++q. At this cut, Block 01 defines H_i(x)=run(p)(x,0^a,0)_ix run(q^-1)(x,0^a,0)_i and proves H_i≤ f, f(x)=1iff∃ i H_i(x)=1. Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives deg(H_i)≤max{TC(p),TC(q)}+1.
Exact mathematical passage
```
H_i\le f,\qquad f(x)=1\iff\exists i\;H_i(x)=1.
```

\operatorname{Im}Use in source · 1 passage

Search aliases: Im operatorname

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Use in source · Read passage · q_visual.tex, line 199
A real three-dimensional view of one amplitude. addcontentslinetocsectionA real three-dimensional view of one amplitude leadThe plot follows an actual complex amplitude as the relative phase advances. At sample n=6, the phase-family amplitude is χ_θ(6)=frace^iθsqrt3. fig16_complex_amplitude_trajectory0.84Coordinates are (θ,Reχ_θ(6),Imχ_θ(6)). In the real-imaginary plane the amplitude traces a radius-1/sqrt3 circle. Advancing the phase coordinate produces the three-dimensional trajectory. Its mag…
Exact mathematical passage
```
(\theta,\operatorname{Re}\chi_\theta(6),\operatorname{Im}\chi_\theta(6))
```

\operatorname{im}Use in source · 1 passage

Search aliases: im operatorname

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Use in source · Read passage · g_main.tex, line 2418
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\mathcal R^*_{n,\le L}
=
\U_n\cap
\bigcup_{\sigma\in\{(t,s):t\ne s\}^L}
\operatorname{im}\Phi_\sigma
```

\inf_{r\ge1}Use in source · 1 passage

Search aliases: inf_r≥1 inf ge

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Use in source · Read passage · g_main.tex, line 2893
[Stable tensor length] The limit τ_*(A) = lim_r→∞ fracellAll(A^⊗ r)r m^r-1 = inf_r≥1 fracellAll(A^⊗ r)r m^r-1 exists. If A is tree tight, then the stable catalytic capacity κ_⊗(A) = ellDown(A)-τ_*(A) satisfies κ_⊗(A)≥DeltaDef(A).
Exact mathematical passage
```
\tau_*(A)
=
\lim_{r\to\infty}
\frac{\ellAll(A^{\otimes r})}{r m^{r-1}}
=
\inf_{r\ge1}
\frac{\ellAll(A^{\otimes r})}{r m^{r-1}}
```

\inf_{x\in\mathcal P_{\mathfrak M}(U)}Use in source · 1 passage

Search aliases: inf_x∈ P_ M(U) inf in mathcal mathfrak

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Use in source · Read passage · g_main.tex, line 97
For a weight vector λ∈R_≥0^r, scalarization gives ℓ_ M^λ(U) = inf_x∈ P_ M(U)λ· x. The scalar defect Δ^λ(U) can vary with λ even when the target and gate alphabets are fixed.
Exact mathematical passage
```
\ell_{\mathfrak M}^{\lambda}(U)
=
\inf_{x\in\mathcal P_{\mathfrak M}(U)}\lambda\cdot x.
```

\mathrm{internal}Use in source · 2 passages

Search aliases: internal mathrm

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Use in source · Read passage · s_main.tex, line 33
The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:
Exact mathematical passage
```
\mathfrak C_n=
\bigl(
N_{\mathrm{packets}},
D_{\mathrm{internal}},
S_{\mathrm{transfer}},
L_{\mathrm{leakage}}
\bigr).
```

JUse in source · 20 passages

Search aliases: J

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Use in source · Read passage · g_appendix.tex, line 65
Binary symplectic form. Pauli exponents are ordered as (x_0,…,x_n-1,z_0,…,z_n-1). The symplectic form is J= 0 · I_n I_n · 0 over F_2. A matrix M is symplectic when M^TJM=J.
Exact mathematical passage
```
J=
\begin{pmatrix}
0&I_n\\
I_n&0
\end{pmatrix}
```

J_+Use in source · 2 passages

Search aliases: J_+

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Use in source · Read passage · s_main.tex, line 940
Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1, J_-ketD_n^k = sqrtk(n-k+1)ketD_n^k-1. Thus co…
Exact mathematical passage
```
J_+=J_x-iJ_y
```

J_-Use in source · 2 passages

Search aliases: J_-

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Use in source · Read passage · s_main.tex, line 940
Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1, J_-ketD_n^k = sqrtk(n-k+1)ketD_n^k-1. Thus co…
Exact mathematical passage
```
J_-=J_x+iJ_y
```

j_1Use in source · 1 passage

Search aliases: j_1

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Use in source · Read passage · g_main.tex, line 4653
itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj_k), item the corresponding packets for row, column, and symbol roles.
Exact mathematical passage
```
\operatorname{tr}(h_{i_1j_1}\cdots h_{i_kj_k}),
```

J_2Use in source · 5 passages

Search aliases: J_2

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Use in source · Read passage · g_main.tex, line 3589
The four exceptional matrices admit one common lower-bound proof. Work in the declared symplectic quotient and entangling metric, with H_i and S_i free and SWAP excluded as a primitive. Put J_2=0 · 1 1 · 0, D=diag(d_0,d_1), d_0,d_1∈F_2, and define M_D=J_2 · 0
Exact mathematical passage
```
J_2=\begin{pmatrix}0&1\\1&0\end{pmatrix},
\qquad D=\operatorname{diag}(d_0,d_1),
\qquad d_0,d_1\in\F_2,
```

J_aUse in source · 2 passages

Search aliases: J_a

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Use in source · Read passage · s_main.tex, line 936
Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1, J_-ketD_n^k = sqrtk(n-k+1)ketD_n^k-1. Thus co…
Exact mathematical passage
```
J_a=\frac12\sum_{q=1}^n\sigma_a^{(q)},
\qquad a\in\{x,y,z\}.
```

j_kUse in source · 1 passage

Search aliases: j_k

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Use in source · Read passage · g_main.tex, line 4653
itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj_k), item the corresponding packets for row, column, and symbol roles.
Exact mathematical passage
```
\operatorname{tr}(h_{i_1j_1}\cdots h_{i_kj_k}),
```

j_rUse in source · 2 passages

Search aliases: j_r

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Use in source · Read passage · g_main.tex, line 781
C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
Q_{>r}e_{i_r}e_{j_r}^{T}P_{<r}
```

J_xUse in source · 3 passages

Search aliases: J_x

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Use in source · Read passage · s_main.tex, line 940
Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1, J_-ketD_n^k = sqrtk(n-k+1)ketD_n^k-1. Thus co…
Exact mathematical passage
```
J_+=J_x-iJ_y
```

J_yUse in source · 3 passages

Search aliases: J_y

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Use in source · Read passage · s_main.tex, line 940
Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1, J_-ketD_n^k = sqrtk(n-k+1)ketD_n^k-1. Thus co…
Exact mathematical passage
```
J_+=J_x-iJ_y
```

J_zUse in source · 2 passages

Search aliases: J_z

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Use in source · Read passage · s_main.tex, line 940
Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1, J_-ketD_n^k = sqrtk(n-k+1)ketD_n^k-1. Thus co…
Exact mathematical passage
```
J_z\ket{D_n^k}=(n/2-k)\ket{D_n^k}
```

KUse in source · 83 passages

Search aliases: K

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Use in source · Read passage · c_main.tex, line 494
The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
K
```

K^{-1}Use in source · 1 passage

Search aliases: K^-1

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Use in source · Read passage · g_main.tex, line 4181
A Gaussian catalytic theorem requires: item a declared primitive library and cost vector; item a target subgroup, such as triangular, number conserving, or local; item an explicit sandwich KHK^-1 returning to the target subgroup; item a lower bound based on symplectic rank, singular values, graph cuts, or mode support; item treatment of approximation error when continuous parameters are synthesized numerically.
Exact mathematical passage
```
KHK^{-1}
```

k^{-1/2}Use in source · 2 passages

Search aliases: k^-1/2

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Use in source · Read passage · s_appendix.tex, line 33
Hypercube quotient coupling. Let ket k = binom nk^-1/2 ∑_|x|=kket x. Every weight-k string has n-k neighbors of weight k+1, while every weight-(k+1) string has k+1 neighbors of weight k. Counting the bipartite edges yields brak+1Aket k = fracbinom nk(n-k) sqrtbinom nkbinom nk+1 = sqrt(k+1)(n-k), proving [source reference].
Exact mathematical passage
```
\ket k
=
\binom nk^{-1/2}
\sum_{|x|=k}\ket x.
```

K^4Use in source · 1 passage

Search aliases: K^4

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Use in source · Read passage · c_visual.tex, line 540
The four indicator functions form a basis of K^4. General m has 2^m such basis elements. A verifier is the sum of the indicators of its accepted witnesses, and pointwise multiplication models intersection of accepting sets.
Exact mathematical passage
```
K^4
```

K^mUse in source · 2 passages

Search aliases: K^m

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Use in source · Read passage · c_main.tex, line 596
The conditional algorithmic consequence. [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The leading coefficient has total degree at most d-r, so it vanishes at a random first (m-1)-tuple with probability at most (d-r)/|K|. Otherwise t…
Exact mathematical passage
```
 \Pr_{z\text{ uniform in }K^m}[P(z)=0]\le\frac d{|K|}.
```

K^WUse in source · 4 passages

Search aliases: K^W

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Use in source · Read passage · c_main.tex, line 495
The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
 \mathcal B=K^W,
```

K_1Use in source · 4 passages

Search aliases: K_1

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Use in source · Read passage · g_main.tex, line 3616
1 · 0) and has rank one. In either directed CNOT or CZ, both diagonal blocks are I_2 and have rank two. Left and right multiplication by local Cliffords preserve these diagonal-block ranks. A circuit with one charged gate has the form K_1 E K_2 with K_1,K_2 local and E a CNOT or CZ, so it cannot equal M_D. A circuit with no charged gate is local and also has rank-two diagonal blocks. The ambient cost is therefore at least two.
Exact mathematical passage
```
K_1 E K_2
```

K_2Use in source · 4 passages

Search aliases: K_2

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Use in source · Read passage · g_main.tex, line 3616
1 · 0) and has rank one. In either directed CNOT or CZ, both diagonal blocks are I_2 and have rank two. Left and right multiplication by local Cliffords preserve these diagonal-block ranks. A circuit with one charged gate has the form K_1 E K_2 with K_1,K_2 local and E a CNOT or CZ, so it cannot equal M_D. A circuit with no charged gate is local and also has rank-two diagonal blocks. The ambient cost is therefore at least two.
Exact mathematical passage
```
K_1 E K_2
```

K_4Use in source · 11 passages

Search aliases: K_4

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Use in source · Read passage · g_main.tex, line 757
[Flagged matroid tour] Let M be a representable matroid with designated initial basis, target basis, and consumer states. A flagged tour is a sequence of basis exchanges or transvections in which the ith carrier remains inside a specified filtration layer. An ambient tour drops this constraint. The six-row atom realizes the graphic regular matroid M(K_4). Its ambient state tour is shorter than every compatible flag lift. Regularity explains why the same dependence pattern survives every field an…
Exact mathematical passage
```
M(K_4)
```

K_\muUse in source · 2 passages

Search aliases: K_μ mu

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Use in source · Read passage · s_main.tex, line 1653
Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.
Exact mathematical passage
```
K_\mu
```

K_\mu^\daggerUse in source · 1 passage

Search aliases: K_μ^† mu dagger

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Use in source · Read passage · s_main.tex, line 1654
Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.
Exact mathematical passage
```
\Phi(\rho)
=
\sum_\mu K_\mu\rho K_\mu^\dagger.
```

\ketUse in source · 55 passages

Search aliases: ket

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 51
Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.
Exact mathematical passage
```
\SUM^{(a)}_{s\to t}
\ket{x_s,x_t}
=
\ket{x_s,x_t+a x_s\bmod d}.
```

\KTopUse in source · 4 passages

Search aliases: KTop

Link to this entry ·

Use in source · Read passage · c_main.tex, line 580
[Weighted top-coefficient identity] Let KTop_w extract the coefficient of w_1⋯ w_m after reduction in the Boolean w algebra. Then P_R(z)=KTop_w[ R(w)∏_i=1^m(1+(1+z_i)w_i) ]. At w_i=0, the ith factor is one. At w_i=1, it is z_i. Lemma~[source reference] therefore converts the right side to exactly the weighted sum defining P_R.
Exact mathematical passage
```
\KTop_w
```

\KTop_wUse in source · 4 passages

Search aliases: KTop_w KTop

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Use in source · Read passage · c_main.tex, line 580
[Weighted top-coefficient identity] Let KTop_w extract the coefficient of w_1⋯ w_m after reduction in the Boolean w algebra. Then P_R(z)=KTop_w[ R(w)∏_i=1^m(1+(1+z_i)w_i) ]. At w_i=0, the ith factor is one. At w_i=1, it is z_i. Lemma~[source reference] therefore converts the right side to exactly the weighted sum defining P_R.
Exact mathematical passage
```
\KTop_w
```

\KTrUse in source · 4 passages

Search aliases: KTr

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Use in source · Read passage · c_main.tex, line 643
[Shared-variable multiplication fails] For a single witness bit, define KTr_z(A)=A(0)+zA(1). Then in the Boolean w algebra, KTr_z(w)=z, KTr_z(1+w)=1, but KTr_z(w(1+w))=0 ≠ z=KTr_z(w)KTr_z(1+w) whenever z≠0. Also, KTr_z(w^2)=z, KTr_z(w)^2=z^2, which differ for field elements z with z^2≠ z. Evaluate w and 1+w at zero and one. In the Boolean quotient w(1+w)=w+w^2=0 and w^2=w. The displayed identities follow directly. In GF(16), any element outside F supplies the second inequality.
Exact mathematical passage
```
\KTr_z(A)=A(0)+zA(1)
```

\KTr_zUse in source · 4 passages

Search aliases: KTr_z KTr

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Use in source · Read passage · c_main.tex, line 643
[Shared-variable multiplication fails] For a single witness bit, define KTr_z(A)=A(0)+zA(1). Then in the Boolean w algebra, KTr_z(w)=z, KTr_z(1+w)=1, but KTr_z(w(1+w))=0 ≠ z=KTr_z(w)KTr_z(1+w) whenever z≠0. Also, KTr_z(w^2)=z, KTr_z(w)^2=z^2, which differ for field elements z with z^2≠ z. Evaluate w and 1+w at zero and one. In the Boolean quotient w(1+w)=w+w^2=0 and w^2=w. The displayed identities follow directly. In GF(16), any element outside F supplies the second inequality.
Exact mathematical passage
```
\KTr_z(A)=A(0)+zA(1)
```

\mathcal LUse in source · 3 passages

Search aliases: L mathcal

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Use in source · Read passage · s_main.tex, line 1680
Lindblad dynamics. A Markovian open system obeys fracdρdt = L(ρ) = -i[H,ρ] + ∑_a ( L_aρ L_a^† - frac12{L_a^† L_a,ρ} ). Packet selection rules of H and L_a determine a sparse directed graph on coherence pairs. If the reachable graph and each block representation remain polynomial, a classical integrator can evolve the reduced system.
Exact mathematical passage
```
\frac{d\rho}{dt}
=
\mathcal L(\rho)
=
-i[H,\rho]
+
\sum_a
\left(
L_a\rho L_a^\dagger
-
\frac12\{L_a^\dagger L_a,\rho\}
\right).
```

lUse in source · 12 passages

Search aliases: l

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Use in source · Read passage · g_main.tex, line 4908
The detailed defect distributions are: c|l model · restricted→ambient pairs n=2 entangling · 3→2:4 n=3 total · 7→6:6,quad8→7:18 n=3 entangling · 3→2:24, 4→3:432, 5→4:1056, 6→4:48, 6→5:64.
Exact mathematical passage
```
\begin{array}{c|l}
\text{model}&\text{restricted}\to\text{ambient pairs}\\
\hline
n=2\text{ entangling}&3\to2:4\\
n=3\text{ total}&7\to6:6,\quad8\to7:18\\
n=3\text{ entangling}&3\to2:24,\ 4\to3:432,\ 5\to4:1056,\ 6\to4:48,\ 6\to5:64.
\end{array}
```

L_1Use in source · 2 passages

Search aliases: L_1

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Use in source · Read passage · c_main.tex, line 432
[Exact nonlinear cost of the whole function] For the family above, M(P_m)=m-1, KTrest(P_m)=2m-3. The linear map x↦(L_1(x),…,L_m(x)) has rank m, since its columns include the standard basis vectors. Extend these forms to an invertible system of linear coordinates. In those coordinates P_m is the product of the first m variables, with the other n-m variables unused.
Exact mathematical passage
```
x\mapsto(L_1(x),\ldots,L_m(x))
```

L_2Use in source · 1 passage

Search aliases: L_2

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Use in source · Read passage · c_visual.tex, line 511
The three linear forms evaluate to L=(0,1,1), so P_3=L_1L_2L_3=0. The quadratic grade has pairs (1,6), (2,5), (3,4), so Q_3=x_1x_6+x_2x_5+x_3x_4=0+0+1=1. Thus Q_3∉ I_P_3: homogeneous extraction does not preserve the positive-support ideal. The whole polynomial is supported on its own accepting set, but its isolated homogeneous components need not be.
Exact mathematical passage
```
P_3=L_1L_2L_3=0
```

L_3Use in source · 1 passage

Search aliases: L_3

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Use in source · Read passage · c_visual.tex, line 511
The three linear forms evaluate to L=(0,1,1), so P_3=L_1L_2L_3=0. The quadratic grade has pairs (1,6), (2,5), (3,4), so Q_3=x_1x_6+x_2x_5+x_3x_4=0+0+1=1. Thus Q_3∉ I_P_3: homogeneous extraction does not preserve the positive-support ideal. The whole polynomial is supported on its own accepting set, but its isolated homogeneous components need not be.
Exact mathematical passage
```
P_3=L_1L_2L_3=0
```

L_aUse in source · 2 passages

Search aliases: L_a

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Use in source · Read passage · s_main.tex, line 1680
Lindblad dynamics. A Markovian open system obeys fracdρdt = L(ρ) = -i[H,ρ] + ∑_a ( L_aρ L_a^† - frac12{L_a^† L_a,ρ} ). Packet selection rules of H and L_a determine a sparse directed graph on coherence pairs. If the reachable graph and each block representation remain polynomial, a classical integrator can evolve the reduced system.
Exact mathematical passage
```
\frac{d\rho}{dt}
=
\mathcal L(\rho)
=
-i[H,\rho]
+
\sum_a
\left(
L_a\rho L_a^\dagger
-
\frac12\{L_a^\dagger L_a,\rho\}
\right).
```

\mathcal L_AUse in source · 1 passage

Search aliases: L_A mathcal

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Use in source · Read passage · s_main.tex, line 1698
Error accounting for truncated channels. Let Π_A project operator space onto an active set of coherence blocks. A reduced evolution uses L_A = Π_A LΠ_A. The discarded transfer is (I-Π_A) LΠ_A. Duhamel's formula yields an error bound analogous to the Hamiltonian leakage theorem. The resulting estimate depends on the induced norm of the discarded superoperator transfer and the integration time.
Exact mathematical passage
```
\mathcal L_A
=
\Pi_A\mathcal L\Pi_A.
```

L_a^\daggerUse in source · 1 passage

Search aliases: L_a^† dagger

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Use in source · Read passage · s_main.tex, line 1680
Lindblad dynamics. A Markovian open system obeys fracdρdt = L(ρ) = -i[H,ρ] + ∑_a ( L_aρ L_a^† - frac12{L_a^† L_a,ρ} ). Packet selection rules of H and L_a determine a sparse directed graph on coherence pairs. If the reachable graph and each block representation remain polynomial, a classical integrator can evolve the reduced system.
Exact mathematical passage
```
\frac{d\rho}{dt}
=
\mathcal L(\rho)
=
-i[H,\rho]
+
\sum_a
\left(
L_a\rho L_a^\dagger
-
\frac12\{L_a^\dagger L_a,\rho\}
\right).
```

L_{\mathrm{leakage}}Use in source · 2 passages

Search aliases: L_leakage mathrm

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Use in source · Read passage · s_main.tex, line 33
The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:
Exact mathematical passage
```
\mathfrak C_n=
\bigl(
N_{\mathrm{packets}},
D_{\mathrm{internal}},
S_{\mathrm{transfer}},
L_{\mathrm{leakage}}
\bigr).
```

L_mUse in source · 1 passage

Search aliases: L_m

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Use in source · Read passage · c_main.tex, line 432
[Exact nonlinear cost of the whole function] For the family above, M(P_m)=m-1, KTrest(P_m)=2m-3. The linear map x↦(L_1(x),…,L_m(x)) has rank m, since its columns include the standard basis vectors. Extend these forms to an invertible system of linear coordinates. In those coordinates P_m is the product of the first m variables, with the other n-m variables unused.
Exact mathematical passage
```
x\mapsto(L_1(x),\ldots,L_m(x))
```

L_PUse in source · 1 passage

Search aliases: L_P

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Use in source · Read passage · s_main.tex, line 327
Hamiltonian leakage. Exact conservation is not the only useful regime. Let P be an orthogonal projector and let U(t)=e^-itH for a bounded self-adjoint Hamiltonian H. The leakage operator from the packet is L_P(t)=(I-P)e^-itHP.
Exact mathematical passage
```
L_P(t)=(I-P)e^{-itH}P.
```

L_{\rm leakage}Use in source · 1 passage

Search aliases: L_rm leakage rm

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Use in source · Read passage · s_main.tex, line 2273
The core research invariant is the complete classical representation record ( discovery cost, N_rm packets, D_rm internal, S_rm transfer, L_rm leakage, measurement cost ). A future compiler that computes and certifies this record can decide when the full state vector is unnecessary, when a hybrid representation is preferable, and when symmetry alone is insufficient.
Exact mathematical passage
```
\bigl(
\text{discovery cost},
N_{\rm packets},
D_{\rm internal},
S_{\rm transfer},
L_{\rm leakage},
\text{measurement cost}
\bigr).
```

L_{v_0}Use in source · 2 passages

Search aliases: L_v_0

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Use in source · Read passage · g_main.tex, line 1963
Suppose the gate is s→ t with s<t. Let v_0 be the lowest node of T whose two children separate s and t. The order of the leaves forces s∈ L_v_0 and t∈ R_v_0. At this one node, a single row of the cross block X[R_v_0,L_v_0] changes, so its rank changes by at most one.
Exact mathematical passage
```
s\in L_{v_0}
```

L_ZUse in source · 4 passages

Search aliases: L_Z

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Use in source · Read passage · g_appendix.tex, line 101
The parabolic subgroup used in the census is generated by S_i, CZ_ij, and CX_i→ j. It stabilizes the Z Lagrangian L_Z=Span(e_n,…,e_2n-1).
Exact mathematical passage
```
L_Z=\Span(e_n,\ldots,e_{2n-1}).
```

\landUse in source · 1 passage

Search aliases: land

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Use in source · Read passage · c_main.tex, line 628
[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independent and the factorization exact.
Exact mathematical passage
```
 P_{A\land B}(z_u,z_v)=P_A(z_u)P_B(z_v).
```

\mathrm{large}Use in source · 1 passage

Search aliases: large mathrm

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Use in source · Read passage · g_main.tex, line 5165
a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). This transfers a catalytic atom to architectures containing it as a retract.
Exact mathematical passage
```
\Delta_{\mathrm{large}}(\iota U)
\ge
\Delta_{\mathrm{small}}(U).
```

\mathrm{large,res}Use in source · 1 passage

Search aliases: large,res mathrm

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Use in source · Read passage · g_main.tex, line 5159
Template VIII: block retraction. Let a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). This transfers a catalytic atom to architectures containing it as a retract.
Exact mathematical passage
```
\ell_{\mathrm{large,res}}(\iota U)
\ge
\ell_{\mathrm{small,res}}(U).
```

\lceilUse in source · 9 passages

Search aliases: lceil

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Use in source · Read passage · c_lean.tex, line 146
The balanced cut is constructed. Because each gate has cost zero or one, a prefix can realize every Toffoli count from zero to T. The source proves this fact and extracts a cut with max{TC(p),TC(q)}≤lceil T/2rceil. The complete safe cut cover then proves β(f)≤lceil T/2rceil+1, 2β(f)≤ T+3. Natural-number arithmetic retains the floor/ceiling and small-T cases. No balance assumption is added to the final theorem.
Exact mathematical passage
```
\max\{\TC(p),\TC(q)\}\le\left\lceil T/2\right\rceil.
```

\mathrm{leakage}Use in source · 2 passages

Search aliases: leakage mathrm

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Use in source · Read passage · s_main.tex, line 33
The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:
Exact mathematical passage
```
\mathfrak C_n=
\bigl(
N_{\mathrm{packets}},
D_{\mathrm{internal}},
S_{\mathrm{transfer}},
L_{\mathrm{leakage}}
\bigr).
```

\liminfUse in source · 1 passage

Search aliases: liminf

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Use in source · Read passage · g_main.tex, line 4059
Tensor and repeated-module questions. For targets U and V, independent synthesis gives ℓ_T(U⊗ V) ≤ ℓ_T(U)+ℓ_T(V) when the systems are disjoint and local Clifford cost is ignored. Joint synthesis may be better because phase identities can couple terms across modules. This motivates a stable regularization τ_T(U)= liminf_r→∞ fracℓ_T(U^⊗ r)r. A one-copy rigid phase target could still display tensor activation.
Exact mathematical passage
```
\tau_T(U)=
\liminf_{r\to\infty}
\frac{\ell_T(U^{\otimes r})}{r}.
```

\liminf_{r\to\infty}Use in source · 1 passage

Search aliases: liminf_r→∞ liminf to infty

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Use in source · Read passage · g_main.tex, line 4059
Tensor and repeated-module questions. For targets U and V, independent synthesis gives ℓ_T(U⊗ V) ≤ ℓ_T(U)+ℓ_T(V) when the systems are disjoint and local Clifford cost is ignored. Joint synthesis may be better because phase identities can couple terms across modules. This motivates a stable regularization τ_T(U)= liminf_r→∞ fracℓ_T(U^⊗ r)r. A one-copy rigid phase target could still display tensor activation.
Exact mathematical passage
```
\tau_T(U)=
\liminf_{r\to\infty}
\frac{\ell_T(U^{\otimes r})}{r}.
```

\limsupUse in source · 1 passage

Search aliases: limsup

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Use in source · Read passage · g_main.tex, line 2193
[Asymptotic multiplicative distortion] For every field, limsup_n→∞ sup_A∈U_n(F), A≠ I fracellDown(A)ellAll(A) ≥frac32.
Exact mathematical passage
```
\limsup_{n\to\infty}
\sup_{A\in\U_n(\F),\,A\ne I}
\frac{\ellDown(A)}{\ellAll(A)}
\ge\frac32.
```

\limsup_{n\to\infty}Use in source · 1 passage

Search aliases: limsup_n→∞ limsup to infty

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Use in source · Read passage · g_main.tex, line 2193
[Asymptotic multiplicative distortion] For every field, limsup_n→∞ sup_A∈U_n(F), A≠ I fracellDown(A)ellAll(A) ≥frac32.
Exact mathematical passage
```
\limsup_{n\to\infty}
\sup_{A\in\U_n(\F),\,A\ne I}
\frac{\ellDown(A)}{\ellAll(A)}
\ge\frac32.
```

\logUse in source · 49 passages

Search aliases: log

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Use in source · Read passage · c_main.tex, line 482
These are exact cost formulas. The large-input entries were not obtained by searching for optimal circuits. The full construction uses 2m(2^m-1-1) CNOTs as well as 2m-3 Toffolis, giving total count O(nlog n). The projected quadratic has n/2 total gates.
Exact mathematical passage
```
O(n\log n)
```

\log_2Use in source · 10 passages

Search aliases: log_2 log

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Use in source · Read passage · g_main.tex, line 1107
[Tensor-amplified snapshot family] Let A=A(u;s_1,…,s_v) have dimension m=u+v+2. For N=m^r, ellDown(A^⊗ r) =r m^r-1(u+2v), while ellAll(A^⊗ r) ≤ r m^r. Consequently, ellDown(A^⊗ r)-ellAll(A^⊗ r) ≥ fracv-2mlog_2mNlog_2N.
Exact mathematical passage
```
\ellDown(A^{\otimes r})-\ellAll(A^{\otimes r})
\ge
\frac{v-2}{m\log_2m}N\log_2N.
```

\log_{21}Use in source · 3 passages

Search aliases: log_21 log

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Use in source · Read passage · g_main.tex, line 2957
[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^r+120 ≤ frac2120n. Also r≥log_21n-1. Therefore S≥ nlog_21n-frac4120n. Multiplying by 8/21 yields D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n.
Exact mathematical passage
```
r=\lfloor\log_{21}n\rfloor
```

\longleftarrowUse in source · 2 passages

Search aliases: longleftarrow

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Use in source · Read passage · g_main.tex, line 919
Let R be a nonzero commutative ring with identity. A row transvection is τ_t,s(a)=I+aE_t,s, t≠ s, a∈ R∖{0}, acting by R_tlongleftarrow R_t+aR_s. The transvection is lower directed when s<t. The lower-directed gates generate U_n(R). We compare ellDown(A) =min{number of lower-directed transvections producing A} with ellAll(A) =min{number of arbitrary row transvections producing A}, A∈U_n(R). For a field F, the first metric is intrinsic to the unipotent radical and the second is induced from the el…
Exact mathematical passage
```
R_t\longleftarrow R_t+aR_s.
```

\LongleftrightarrowUse in source · 6 passages

Search aliases: Longleftrightarrow

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Use in source · Read passage · c_lean.tex, line 236
In the generic C01 theorem, completeness is the existential statement f(x)=1 Longleftrightarrow ∃ i:W, H_i(x)=1. No finite disjunction is needed there. Specializing to the canonical finite wires yields the finite complete family used in the main paper. Completeness does not say that the XOR of all packets reconstructs f; detecting packets can cancel under XOR. It states existence of a detecting coordinate at every positive input.
Exact mathematical passage
```
 f(x)=1\quad\Longleftrightarrow\quad\exists i:W,\ H_i(x)=1.
```

\LongrightarrowUse in source · 1 passage

Search aliases: Longrightarrow

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Use in source · Read passage · c_supplement.tex, line 26

Dependency structure. The main proof dependencies are: restored oracle convention Longrightarrow cut comparison, cut comparison+degree per multiplication Longrightarrow complete-cover lower bound, safe ideal+finite linear duality Longrightarrow evaluation-span certificates, monotone ideal+affine input/output changes Longrightarrow calibration formulas, affine restriction+quadratic polar rank Longrightarrow projection-cost lower bound, orthogonal idempotents Longrightarrow gluing dimension, Boole…

Exact mathematical passage


\begin{gathered}
\text{restored oracle convention}\ \Longrightarrow\ \text{cut comparison},\\
\text{cut comparison}+\text{degree per multiplication}\
\Longrightarrow\ \text{complete-cover lower bound},\\
\text{safe ideal}+\text{finite linear duality}\
\Longrightarrow\ \text{evaluation-span certificates},\\
\text{monotone ideal}+\text{affine input/output changes}\
\Longrightarrow\ \text{calibration formulas},\\
\text{affine restriction}+\text{quadratic polar rank}\
\Longrightarrow\ \text{projection-cost lower bound},\\
\text{orthogonal idempotents}\ \Longrightarrow\ \text{gluing dimension},\\
\text{Boolean coefficient identity}+\text{root bound}\
\Longrightarrow\ \text{conditional fingerprint test}.
\end{gathered}

\lorUse in source · 3 passages

Search aliases: lor

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Use in source · Read passage · c_main.tex, line 8
rScope, notation, and computational model Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies alor b=a+b+ab. A statement h≤ f is pointwise and means that h has no positive value at a negative input of f.
Exact mathematical passage
```
\lor
```

m^jUse in source · 1 passage

Search aliases: m^j

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Use in source · Read passage · g_main.tex, line 2945
[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^r+120 ≤ frac2120n. Also r≥log_21n-1. Therefore S≥ nlog_21n-frac4120n. Multiplying by 8/21 yields D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n.
Exact mathematical passage
```
n=\sum_{j=0}^{r}d_jm^j,
\qquad
0\le d_j<m.
```

m^NUse in source · 1 passage

Search aliases: m^N

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Use in source · Read passage · s_main.tex, line 1296
Bosonic occupation space. For N indistinguishable bosons in m modes, a Fock basis state is an occupation vector (n_1,…,n_m), n_1+⋯+n_m=N. The dimension is D_rm boson(m,N) = binomN+m-1N. For fixed N, this is O(m^N).
Exact mathematical passage
```
O(m^N)
```

M^{N-1}Use in source · 1 passage

Search aliases: M^N-1

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Use in source · Read passage · g_main.tex, line 2572
The same counting argument also gives Prob_A∈U_n(F_q)[ellAll(A)=ellDown(A)=N] ≥ 1-M^N-1q^-1, whenever the right side is positive. Thus maximal ambient and intrinsic length is not only Zariski generic over infinite fields but asymptotically typical over growing finite fields at fixed dimension.
Exact mathematical passage
```
\Prob_{A\in\U_n(\F_q)}[\ellAll(A)=\ellDown(A)=N]
\ge
1-M^{N-1}q^{-1},
```

M^{N-d}Use in source · 2 passages

Search aliases: M^N-d

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Use in source · Read passage · g_main.tex, line 1085
[Finite-field sparsity] Let N=binom n2 and M=n(n-1). For 1≤ d≤ N, Prob_A∈U_n(F_q)[DeltaDef(A)≥ d] ≤ min{1,M^N-dq^-d}. For fixed n, a uniformly random target therefore has defect zero with probability 1-O_n(q^-1) as q→∞.
Exact mathematical passage
```
\Prob_{A\in\U_n(\F_q)}[\DeltaDef(A)\ge d]
\le
\min\{1,M^{N-d}q^{-d}\}.
```

m^rUse in source · 5 passages

Search aliases: m^r

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Use in source · Read passage · g_main.tex, line 1096
[Tensor-amplified snapshot family] Let A=A(u;s_1,…,s_v) have dimension m=u+v+2. For N=m^r, ellDown(A^⊗ r) =r m^r-1(u+2v), while ellAll(A^⊗ r) ≤ r m^r. Consequently, ellDown(A^⊗ r)-ellAll(A^⊗ r) ≥ fracv-2mlog_2mNlog_2N.
Exact mathematical passage
```
N=m^r
```

m^sUse in source · 1 passage

Search aliases: m^s

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Use in source · Read passage · g_main.tex, line 5207
Template XI: stable capacity. If a_r+s≤ m^s a_r+m^r a_s, then b_r=fraca_rm^r-1 is subadditive and lim_r→∞fraca_rr m^r-1 exists by Fekete's lemma. This defines a regularized ambient cost and exposes possible tensor activation.
Exact mathematical passage
```
a_{r+s}\le m^s a_r+m^r a_s,
```

M^TUse in source · 1 passage

Search aliases: M^T

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Use in source · Read passage · g_appendix.tex, line 73
I_n · 0 over F_2. A matrix M is symplectic when M^TJM=J.
Exact mathematical passage
```
M^TJM=J.
```

m^\alphaUse in source · 1 passage

Search aliases: m^α alpha

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Use in source · Read passage · s_main.tex, line 1460
[Few-particle scaling] With the number of modes as the input-size parameter: item fixed N gives polynomial explicit-amplitude simulation; item N=O(log m) gives quasipolynomial sector size; item N=O(m^α), 0<α<1, gives subexponential sector size; item finite-density fermionic sectors remain exponential. For bosons, the statements hold in the dilute regime with the same occupation-basis access assumptions.
Exact mathematical passage
```
N=O(m^\alpha)
```

M_0Use in source · 1 passage

Search aliases: M_0

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Use in source · Read passage · g_main.tex, line 3607
ubit orbit and exact costs] Each of the four distinct matrices M_D lies in P_2 and has restricted entangling cost three and ambient entangling cost two. They form one orbit under left multiplication by the free restricted subgroup generated by S_0,S_1. The ambient sequence H_1, CX_1→0, CX_0→1, H_0 has matrix M_0=(J_2 · 0 I_2 · J_2). Applying S_0^d_0S_1^d_1 afterwards left multiplies by (I_2 · 0
Exact mathematical passage
```
M_0=\left(\begin{smallmatrix}J_2&0\\I_2&J_2\end{smallmatrix}\right)
```

M_2Use in source · 3 passages

Search aliases: M_2

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Use in source · Read passage · g_appendix.tex, line 107
Two-qubit Clifford witness. The target symplectic matrix is M_2= 0 · 1 · 0 · 0 1 · 0 · 0 · 0 1 · 1 · 0 · 1 1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.
Exact mathematical passage
```
M_2=
\begin{pmatrix}
0&1&0&0\\
1&0&0&0\\
1&1&0&1\\
1&1&1&0
\end{pmatrix}.
```

M_3Use in source · 2 passages

Search aliases: M_3

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Use in source · Read passage · g_appendix.tex, line 128
Three-qubit Clifford witness. The target symplectic matrix is M_3= 0 · 1 · 0 · 0 · 0 · 0 0 · 0 · 1 · 0 · 0 · 0 1 · 0 · 0 · 0 · 0 · 0 1 · 0 · 1 · 0 · 1 · 0 0 · 1 · 0 · 0 · 0 · 1 0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.
Exact mathematical passage
```
M_3=
\begin{pmatrix}
0&1&0&0&0&0\\
0&0&1&0&0&0\\
1&0&0&0&0&0\\
1&0&1&0&1&0\\
0&1&0&0&0&1\\
0&1&0&1&0&0
\end{pmatrix}.
```

M_DUse in source · 7 passages

Search aliases: M_D

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Use in source · Read passage · g_main.tex, line 3595
1 · 0, D=diag(d_0,d_1), d_0,d_1∈F_2, and define M_D=J_2 · 0 I_2+DJ_2 · J_2.
Exact mathematical passage
```
M_D=\begin{pmatrix}J_2&0\\I_2+DJ_2&J_2\end{pmatrix}.
```

m_iUse in source · 2 passages

Search aliases: m_i

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Use in source · Read passage · g_main.tex, line 2615
_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in every factor. If m_i gates survive in factor i, then the total word length is at least ∑_i m_i ≥ ∑_iellDown(A_i).
Exact mathematical passage
```
m_i
```

M_{I_2}Use in source · 1 passage

Search aliases: M_I_2

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Use in source · Read passage · g_main.tex, line 3607
D · I_2) and gives M_D. These matrices are distinct because J_2 is invertible. This also proves their orbit statement and the ambient upper bound two. The displayed restricted sequence for M_2 above gives M_I_2 using three CNOT gates and one free S_1. Free final phase gates then give every M_D, proving the restricted upper bound three.
Exact mathematical passage
```
M_{I_2}
```

M_tUse in source · 4 passages

Search aliases: M_t

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Use in source · Read passage · s_main.tex, line 450
Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.
Exact mathematical passage
```
M_t=B_t^*G_tB_{t-1}.
\label{s:eq:dynamic-coordinate-update}
```

M_\lambdaUse in source · 2 passages

Search aliases: M_λ lambda

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Use in source · Read passage · s_main.tex, line 1132
A permutation-invariant unitary has the form U = ⊕_λ U_λ⊗ I_ V_λ, and a permutation-invariant measurement operator has the analogous form M = ⊕_λ M_λ⊗ I_ V_λ. The output probability is Tr(MUρ U^*) = ∑_λ p_λ Tr(M_λ U_λρ_λ U_λ^*).
Exact mathematical passage
```
M
=
\bigoplus_\lambda
M_\lambda\otimes I_{\mathcal V_\lambda}.
```

\mathbinUse in source · 2 passages

Search aliases: mathbin

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Use in source · Read passage · c_lean.tex, line 119
pageheadFrom the cut to a numerical lower boundEXISTENCE, BALANCE, AND COMPLETE COVERAGE Write c=pmathbin++q. At this cut, Block 01 defines H_i(x)=run(p)(x,0^a,0)_ix run(q^-1)(x,0^a,0)_i and proves H_i≤ f, f(x)=1iff∃ i H_i(x)=1. Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives deg(H_i)≤max{TC(p),TC(q)}+1.
Exact mathematical passage
```
c=p\mathbin{++}q
```

\maxUse in source · 24 passages

Search aliases: max

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Use in source · Read passage · c_lean.tex, line 41
Degree and complete-cover bound Deg(T+1,f) and 2β^*(f)≤ T+3. pathrefCircuitObstructions.restored_degree_and_cover_bound The first conclusion is an explicit certificate that f has a Boolean polynomial representation of degree at most T+1. The second is the affine-optimized complete safe-cover lower bound, equivalently T≥max{0,2β^*(f)-3}.
Exact mathematical passage
```
T\ge\max\{0,2\beta^*(f)-3\}
```

\max_{q\ne r}Use in source · 1 passage

Search aliases: max_q≠ r max ne

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Use in source · Read passage · g_main.tex, line 4348
Sector-flow invariants. Let P_q be the projector onto sector q. For an intermediate unitary U_t, define transfer blocks T_q,r(t)=P_q U_t P_r. A restricted path has T_q,r=0 for q≠ r. Ambient paths may populate off-diagonal blocks. Possible excursion measures include ∑_q≠ rrank T_q,r, max_q≠ r T_q,r , bandwidth in |q-r|. These are analogues of strict-upper rank and bandwidth in the unitriangular theory.
Exact mathematical passage
```
\sum_{q\ne r}\rank T_{q,r},
\qquad
\max_{q\ne r}\|T_{q,r}\|,
\qquad
\text{bandwidth in }|q-r|.
```

\max_tUse in source · 2 passages

Search aliases: max_t max

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Use in source · Read passage · g_main.tex, line 353
Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.
Exact mathematical passage
```
w_{\mathrm{cat}}=
\max_t
\#\{j:t\in I_j\}.
```

\max_{U}Use in source · 1 passage

Search aliases: max_U max

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Use in source · Read passage · g_main.tex, line 359
Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.
Exact mathematical passage
```
D_n^{(w)}
=
\max_{U}
\left(
\ellRes(U)-\ellAmb^{(w)}(U)
\right),
```

\max_{v:f(v)=1}Use in source · 2 passages

Search aliases: max_v:f(v)=1 max

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Use in source · Read passage · c_main.tex, line 143
For a positive input v, define instead β_v(f)=min{deg h:h∈ I_f, h(v)=1}, β(f)=max_v:f(v)=1β_v(f). Set β(0)=0. Thus β(f) is the degree needed before safe polynomials collectively detect every positive input. Finally, put β^*(f)=max_ℓ∈Aff_nβ(f+ℓ), where Aff_n is the set of affine Boolean functions on the logical inputs.
Exact mathematical passage
```
 \beta_v(f)=\min\{\deg h:h\in I_f,\ h(v)=1\},\qquad
 \beta(f)=\max_{v:f(v)=1}\beta_v(f).
```

\max_\ellUse in source · 3 passages

Search aliases: max_ℓ max ell

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Use in source · Read passage · c_main.tex, line 339
@rrrrr@ Inputs n · max_ℓα_1 · β^* · One-packet bound · Complete-cover bound
Exact mathematical passage
```
\max_\ell\alpha_1
```

\max_{\ell}Use in source · 1 passage

Search aliases: max_ℓ max ell

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Use in source · Read passage · c_main.tex, line 334
The preceding combined bound max{0,deg F_k-1,2max_ℓα_1(F_k+ℓ)-3} gives n-1 for this family; its immunity-only term is the constant 3. The complete-cover bound gives 2n-5. The stronger estimate can also be recognized by restriction to b=1 followed by a free affine output adjustment. No exact optimum is claimed when the displayed endpoints differ.
Exact mathematical passage
```
\max\{0,\deg F_k-1,2\max_{\ell}\alpha_1(F_k+\ell)-3\}
```

\max_{\ell\text{ affine}}Use in source · 1 passage

Search aliases: max_ℓ affine max ell text

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Use in source · Read passage · c_visual.tex, line 276
An affine output mask ℓ(x) costs zero additional Toffolis: append CNOTs from restored inputs and, if needed, a target NOT. This constructs a restored oracle for f+ℓ, so the bound applies to every affine mask. Define β^*(f)=max_ℓ affineβ(f+ℓ). Together with the output degree bound, T≥max{0,deg f-1,2β^*(f)-3}.
Exact mathematical passage
```
 \beta^*(f)=\max_{\ell\text{ affine}}\beta(f+\ell).
```

\max_{\ell\in\Aff_n}Use in source · 2 passages

Search aliases: max_ℓ∈Aff_n max ell in Aff

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Use in source · Read passage · c_main.tex, line 148
For a positive input v, define instead β_v(f)=min{deg h:h∈ I_f, h(v)=1}, β(f)=max_v:f(v)=1β_v(f). Set β(0)=0. Thus β(f) is the degree needed before safe polynomials collectively detect every positive input. Finally, put β^*(f)=max_ℓ∈Aff_nβ(f+ℓ), where Aff_n is the set of affine Boolean functions on the logical inputs.
Exact mathematical passage
```
 \beta^*(f)=\max_{\ell\in\Aff_n}\beta(f+\ell),
```

\mathrm{metric}Use in source · 1 passage

Search aliases: metric mathrm

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Use in source · Read passage · g_main.tex, line 148
Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.
Exact mathematical passage
```
\tau_{\mathrm{metric}}&=\min\{n:\exists U,\ 0<\Delta(U)<\infty\},\\
\tau_{\mathrm{exist}}&=\min\{n:\exists U,\ \ellRes(U)=\infty,\ \ellAmb(U)<\infty\},\\
\tau_{q}&=\min\{n:\exists U,\ \ell_{\le q}(U)<\ell_{\le0}(U)\},
```

\midUse in source · 22 passages

Search aliases: mid

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Use in source · Read passage · g_main.tex, line 3387
Binary symplectic representation. Modulo global phase and Pauli translations, an n-qubit Clifford operation acts on Pauli exponent vectors v=(x_0,…,x_n-1mid z_0,…,z_n-1)^T∈F_2^2n through a symplectic matrix M∈Sp(2n,2). The elementary gates used here act as follows: H_i · : (x_i,z_i)↦(z_i,x_i),
Exact mathematical passage
```
v=(x_0,\ldots,x_{n-1}\mid z_0,\ldots,z_{n-1})^T\in\F_2^{2n}
```

\min_{c\in\mathcal C_{\mathrm{amb}}(U)}Use in source · 1 passage

Search aliases: min_c∈ C_amb(U) min in mathcal mathrm

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Use in source · Read passage · g_main.tex, line 4006
Scalarization by positive weights w gives Δ_w(U)= min_c∈ C_res(U)w· c - min_c∈ C_amb(U)w· c. Different w can change the magnitude of the defect and whether it is zero or positive. It remains nonnegative because every restricted realization is also an ambient realization with the same resource vector.
Exact mathematical passage
```
\Delta_w(U)=
\min_{c\in\mathcal C_{\mathrm{res}}(U)}w\cdot c
-
\min_{c\in\mathcal C_{\mathrm{amb}}(U)}w\cdot c.
```

\min_{c\in\mathcal C_{\mathrm{res}}(U)}Use in source · 1 passage

Search aliases: min_c∈ C_res(U) min in mathcal mathrm

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Use in source · Read passage · g_main.tex, line 4006
Scalarization by positive weights w gives Δ_w(U)= min_c∈ C_res(U)w· c - min_c∈ C_amb(U)w· c. Different w can change the magnitude of the defect and whether it is zero or positive. It remains nonnegative because every restricted realization is also an ambient realization with the same resource vector.
Exact mathematical passage
```
\Delta_w(U)=
\min_{c\in\mathcal C_{\mathrm{res}}(U)}w\cdot c
-
\min_{c\in\mathcal C_{\mathrm{amb}}(U)}w\cdot c.
```

\NUse in source · 1 passage

Search aliases: N

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Use in source · Read passage · g_main.tex, line 4979
Multiobjective search. Let each gate have a vector cost in N^r. At each state, retain only nondominated labels. The algorithm terminates for finite state spaces with a bounded search region, but the frontier can grow exponentially.
Exact mathematical passage
```
\N^r
```

N^{-1/2}Use in source · 1 passage

Search aliases: N^-1/2

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Use in source · Read passage · q_main.tex, line 411
Let A be the cyclic shift (Ax)(n)=x(n-1). A Fourier column f_a(n)=N^-1/2ζ_N^na satisfies Af_a=ζ_N^-af_a. Therefore
Exact mathematical passage
```
f_a(n)=N^{-1/2}\zeta_N^{na}
```

N^2Use in source · 1 passage

Search aliases: N^2

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Use in source · Read passage · s_main.tex, line 1889
Forrelation amplitude. Let f,g:{0,1}^n→{-1,1} and N=2^n. The circuit with alternating Hadamards and phase oracles has zero-state amplitude Φ(f,g) = frac1N^3/2 ∑_x,y f(x)(-1)^x· yg(y). The circuit does not evaluate the N^2 terms separately. It preserves phases and lets them interfere.
Exact mathematical passage
```
N^2
```

n^2Use in source · 5 passages

Search aliases: n^2

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Use in source · Read passage · g_main.tex, line 3031
[Correct growth of the maximum defect] Is D_n(F) of order nlog n, or can interacting catalysts produce a larger power of n? The current bounds are Ω(nlog n) ≤ D_n(F) ≤ O(n^2).
Exact mathematical passage
```
\Omega(n\log n)
\le
D_n(\F)
\le
O(n^2).
```

n^{20}Use in source · 1 passage

Search aliases: n^20

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Use in source · Read passage · s_main.tex, line 1496
Complexity class versus practical runtime. A polynomial sector can still have a large exponent. Fixed k=20 gives O(n^20) explicit amplitudes, which is in P but may be impractical. Conversely, a subexponential sector may be useful at moderate size without changing the formal complexity class.
Exact mathematical passage
```
O(n^{20})
```

n^3Use in source · 3 passages

Search aliases: n^3

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Use in source · Read passage · s_main.tex, line 1197
Qubit corollary. For qubits, Schur labels are total spins j. The carrier dimension is u_j=2j+1≤ n+1, and there are at most lfloor n/2rfloor+1 labels. Hence ∑_j(2j+1)^2=O(n^3).
Exact mathematical passage
```
\sum_j(2j+1)^2=O(n^3).
\label{s:eq:qubit-commutant-size}
```

N^{3/2}Use in source · 1 passage

Search aliases: N^3/2

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Use in source · Read passage · s_main.tex, line 1881
Forrelation amplitude. Let f,g:{0,1}^n→{-1,1} and N=2^n. The circuit with alternating Hadamards and phase oracles has zero-state amplitude Φ(f,g) = frac1N^3/2 ∑_x,y f(x)(-1)^x· yg(y). The circuit does not evaluate the N^2 terms separately. It preserves phases and lets them interfere.
Exact mathematical passage
```
\Phi(f,g)
=
\frac1{N^{3/2}}
\sum_{x,y}
 f(x)(-1)^{x\cdot y}g(y).
\label{s:eq:forrelation}
```

n^{d(d-1)/2}Use in source · 1 passage

Search aliases: n^d(d-1)/2

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Use in source · Read passage · s_main.tex, line 1157
Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).
Exact mathematical passage
```
u_\lambda
=
\prod_{1\le i<j\le d}
\frac{\lambda_i-\lambda_j+j-i}{j-i}
=O_d\left(n^{d(d-1)/2}\right).
\label{s:eq:weyl-dimension-bound}
```

n^{d-1}Use in source · 2 passages

Search aliases: n^d-1

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Use in source · Read passage · s_main.tex, line 1150
Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).
Exact mathematical passage
```
\#\{\lambda:\ell(\lambda)\le d\}
\le
\binom{n+d-1}{d-1}
=O_d(n^{d-1}).
```

n^{d^2-1}Use in source · 1 passage

Search aliases: n^d^2-1

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Use in source · Read passage · s_main.tex, line 1166
umber of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).
Exact mathematical passage
```
\sum_\lambda u_\lambda^2
=O_d\left(n^{d^2-1}\right).
\label{s:eq:commutant-parameter-bound}
```

n^KUse in source · 2 passages

Search aliases: n^K

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Use in source · Read passage · s_main.tex, line 589
The active dimension is ∑_k=0^Kbinom nk =O(n^K). Let P project onto Hh_≤ K. By the hypothesis on the actual circuit prefixes, Pψ_t=ψ_t at every step. Consequently PG_tPψ_t-1=G_tψ_t-1 on the evolving state, even if the gate does not preserve all of PHh. The restricted update therefore reproduces that trajectory exactly. The hypothesis supplies efficient initialization, updates, and measurement in these polynomially many coordinates.
Exact mathematical passage
```
\sum_{k=0}^{K}\binom nk
=O(n^K).
```

n^kUse in source · 1 passage

Search aliases: n^k

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Use in source · Read passage · s_main.tex, line 767
For fixed k, dimHh_n,k=binom nk=O(n^k). The weight-k basis is efficiently indexed by k-element subsets. A local number-preserving gate changes only basis states containing the affected occupation pattern. Apply [source reference].
Exact mathematical passage
```
\dim\Hh_{n,k}=\binom nk=O(n^k).
```

\N^rUse in source · 1 passage

Search aliases: N^r N

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Use in source · Read passage · g_main.tex, line 4979
Multiobjective search. Let each gate have a vector cost in N^r. At each state, retain only nondominated labels. The algorithm terminates for finite state spaces with a bounded search region, but the frontier can grow exponentially.
Exact mathematical passage
```
\N^r
```

n^rUse in source · 1 passage

Search aliases: n^r

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Use in source · Read passage · s_main.tex, line 1402
The upper bounds follow from [source reference]. If r=O(1), then binom nr=O(n^r). Conversely, suppose r→∞ and r≤ n/2. If r≤sqrt n, then logbinom nr ≥ rlog(n/r) ≥ frac r2log n, whose ratio to log n diverges. If r>sqrt n, then logbinom nr ≥ rlog2 ≥ sqrt nlog2, which also dominates every constant multiple of log n. For the final assertion, p_n=k(n)/n stays in a compact subinterval of (0,1). Stirling's formula gives log_2binom nk(n) =nH_2(p_n)-frac12log_2 (2π n p_n(1-p_n))+O(1/n). The logarithmic te…
Exact mathematical passage
```
\binom nr=O(n^r)
```

n^\alphaUse in source · 2 passages

Search aliases: n^α alpha

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1384
d-charge phase diagram] For explicit amplitude storage in a weight-k(n) packet: item the packet dimension is polynomial in n if r(n)=O(1); item the packet dimension is superpolynomial whenever r(n)→∞; item if r(n)=O(log n), then D_n,k=2^O((log n)^2), so the explicit packet simulation is quasipolynomial; item if r(n)=O(n^α) for a fixed 0<α<1, then D_n,k=2^O(n^αlog n), which is subexponential; item if k(n)/n→ p∈(0,1), then log_2 D_n,k = nH_2(k(n)/n)+O(log n) = nH_2(p)+o(n), where H_2 is binary ent…
Exact mathematical passage
```
r(n)=O(n^\alpha)
```

n_0Use in source · 3 passages

Search aliases: n_0

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Use in source · Read passage · g_main.tex, line 5253
Template XV: exhaustive threshold classification. A finite first-threshold theorem should contain: item no defects below size n_0; item at least one defect at n_0; item exact classification of every defect at n_0 modulo declared equivalence; item analytic generation of the exceptional motifs; item equality between the analytic set and computational census; item witness and completeness certificates. This is the standard achieved by the six-wire linear core and is the target standard for future g…
Exact mathematical passage
```
n_0
```

n_1Use in source · 4 passages

Search aliases: n_1

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Use in source · Read passage · g_main.tex, line 2611
For the lower bound, view an element of U_n_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in every factor. If m_i gates survive in factor i, then the total word length is at least ∑_i m_i ≥ ∑_iellDow…
Exact mathematical passage
```
\U_{n_1+\cdots+n_k}(\F)
```

N_{1,3}Use in source · 1 passage

Search aliases: N_1,3

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Use in source · Read passage · g_main.tex, line 1598
[Motif count] For a split (L,R)=(|L|,|R|) with L+R=4, the number of minimal phase-separated targets is N_L,R =L(2^R-2)+R(2^L-2). For the three possible splits, N_1,3=6, N_2,2=8, N_3,1=6. Hence there are twenty targets.
Exact mathematical passage
```
N_{1,3}=6,
\qquad
N_{2,2}=8,
\qquad
N_{3,1}=6.
```

N_{2,2}Use in source · 1 passage

Search aliases: N_2,2

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Use in source · Read passage · g_main.tex, line 1598
[Motif count] For a split (L,R)=(|L|,|R|) with L+R=4, the number of minimal phase-separated targets is N_L,R =L(2^R-2)+R(2^L-2). For the three possible splits, N_1,3=6, N_2,2=8, N_3,1=6. Hence there are twenty targets.
Exact mathematical passage
```
N_{1,3}=6,
\qquad
N_{2,2}=8,
\qquad
N_{3,1}=6.
```

N_{3,1}Use in source · 1 passage

Search aliases: N_3,1

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Use in source · Read passage · g_main.tex, line 1598
[Motif count] For a split (L,R)=(|L|,|R|) with L+R=4, the number of minimal phase-separated targets is N_L,R =L(2^R-2)+R(2^L-2). For the three possible splits, N_1,3=6, N_2,2=8, N_3,1=6. Hence there are twenty targets.
Exact mathematical passage
```
N_{1,3}=6,
\qquad
N_{2,2}=8,
\qquad
N_{3,1}=6.
```

n_dUse in source · 2 passages

Search aliases: n_d

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Use in source · Read passage · s_appendix.tex, line 52
Symmetric-subspace dimension. A symmetric basis state of n qudits is indexed by occupation numbers (n_1,…,n_d), n_1+⋯+n_d=n. The stars-and-bars count is binomn+d-1d-1, proving [source reference].
Exact mathematical passage
```
(n_1,\ldots,n_d),
\qquad
n_1+\cdots+n_d=n.
```

n_iUse in source · 1 passage

Search aliases: n_i

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Use in source · Read passage · g_main.tex, line 2600
[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).
Exact mathematical passage
```
A_i\in\U_{n_i}(\F)
```

n_kUse in source · 1 passage

Search aliases: n_k

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Use in source · Read passage · g_main.tex, line 2611
For the lower bound, view an element of U_n_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in every factor. If m_i gates survive in factor i, then the total word length is at least ∑_i m_i ≥ ∑_iellDow…
Exact mathematical passage
```
\U_{n_1+\cdots+n_k}(\F)
```

N_{L,R}Use in source · 1 passage

Search aliases: N_L,R

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Use in source · Read passage · g_main.tex, line 1593
[Motif count] For a split (L,R)=(|L|,|R|) with L+R=4, the number of minimal phase-separated targets is N_L,R =L(2^R-2)+R(2^L-2). For the three possible splits, N_1,3=6, N_2,2=8, N_3,1=6. Hence there are twenty targets.
Exact mathematical passage
```
N_{L,R}
=L(2^R-2)+R(2^L-2).
```

n_mUse in source · 1 passage

Search aliases: n_m

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Use in source · Read passage · s_main.tex, line 1284
Bosonic occupation space. For N indistinguishable bosons in m modes, a Fock basis state is an occupation vector (n_1,…,n_m), n_1+⋯+n_m=N. The dimension is D_rm boson(m,N) = binomN+m-1N. For fixed N, this is O(m^N).
Exact mathematical passage
```
(n_1,\ldots,n_m),
\qquad
n_1+\cdots+n_m=N.
```

N_{\mathrm{packets}}Use in source · 2 passages

Search aliases: N_packets mathrm

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Use in source · Read passage · s_main.tex, line 33
The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:
Exact mathematical passage
```
\mathfrak C_n=
\bigl(
N_{\mathrm{packets}},
D_{\mathrm{internal}},
S_{\mathrm{transfer}},
L_{\mathrm{leakage}}
\bigr).
```

N_{\rm packets}Use in source · 1 passage

Search aliases: N_rm packets rm

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Use in source · Read passage · s_main.tex, line 2273
The core research invariant is the complete classical representation record ( discovery cost, N_rm packets, D_rm internal, S_rm transfer, L_rm leakage, measurement cost ). A future compiler that computes and certifies this record can decide when the full state vector is unnecessary, when a hybrid representation is preferable, and when symmetry alone is insufficient.
Exact mathematical passage
```
\bigl(
\text{discovery cost},
N_{\rm packets},
D_{\rm internal},
S_{\rm transfer},
L_{\rm leakage},
\text{measurement cost}
\bigr).
```

\nmidUse in source · 1 passage

Search aliases: nmid

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Use in source · Read passage · q_main.tex, line 161
∑_n=0^q-1ζ_q^hn =q, · qmid h, 0, · qnmid h.
Exact mathematical passage
```
\sum_{n=0}^{q-1}\zeta_q^{hn}
=\begin{cases}q,&q\mid h,\\0,&q\nmid h.\end{cases}
```

OUse in source · 50 passages

Search aliases: O

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Use in source · Read passage · c_main.tex, line 482
These are exact cost formulas. The large-input entries were not obtained by searching for optimal circuits. The full construction uses 2m(2^m-1-1) CNOTs as well as 2m-3 Toffolis, giving total count O(nlog n). The projected quadratic has n/2 total gates.
Exact mathematical passage
```
O(n\log n)
```

\mathcal OUse in source · 3 passages

Search aliases: O mathcal

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Use in source · Read passage · s_main.tex, line 1226
Orbit states. Let a group G act by automorphisms on a finite graph with vertex set V. The orbits are O_1,…, O_r. Define normalized orbit states ket O_a = frac1sqrt| O_a| ∑_x∈ O_aket x. If the adjacency or Hamiltonian H commutes with the group action, the span of these states is invariant whenever the partition is equitable.
Exact mathematical passage
```
\mathcal O_1,\ldots,\mathcal O_r
```

oUse in source · 9 passages

Search aliases: o

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Use in source · Read passage · c_main.tex, line 105
d oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with target z, correctness of the full circuit gives R(x,z)=P(x,z+f(x)). When f(x)=0, this says B(x)=A(x). When f(x)=1, it says B(x)=P(x,1), and the discr…
Exact mathematical passage
```
\label{c:eq:cover}
 H_i\le f\quad\hbox{for every }i,\qquad
 \bigvee_iH_i=f.
```

O^TUse in source · 2 passages

Search aliases: O^T

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Use in source · Read passage · g_main.tex, line 4221
Covariance-matrix lower bounds. For a Gaussian state, the covariance matrix Γ transforms as Γ⟼ SΓ S^T in the bosonic case and Γ⟼ OΓ O^T in the fermionic case. Possible restricted lower bounds include: item rank change of off-diagonal covariance blocks, item symplectic singular-value obligations, item growth of mode support, item graph bandwidth and cut rank, item number of nontrivial canonical angles between invariant subspaces. One elementary gate changes only a controlled portion of these data…
Exact mathematical passage
```
\Gamma\longmapsto O\Gamma O^T
```

\mathcal O_1Use in source · 1 passage

Search aliases: O_1 mathcal

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Use in source · Read passage · s_main.tex, line 1226
Orbit states. Let a group G act by automorphisms on a finite graph with vertex set V. The orbits are O_1,…, O_r. Define normalized orbit states ket O_a = frac1sqrt| O_a| ∑_x∈ O_aket x. If the adjacency or Hamiltonian H commutes with the group action, the span of these states is invariant whenever the partition is equitable.
Exact mathematical passage
```
\mathcal O_1,\ldots,\mathcal O_r
```

\mathcal O_aUse in source · 2 passages

Search aliases: O_a mathcal

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Use in source · Read passage · s_main.tex, line 1227
Orbit states. Let a group G act by automorphisms on a finite graph with vertex set V. The orbits are O_1,…, O_r. Define normalized orbit states ket O_a = frac1sqrt| O_a| ∑_x∈ O_aket x. If the adjacency or Hamiltonian H commutes with the group action, the span of these states is invariant whenever the partition is equitable.
Exact mathematical passage
```
\ket{\mathcal O_a}
=
\frac1{\sqrt{|\mathcal O_a|}}
\sum_{x\in\mathcal O_a}\ket x.
```

\mathcal O_bUse in source · 1 passage

Search aliases: O_b mathcal

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Use in source · Read passage · s_main.tex, line 1236
The quotient matrix is widetilde H_ab = bra O_aHket O_b.
Exact mathematical passage
```
\widetilde H_{ab}
=
\bra{\mathcal O_a}H\ket{\mathcal O_b}.
```

O_dUse in source · 3 passages

Search aliases: O_d

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Use in source · Read passage · s_main.tex, line 1150
Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).
Exact mathematical passage
```
\#\{\lambda:\ell(\lambda)\le d\}
\le
\binom{n+d-1}{d-1}
=O_d(n^{d-1}).
```

O_nUse in source · 2 passages

Search aliases: O_n

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Use in source · Read passage · g_main.tex, line 1090
[Finite-field sparsity] Let N=binom n2 and M=n(n-1). For 1≤ d≤ N, Prob_A∈U_n(F_q)[DeltaDef(A)≥ d] ≤ min{1,M^N-dq^-d}. For fixed n, a uniformly random target therefore has defect zero with probability 1-O_n(q^-1) as q→∞.
Exact mathematical passage
```
1-O_n(q^{-1})
```

O_{N,M}Use in source · 1 passage

Search aliases: O_N,M

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Use in source · Read passage · g_main.tex, line 600
Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estimate can be sharpened by orbit counting, stabilizers, and the geometry of word-map fibers.
Exact mathematical passage
```
\Prob[\Delta\ge d]
=O_{N,M}(q^{-d}).
```

\mathcal O_rUse in source · 1 passage

Search aliases: O_r mathcal

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Use in source · Read passage · s_main.tex, line 1226
Orbit states. Let a group G act by automorphisms on a finite graph with vertex set V. The orbits are O_1,…, O_r. Define normalized orbit states ket O_a = frac1sqrt| O_a| ∑_x∈ O_aket x. If the adjacency or Hamiltonian H commutes with the group action, the span of these states is invariant whenever the partition is equitable.
Exact mathematical passage
```
\mathcal O_1,\ldots,\mathcal O_r
```

\mathrm{open}Use in source · 3 passages

Search aliases: open mathrm

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Use in source · Read passage · g_main.tex, line 343
Catalytic interval geometry. Associate to every matched opening and closing operation an interval I_j=[t_j^open,t_j^close]. The family of intervals can be disjoint, nested, laminar, or crossing.
Exact mathematical passage
```
I_j=[t_j^{\mathrm{open}},t_j^{\mathrm{close}}].
```

\mathrm{OR}Use in source · 2 passages

Search aliases: OR mathrm

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Use in source · Read passage · c_visual.tex, line 234
guidepageOne safe packet need not cover every accepting point For k=3 consider F_3(a,b,z_1,z_2,z_3)=ab OR z_1z_2z_3. The packet ab is safe and has degree two. It detects every teal point below. It misses three additional positives, which require the other packet z_1z_2z_3.
Exact mathematical passage
```
 F_3(a,b,z_1,z_2,z_3)=ab\ \mathrm{OR}\ z_1z_2z_3.
```

\operatorname{ord}Use in source · 1 passage

Search aliases: ord operatorname

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Use in source · Read passage · q_visual.tex, line 23
exact order twelve. Set ζ=e^2π i/12. The phase ζ^a first returns to 1 after 12/gcd(a,12) powers. Thus the coprime labels 1,5,7,11 select exactly the primitive twelfth roots. fig01_primitive_roots0.91The circle is the complex unit circle. Teal points are the four primitive roots; pale points are the other eight roots. Labels are residues a, not measured outcomes. gcd(a,12)=1 Longleftrightarrow ord(ζ^a)=12. Arithmetic has supplied a precise selector. Nothing probabilistic has happened at this stag…
Exact mathematical passage
```
 \gcd(a,12)=1\quad\Longleftrightarrow\quad\operatorname{ord}(\zeta^a)=12.
```

\ordUse in source · 5 passages

Search aliases: ord

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Use in source · Read passage · g_main.tex, line 468
[Endpoint Green valuation] The word distance is d_S(e,g) = ord_z=0 ⟨ δ_g,(I-zA_S)^-1δ_e ⟩. Expand the resolvent as a formal power series: (I-zA_S)^-1=∑_k≥0z^kA_S^k. The first nonzero endpoint coefficient occurs at the shortest word length.
Exact mathematical passage
```
d_S(e,g)
=
\ord_{z=0}
\left\langle
\delta_g,(I-zA_S)^{-1}\delta_e
\right\rangle.
```

\ord_0Use in source · 2 passages

Search aliases: ord_0 ord

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Use in source · Read passage · g_main.tex, line 1903
G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.
Exact mathematical passage
```
\ord_0\mathcal G_{\downarrow,C}
-
\ord_0\mathcal G_{*,C}
=1.
```

\ord_{z=0}Use in source · 3 passages

Search aliases: ord_z=0 ord

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Use in source · Read passage · g_main.tex, line 468
[Endpoint Green valuation] The word distance is d_S(e,g) = ord_z=0 ⟨ δ_g,(I-zA_S)^-1δ_e ⟩. Expand the resolvent as a formal power series: (I-zA_S)^-1=∑_k≥0z^kA_S^k. The first nonzero endpoint coefficient occurs at the shortest word length.
Exact mathematical passage
```
d_S(e,g)
=
\ord_{z=0}
\left\langle
\delta_g,(I-zA_S)^{-1}\delta_e
\right\rangle.
```

\overline{\mathcal D_{n,\ge d}}Use in source · 3 passages

Search aliases: overline D_n,≥ d overline mathcal ge

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Use in source · Read passage · g_main.tex, line 2508
[Defect--codimension bound] Let K be algebraically closed. For every 1≤ d≤ N, dimoverline D_n,≥ d≤ N-d. Equivalently, codim_U_noverline D_n,≥ d≥ d. More finely, on the stratum where ellDown(A)=L and DeltaDef(A)≥ d, the closure has dimension at most L-d.
Exact mathematical passage
```
\dim\overline{\mathcal D_{n,\ge d}}\le N-d.
```

\overline{\mathcal D_{n,\ge d}(\overline K)}Use in source · 1 passage

Search aliases: overline D_n,≥ d(overline K) overline mathcal ge overline

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Use in source · Read passage · g_main.tex, line 1077
-rigidity-intro Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline K overline D_n,≥ d(overline K)≥ d.
Exact mathematical passage
```
\codim_{(\U_n)_{\overline K}}
\overline{\mathcal D_{n,\ge d}(\overline K)}\ge d.
```

\overline{\mathcal D_{\ge d}}Use in source · 1 passage

Search aliases: overline D_≥ d overline mathcal ge

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Use in source · Read passage · g_main.tex, line 5143
Template VI: defect codimension. Assume every target has restricted cost at most M. A defect of size at least d implies ambient cost at most M-d. If the (M-d)-gate ambient locus has dimension at most N-d, then codimoverline D_≥ d≥ d. This is the general mechanism behind algebraic rarity.
Exact mathematical passage
```
\codim\overline{\mathcal D_{\ge d}}\ge d.
```

\overline KUse in source · 4 passages

Search aliases: overline K overline

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Use in source · Read passage · g_main.tex, line 1071
[Generic rigidity and defect codimension] Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline K overline D_n,≥ d(overline K)≥ d.
Exact mathematical passage
```
\overline K
```

\overline{\mathcal R^*_{n,\le L}}Use in source · 1 passage

Search aliases: overline R^*_n,≤ L overline mathcal le

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Use in source · Read passage · g_main.tex, line 2441
[Dimension of bounded-length loci] For every L≥0, the sets R^*_n,≤ L and R^downarrow_n,≤ L are constructible and dimoverline R^*_n,≤ L ≤ L, dimoverline R^downarrow_n,≤ L ≤ L.
Exact mathematical passage
```
\dim\overline{\mathcal R^*_{n,\le L}}
\le L,
\qquad
\dim\overline{\mathcal R^\downarrow_{n,\le L}}
\le L.
```

\overline{\mathcal R^*_{n,\le N-1}}Use in source · 1 passage

Search aliases: overline R^*_n,≤ N-1 overline mathcal le

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Use in source · Read passage · g_main.tex, line 2487
After base change to an algebraic closure overline K, [source reference] gives dimoverline R^*_n,≤ N-1≤ N-1<N=dimU_n. Because the word maps are defined over K, this proper closed set descends to a proper closed subset of U_n over K. Its complement is a nonempty Zariski-open set Ω_n. Since K is infinite and U_n≅Aff^N, the set Ω_n(K) is nonempty and Zariski dense. For any A∈Ω_n(K), one has ellAll(A)≥ N. The inequalities ellAll(A)≤ellDown(A)≤ N force equality. Replacing each image by its Zariski cl…
Exact mathematical passage
```
\dim\overline{\mathcal R^*_{n,\le N-1}}\le N-1<N=\dim\U_n.
```

\overline{\mathcal R^\downarrow_{n,\le L}}Use in source · 1 passage

Search aliases: overline R^downarrow_n,≤ L overline mathcal downarrow le

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Use in source · Read passage · g_main.tex, line 2441
[Dimension of bounded-length loci] For every L≥0, the sets R^*_n,≤ L and R^downarrow_n,≤ L are constructible and dimoverline R^*_n,≤ L ≤ L, dimoverline R^downarrow_n,≤ L ≤ L.
Exact mathematical passage
```
\dim\overline{\mathcal R^*_{n,\le L}}
\le L,
\qquad
\dim\overline{\mathcal R^\downarrow_{n,\le L}}
\le L.
```

\overline{V\cap X^{(L)}}Use in source · 1 passage

Search aliases: overlineV∩ X^(L) overline cap

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Use in source · Read passage · g_main.tex, line 2464
[Dimension per gate] Let G be an algebraic group, let V⊆ G be an irreducible subvariety of dimension D, and let X⊆ G be a finite union of locally closed subvarieties, each of dimension at most r, with the identity in X. Write X^(L) = {x_L⋯ x_1:x_1,…,x_L∈ X}. Then dimoverlineV∩ X^(L)≤ rL. In particular, if rL<D, a Zariski-generic point of V has X-product length greater than L.
Exact mathematical passage
```
\dim\overline{V\cap X^{(L)}}\le rL.
```

\overline{x(n)}Use in source · 1 passage

Search aliases: overlinex(n) overline

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Use in source · Read passage · q_main.tex, line 116
⟨ x,y⟩=∑_n=0^q-1overlinex(n)y(n), x ^2=⟨ x,x⟩.
Exact mathematical passage
```
\langle x,y\rangle=\sum_{n=0}^{q-1}\overline{x(n)}y(n),
\qquad \|x\|^2=\langle x,x\rangle.
```

\overline{X^L}Use in source · 2 passages

Search aliases: overlineX^L overline

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Use in source · Read passage · g_main.tex, line 700
Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but tensor-regularized defect positive, item ambient and restricted border ranks both smaller than exact ranks but by different amounts. Each requires a se…
Exact mathematical passage
```
\underline{\operatorname{rk}}_{X}(U)
=
\min\{L:U\in\overline{X^L}\}.
```

\overline{\{U:\Delta(U)\ge d\}}Use in source · 1 passage

Search aliases: overline{U:Δ(U)≥ d} overline Delta ge

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Use in source · Read passage · g_main.tex, line 582
Defect codimension. If every target has restricted cost at most N, then defect at least d forces ambient cost at most N-d. The dimension-per-gate theorem gives codim_Voverline{U:Δ(U)≥ d} ≥ d when each ambient gate carries one parameter.
Exact mathematical passage
```
\codim_V\overline{\{U:\Delta(U)\ge d\}}
\ge d
```

\overline{\chi(g)}Use in source · 1 passage

Search aliases: overlineχ(g) overline chi

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Use in source · Read passage · s_main.tex, line 158
Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).
Exact mathematical passage
```
P_\chi
=
\frac{d_\chi}{|G|}
\sum_{g\in G}
\overline{\chi(g)}U_g.
\label{s:eq:character-projector}
```

P^{(1)}_{\alpha_1}Use in source · 1 passage

Search aliases: P^(1)_α_1 alpha

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Use in source · Read passage · s_main.tex, line 182
Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential preprocessing merely relocates the complexity.
Exact mathematical passage
```
P_{\bm\alpha}
=
P^{(1)}_{\alpha_1}\cdots P^{(s)}_{\alpha_s}.
```

P^{(s)}_{\alpha_s}Use in source · 1 passage

Search aliases: P^(s)_α_s alpha

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Use in source · Read passage · s_main.tex, line 182
Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential preprocessing merely relocates the complexity.
Exact mathematical passage
```
P_{\bm\alpha}
=
P^{(1)}_{\alpha_1}\cdots P^{(s)}_{\alpha_s}.
```

P^2Use in source · 4 passages

Search aliases: P^2

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Use in source · Read passage · q_lean.tex, line 1119
projector12_idempotent · Exact rational multiplication gives P^2=P
Exact mathematical passage
```
P^2=P
```

P^\daggerUse in source · 4 passages

Search aliases: P^† dagger

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Use in source · Read passage · q_main.tex, line 754
P=FDF^†, P^2=P, P^†=P, rankP=trP=4.
Exact mathematical passage
```
P=FDF^\dagger,
\qquad P^2=P,
\qquad P^\dagger=P,
\qquad \operatorname{rank}P=\operatorname{tr}P=4.
```

P_{\mathcal D}Use in source · 1 passage

Search aliases: P_ D mathcal

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Use in source · Read passage · q_main.tex, line 445
P_ D=∑_d∈ DP_d^(N)
Exact mathematical passage
```
P_{\mathcal D}=\sum_{d\in\mathcal D}P_d^{(N)}
```

p_1Use in source · 2 passages

Search aliases: p_1

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Use in source · Read passage · g_appendix.tex, line 167
Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).
Exact mathematical passage
```
(q_1,\ldots,q_n,p_1,\ldots,p_n)
```

P_{10}Use in source · 1 passage

Search aliases: P_10

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Use in source · Read passage · c_visual.tex, line 483
visualfigureprojection_m10Two different resource comparisons at the same input length. The right panel reports the totals of the stated constructions, not an asserted optimal total for P_10.
Exact mathematical passage
```
P_{10}
```

P_{12}^{(12)}Use in source · 1 passage

Search aliases: P_12^(12)

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Use in source · Read passage · q_main.tex, line 735
Put P=P_12^(12). Its entries are
Exact mathematical passage
```
P=P_{12}^{(12)}
```

P_2Use in source · 2 passages

Search aliases: P_2

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Use in source · Read passage · g_main.tex, line 3600
[Two-qubit orbit and exact costs] Each of the four distinct matrices M_D lies in P_2 and has restricted entangling cost three and ambient entangling cost two. They form one orbit under left multiplication by the free restricted subgroup generated by S_0,S_1. The ambient sequence H_1, CX_1→0, CX_0→1, H_0 has matrix M_0=(J_2 · 0
Exact mathematical passage
```
P_2
```

P_3Use in source · 4 passages

Search aliases: P_3

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Use in source · Read passage · c_main.tex, line 462
[Projection does not preserve safe support] The quadratic extraction [P_m]_2 need not be supported on the positive set of P_m. Take m=3, identify the columns with the integers 1,…,6 using their three-bit expansions, and set x_1=x_3=x_4=1 with all other inputs zero. Their column sum is 1+3+4=6 under bitwise XOR, not a=7, so P_3=0. Among the complementary pairs (1,6),(2,5),(3,4), exactly (3,4) has both inputs one. Thus Q_3=1. In the source ordering this is the logical assignment encoded by integer…
Exact mathematical passage
```
P_3=0
```

P_6Use in source · 1 passage

Search aliases: P_6

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Use in source · Read passage · g_main.tex, line 4698
This is naturally expressed as Real_product(P_6)=∅, Real_entangled(P_6)≠∅. It is an existence defect, not a gate-count defect.
Exact mathematical passage
```
\operatorname{Real}_{\mathrm{product}}(P_6)=\varnothing,
\qquad
\operatorname{Real}_{\mathrm{entangled}}(P_6)\ne\varnothing.
```

P_{<r}Use in source · 2 passages

Search aliases: P_<r

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Use in source · Read passage · g_main.tex, line 781
C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
Q_{>r}e_{i_r}e_{j_r}^{T}P_{<r}
```

P_AUse in source · 1 passage

Search aliases: P_A

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Use in source · Read passage · c_main.tex, line 628
[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independent and the factorization exact.
Exact mathematical passage
```
 P_{A\land B}(z_u,z_v)=P_A(z_u)P_B(z_v).
```

P_{A\land B}Use in source · 1 passage

Search aliases: P_Aland B land

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Use in source · Read passage · c_main.tex, line 628
[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independent and the factorization exact.
Exact mathematical passage
```
 P_{A\land B}(z_u,z_v)=P_A(z_u)P_B(z_v).
```

P_BUse in source · 1 passage

Search aliases: P_B

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Use in source · Read passage · c_main.tex, line 628
[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independent and the factorization exact.
Exact mathematical passage
```
 P_{A\land B}(z_u,z_v)=P_A(z_u)P_B(z_v).
```

p_CUse in source · 2 passages

Search aliases: p_C

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Use in source · Read passage · s_main.tex, line 1942
A representation-relative invariant. For a declared classical representation class R, define TC^ R_ε(C) = min{ cost(R): R∈ R, |R(C)-p_C|≤ε }, where p_C is the target acceptance probability. This permits a disciplined ladder:
Exact mathematical passage
```
\operatorname{TC}^{\mathcal R}_\varepsilon(C)
=
\min\left\{
\operatorname{cost}(R):
R\in\mathcal R,
\ |R(C)-p_C|\le\varepsilon
\right\},
```

P_dUse in source · 3 passages

Search aliases: P_d

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Use in source · Read passage · q_visual.tex, line 128
six order classes leadThe primitive-twelve projector is one packet in a complete orthogonal decomposition. A frequency label k has root order d=12/gcd(k,12). The possible orders are the divisors 1,2,3,4,6,12. fig08_order_partition1.0The number inside each tile is its root order d. The horizontal label is frequency k. Every label belongs to one class; no labels are dropped. P_d_nm=fracc_d(n-m)12, P_dP_e=δ_deP_d, ∑_dmid12P_d=I. These are character-order projectors. They select Fourier modes by roo…
Exact mathematical passage
```
 P_d{}_{nm}=\frac{c_d(n-m)}{12},\quad
 P_dP_e=\delta_{de}P_d,\quad
 \sum_{d\mid12}P_d=I.
```

P_d^{(N)}Use in source · 6 passages

Search aliases: P_d^(N)

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Use in source · Read passage · q_main.tex, line 344
P_d^(N)=F_ND_dF_N^†.
Exact mathematical passage
```
P_d^{(N)}=F_ND_dF_N^\dagger.
```

P_eUse in source · 1 passage

Search aliases: P_e

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Use in source · Read passage · q_visual.tex, line 128
six order classes leadThe primitive-twelve projector is one packet in a complete orthogonal decomposition. A frequency label k has root order d=12/gcd(k,12). The possible orders are the divisors 1,2,3,4,6,12. fig08_order_partition1.0The number inside each tile is its root order d. The horizontal label is frequency k. Every label belongs to one class; no labels are dropped. P_d_nm=fracc_d(n-m)12, P_dP_e=δ_deP_d, ∑_dmid12P_d=I. These are character-order projectors. They select Fourier modes by roo…
Exact mathematical passage
```
 P_d{}_{nm}=\frac{c_d(n-m)}{12},\quad
 P_dP_e=\delta_{de}P_d,\quad
 \sum_{d\mid12}P_d=I.
```

P_e^{(N)}Use in source · 1 passage

Search aliases: P_e^(N)

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Use in source · Read passage · q_main.tex, line 363

(P_d^(N))^† · =P_d^(N), P_d^(N)P_e^(N) · =δ_deP_d^(N), ∑_dmid NP_d^(N) · =Id_N, rankP_d^(N) · =trP_d^(N)=φ(d).

Exact mathematical passage


\begin{aligned}
(P_d^{(N)})^\dagger&=P_d^{(N)},\\
P_d^{(N)}P_e^{(N)}&=\delta_{de}P_d^{(N)},\\
\sum_{d\mid N}P_d^{(N)}&=\operatorname{Id}_N,\\
\operatorname{rank}P_d^{(N)}&=\operatorname{tr}P_d^{(N)}=\varphi(d).
\end{aligned}

P_{\mathrm{final}}Use in source · 1 passage

Search aliases: P_final mathrm

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Use in source · Read passage · g_main.tex, line 791
endpmatrix. A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
P_{\mathrm{final}}=I_s,
\qquad
Q_{\mathrm{final}}=I_t.
```

p_gUse in source · 2 passages

Search aliases: p_g

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Use in source · Read passage · g_main.tex, line 4449
For stochastic error rates p_g, a first-order weighted cost is C_err(W)=∑_g∈ W-log(1-p_g). This additive approximation can be inserted directly into Dijkstra search. It does not capture coherent or correlated errors, but it produces a declared and reproducible metric.
Exact mathematical passage
```
p_g
```

P_iUse in source · 3 passages

Search aliases: P_i

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Use in source · Read passage · s_main.tex, line 111
[Invariant quantum packets] If a unitary U satisfies [U,Q]=0, then P_iUP_j=0 (i≠ j), and U=⊕_i U_i, U_i=P_iUP_i|_P_iHh. If P_kketψ=ketψ, then P_kUketψ=Uketψ.
Exact mathematical passage
```
P_iUP_j=0\qquad(i\neq j),
```

P_jUse in source · 5 passages

Search aliases: P_j

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Use in source · Read passage · c_main.tex, line 602
tional algorithmic consequence [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The leading coefficient has total degree at most d-r, so it vanishes at a random first (m-1)-tuple with probability at most (d-r)/|K|. Otherwise the remaini…
Exact mathematical passage
```
 P=\sum_{j=0}^r P_j(z_1,\ldots,z_{m-1})z_m^j,
 \qquad P_r\ne0.
```

P_kUse in source · 6 passages

Search aliases: P_k

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Use in source · Read passage · s_main.tex, line 103
Conserved observables. Let Q=∑_k q_kP_k be a self-adjoint observable with spectral projectors P_k.
Exact mathematical passage
```
Q=\sum_k q_kP_k
```

p_nUse in source · 6 passages

Search aliases: p_n

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Use in source · Read passage · g_appendix.tex, line 167
Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).
Exact mathematical passage
```
(q_1,\ldots,q_n,p_1,\ldots,p_n)
```

P_{n,k}Use in source · 2 passages

Search aliases: P_n,k

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Use in source · Read passage · s_main.tex, line 724
Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.
Exact mathematical passage
```
P_{n,k}
=
\sum_{|x|=k}\ket x\bra x,
```

P_{nm}Use in source · 2 passages

Search aliases: P_nm

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Use in source · Read passage · q_main.tex, line 737
P_nm=fracc(n-m)12.
Exact mathematical passage
```
P_{nm}=\frac{c(n-m)}{12}.
```

P_{nn}Use in source · 1 passage

Search aliases: P_nn

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Use in source · Read passage · q_main.tex, line 1403
tr(Pρ_diag) =∑_n P_nnfracc(n)^248 =frac13∑_nfracc(n)^248 =frac13. square
Exact mathematical passage
```
\operatorname{tr}(P\rho_{\mathrm{diag}})
=\sum_n P_{nn}\frac{c(n)^2}{48}
=\frac13\sum_n\frac{c(n)^2}{48}
=\frac13.
```

P_qUse in source · 4 passages

Search aliases: P_q

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Use in source · Read passage · g_main.tex, line 4343
Sector-flow invariants. Let P_q be the projector onto sector q. For an intermediate unitary U_t, define transfer blocks T_q,r(t)=P_q U_t P_r. A restricted path has T_q,r=0 for q≠ r. Ambient paths may populate off-diagonal blocks. Possible excursion measures include ∑_q≠ rrank T_q,r, max_q≠ r T_q,r , bandwidth in |q-r|. These are analogues of strict-upper rank and bandwidth in the unitriangular theory.
Exact mathematical passage
```
P_q
```

P_RUse in source · 9 passages

Search aliases: P_R

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Use in source · Read passage · c_main.tex, line 554
Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.
Exact mathematical passage
```
 P_R(z_1,\ldots,z_m)=
 \sum_{w\in\B^m}R(w)\prod_{i=1}^m z_i^{w_i}\in K[z_1,\ldots,z_m].
```

P_rUse in source · 4 passages

Search aliases: P_r

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Use in source · Read passage · c_main.tex, line 602
tional algorithmic consequence [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The leading coefficient has total degree at most d-r, so it vanishes at a random first (m-1)-tuple with probability at most (d-r)/|K|. Otherwise the remaini…
Exact mathematical passage
```
 P=\sum_{j=0}^r P_j(z_1,\ldots,z_{m-1})z_m^j,
 \qquad P_r\ne0.
```

P_SUse in source · 8 passages

Search aliases: P_S

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Use in source · Read passage · q_main.tex, line 461
For any set S⊆ I_N, let D_S be its diagonal indicator and P_S=F_ND_SF_N^†. This construction does not require S to be an order class. Rationality is special, but orthogonal projection is general.
Exact mathematical passage
```
P_S=F_ND_SF_N^\dagger
```

P_sUse in source · 1 passage

Search aliases: P_s

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Use in source · Read passage · s_main.tex, line 1513
Syndrome projectors. Let S_1,…,S_m be commuting Pauli stabilizers with eigenvalues ±1. A syndrome s∈{0,1}^m defines P_s = ∏_j=1^m fracI+(-1)^s_jS_j2. These projectors are orthogonal and sum to the identity. Syndrome measurement selects one packet, while recovery applies a syndrome-dependent transfer toward the code packet.
Exact mathematical passage
```
P_s
=
\prod_{j=1}^m
\frac{I+(-1)^{s_j}S_j}{2}.
```

P_S^2Use in source · 1 passage

Search aliases: P_S^2

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Use in source · Read passage · q_main.tex, line 483
Proof. The first equality is distributivity. Since P_S^†=P_S and P_S^2=P_S,
Exact mathematical passage
```
P_S^2=P_S
```

P_S^\daggerUse in source · 1 passage

Search aliases: P_S^† dagger

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Use in source · Read passage · q_main.tex, line 483
Proof. The first equality is distributivity. Since P_S^†=P_S and P_S^2=P_S,
Exact mathematical passage
```
P_S^\dagger=P_S
```

P_\alphaUse in source · 24 passages

Search aliases: P_α alpha

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Use in source · Read passage · glossary.tex, line 15
Coherence block · P_αρ P_β with possibly distinct left and right packet labels.
Exact mathematical passage
```
P_\alpha\rho P_\beta
```

P_{\bm\alpha}Use in source · 1 passage

Search aliases: P_α bm alpha

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Use in source · Read passage · s_main.tex, line 182
Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential preprocessing merely relocates the complexity.
Exact mathematical passage
```
P_{\bm\alpha}
=
P^{(1)}_{\alpha_1}\cdots P^{(s)}_{\alpha_s}.
```

p_\alphaUse in source · 3 passages

Search aliases: p_α alpha

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Use in source · Read passage · s_main.tex, line 1722
Projective measurement. For projectors {P_α}, the probability of outcome α is p_α = Tr(P_αρ), and the conditioned state is ρ_α = fracP_αρ P_αp_α when p_α>0. Measurement is therefore packet selection followed by normalization.
Exact mathematical passage
```
p_\alpha
=
\Tr(P_\alpha\rho),
```

\mathcal P_{\alpha\beta}Use in source · 2 passages

Search aliases: P_αβ mathcal alpha beta

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Use in source · Read passage · s_main.tex, line 1660
Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.
Exact mathematical passage
```
\mathcal P_{\alpha\beta}(X)
=
P_\alpha X P_\beta,
```

P_\betaUse in source · 14 passages

Search aliases: P_β beta

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Use in source · Read passage · glossary.tex, line 15
Coherence block · P_αρ P_β with possibly distinct left and right packet labels.
Exact mathematical passage
```
P_\alpha\rho P_\beta
```

\mathcal P_{\gamma\delta}Use in source · 1 passage

Search aliases: P_γδ mathcal gamma delta

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Use in source · Read passage · s_main.tex, line 1666
Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.
Exact mathematical passage
```
\Phi_{\gamma\delta,\alpha\beta}
=
\mathcal P_{\gamma\delta}
\,\Phi\,
\mathcal P_{\alpha\beta}.
```

p_\lambdaUse in source · 10 passages

Search aliases: p_λ lambda

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Use in source · Read passage · s_main.tex, line 1104
[Permutation-invariant state normal form] A density operator ρ commutes with every permutation if and only if it can be written as ρ = ⊕_λ p_λ ρ_λ⊗ fracI_ V_λf_λ, where p_λ≥0, ∑_λ p_λ=1, and each ρ_λ is a density operator on U_λ whenever p_λ>0. The weights p_λ and the unnormalized blocks p_λρ_λ are unique. The normalized block ρ_λ is unique only when p_λ>0 and is omitted when p_λ=0.
Exact mathematical passage
```
\rho
=
\bigoplus_\lambda
p_\lambda\,\rho_\lambda\otimes
\frac{I_{\mathcal V_\lambda}}{f_\lambda},
\label{s:eq:commutant-state-record}
```

P_{\rho_i\rho_j^{-1}}Use in source · 1 passage

Search aliases: P_ρ_iρ_j^-1 rho rho

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Use in source · Read passage · g_main.tex, line 4612
Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.
Exact mathematical passage
```
\label{g:eq:transition-fusion}
\sum_{i<j}
\left(
P_{\rho_i\rho_j^{-1}}
+
P_{\rho_j\rho_i^{-1}}
\right)
=n(J-I).
```

P_{\rho_j\rho_i^{-1}}Use in source · 1 passage

Search aliases: P_ρ_jρ_i^-1 rho rho

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Use in source · Read passage · g_main.tex, line 4612
Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.
Exact mathematical passage
```
\label{g:eq:transition-fusion}
\sum_{i<j}
\left(
P_{\rho_i\rho_j^{-1}}
+
P_{\rho_j\rho_i^{-1}}
\right)
=n(J-I).
```

P_\chiUse in source · 4 passages

Search aliases: P_χ chi

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Use in source · Read passage · s_main.tex, line 158
Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).
Exact mathematical passage
```
P_\chi
=
\frac{d_\chi}{|G|}
\sum_{g\in G}
\overline{\chi(g)}U_g.
\label{s:eq:character-projector}
```

P_\psiUse in source · 3 passages

Search aliases: P_ψ psi

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Use in source · Read passage · s_main.tex, line 167
Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).
Exact mathematical passage
```
P_\chi P_\psi=\delta_{\chi\psi}P_\chi,
\qquad
\sum_\chi P_\chi=I
```

\mathrm{packet}Use in source · 1 passage

Search aliases: packet mathrm

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Use in source · Read passage · s_main.tex, line 227
State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectorization, and basis compilation determine actual runtime. The asymptotic distinction is exact when D is polynomial and 2^n is exponential.
Exact mathematical passage
```
C_{\mathrm{packet}}=O(dnD)+C_{\mathrm{compile}}.
```

\mathrm{packets}Use in source · 2 passages

Search aliases: packets mathrm

Link to this entry ·

Use in source · Read passage · s_main.tex, line 33
The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:
Exact mathematical passage
```
\mathfrak C_n=
\bigl(
N_{\mathrm{packets}},
D_{\mathrm{internal}},
S_{\mathrm{transfer}},
L_{\mathrm{leakage}}
\bigr).
```

\operatorname{Pareto}Use in source · 2 passages

Search aliases: Pareto operatorname

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Use in source · Read passage · g_main.tex, line 3999
A vector-valued defect. For a target U in a restricted phase model, define C_res(U) = { c( W): W is a restricted circuit for U} and similarly C_amb(U). Their Pareto frontiers are denoted Pareto_res(U) and Pareto_amb(U).
Exact mathematical passage
```
\operatorname{Pareto}_{\mathrm{res}}(U)
```

\operatorname{Pareto}_{\mathrm{amb}}Use in source · 1 passage

Search aliases: Pareto_amb operatorname mathrm

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Use in source · Read passage · g_main.tex, line 3999
A vector-valued defect. For a target U in a restricted phase model, define C_res(U) = { c( W): W is a restricted circuit for U} and similarly C_amb(U). Their Pareto frontiers are denoted Pareto_res(U) and Pareto_amb(U).
Exact mathematical passage
```
\operatorname{Pareto}_{\mathrm{amb}}(U)
```

\operatorname{Pareto}_{\mathrm{res}}Use in source · 1 passage

Search aliases: Pareto_res operatorname mathrm

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Use in source · Read passage · g_main.tex, line 3999
A vector-valued defect. For a target U in a restricted phase model, define C_res(U) = { c( W): W is a restricted circuit for U} and similarly C_amb(U). Their Pareto frontiers are denoted Pareto_res(U) and Pareto_amb(U).
Exact mathematical passage
```
\operatorname{Pareto}_{\mathrm{res}}(U)
```

\operatorname{PG}Use in source · 2 passages

Search aliases: PG operatorname

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1140
Over F_2, the carrier states used by the canonical word form six of the seven points of PG(2,2). The four projective lines supported on those six points form a Pasch configuration. The same six vectors represent the graphic matroid M(K_4) over every field; this regular-matroid structure isolates the characteristic-independent core of the atom. At the spectral level, the distance to a target is the first nonzero coefficient of an endpoint Green kernel. For C, the ambient kernel begins in degree s…
Exact mathematical passage
```
\operatorname{PG}(2,2)
```

\pmodUse in source · 5 passages

Search aliases: pmod

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Use in source · Read passage · g_main.tex, line 4043
Moment and derivative packets. For a phase function P, finite differences D_uP(x)=P(x+u)-P(x)pmod8 separate affine, quadratic, and higher interactions. A hierarchy of derivative tensors can record temporary flow into nonlinear phase packets.
Exact mathematical passage
```
D_uP(x)=P(x+u)-P(x)\pmod8
```

\polyUse in source · 7 passages

Search aliases: poly

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Use in source · Read passage · s_main.tex, line 199
Circuit families. A quantum decision procedure is a polynomial-time uniform family of circuits {C_x}, where the input string x determines a circuit on n=n(|x|) qubits with at most poly(|x|) gates. Let C_x=G_m⋯ G_1. A classical simulation theorem must describe an algorithm receiving the same succinct description of x and the circuit. It is not enough to assume that an exponentially large matrix or a complete orbit list has already been supplied.
Exact mathematical passage
```
\poly(|x|)
```

\operatorname{polyEval}Use in source · 1 passage

Search aliases: polyEval operatorname

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Use in source · Read passage · c_lean.tex, line 75
Polynomial certificates are not postulated degree oracles. The inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator polyEval(p,x)=⊕_L∈ p∏_i∈ Lx_i by a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a certificate.
Exact mathematical passage
```
\operatorname{polyEval}(p,x)=\bigoplus_{L\in p}\prod_{i\in L}x_i
```

\PrUse in source · 3 passages

Search aliases: Pr

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Use in source · Read passage · c_main.tex, line 596
The conditional algorithmic consequence. [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The leading coefficient has total degree at most d-r, so it vanishes at a random first (m-1)-tuple with probability at most (d-r)/|K|. Otherwise t…
Exact mathematical passage
```
 \Pr_{z\text{ uniform in }K^m}[P(z)=0]\le\frac d{|K|}.
```

\Pr_{F^\dagger\chi}Use in source · 1 passage

Search aliases: Pr_F^†χ Pr dagger chi

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Use in source · Read passage · q_main.tex, line 985
Pr_F^†ψ(k∈ U_12)=1, Pr_F^†χ(k∈ U_12)=0.
Exact mathematical passage
```
\Pr_{F^\dagger\psi}(k\in U_{12})=1,
\qquad
\Pr_{F^\dagger\chi}(k\in U_{12})=0.
```

\Pr_{F^\dagger\chi_\theta}Use in source · 1 passage

Search aliases: Pr_F^†χ_θ Pr dagger chi theta

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Use in source · Read passage · q_main.tex, line 1330
Pr_F^†χ_θ(k∈ U_12) = Pχ_θ ^2=sin^2(θ/2).
Exact mathematical passage
```
\boxed{\Pr_{F^\dagger\chi_\theta}(k\in U_{12})
=\|P\chi_\theta\|^2=\sin^2(\theta/2).}
```

\Pr_{F^\dagger\psi}Use in source · 1 passage

Search aliases: Pr_F^†ψ Pr dagger psi

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Use in source · Read passage · q_main.tex, line 985
Pr_F^†ψ(k∈ U_12)=1, Pr_F^†χ(k∈ U_12)=0.
Exact mathematical passage
```
\Pr_{F^\dagger\psi}(k\in U_{12})=1,
\qquad
\Pr_{F^\dagger\chi}(k\in U_{12})=0.
```

\Pr_{z\text{ uniform in }K^m}Use in source · 1 passage

Search aliases: Pr_z uniform in K^m Pr text

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Use in source · Read passage · c_main.tex, line 596
The conditional algorithmic consequence. [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The leading coefficient has total degree at most d-r, so it vanishes at a random first (m-1)-tuple with probability at most (d-r)/|K|. Otherwise t…
Exact mathematical passage
```
 \Pr_{z\text{ uniform in }K^m}[P(z)=0]\le\frac d{|K|}.
```

\precUse in source · 3 passages

Search aliases: prec

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Use in source · Read passage · g_main.tex, line 539
The trivial representation records only the total number of loops returning to the subgroup. Nontrivial H-types retain their holonomy. This yields a hierarchy: spherical return data prec type-resolved holonomy packets prec exact endpoint realization.
Exact mathematical passage
```
\text{spherical return data}
\prec
\text{type-resolved holonomy packets}
\prec
\text{exact endpoint realization}.
```

\preceqUse in source · 1 passage

Search aliases: preceq

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Use in source · Read passage · g_main.tex, line 5222
Template XII: vector costs. Let gates carry costs in R_+^k. A lower bound may be a vector inequality F(U)preceq∑_g∈ Wc(g) or a family of scalar inequalities indexed by dual weights w∈R_+^k: w· F(U)≤ w· c(W). A complete Pareto proof may require enough supporting weights to separate the restricted and ambient convex lower hulls.
Exact mathematical passage
```
F(U)\preceq\sum_{g\in W}c(g)
```

\ProbUse in source · 6 passages

Search aliases: Prob

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Use in source · Read passage · g_main.tex, line 600
Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estimate can be sharpened by orbit counting, stabilizers, and the geometry of word-map fibers.
Exact mathematical passage
```
\Prob[\Delta\ge d]
=O_{N,M}(q^{-d}).
```

\Prob_{A\in\U_n(\F_q)}Use in source · 3 passages

Search aliases: Prob_A∈U_n(F_q) Prob in U F

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Use in source · Read passage · g_main.tex, line 1085
[Finite-field sparsity] Let N=binom n2 and M=n(n-1). For 1≤ d≤ N, Prob_A∈U_n(F_q)[DeltaDef(A)≥ d] ≤ min{1,M^N-dq^-d}. For fixed n, a uniformly random target therefore has defect zero with probability 1-O_n(q^-1) as q→∞.
Exact mathematical passage
```
\Prob_{A\in\U_n(\F_q)}[\DeltaDef(A)\ge d]
\le
\min\{1,M^{N-d}q^{-d}\}.
```

\mathrm{product}Use in source · 1 passage

Search aliases: product mathrm

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Use in source · Read passage · g_main.tex, line 4698
This is naturally expressed as Real_product(P_6)=∅, Real_entangled(P_6)≠∅. It is an existence defect, not a gate-count defect.
Exact mathematical passage
```
\operatorname{Real}_{\mathrm{product}}(P_6)=\varnothing,
\qquad
\operatorname{Real}_{\mathrm{entangled}}(P_6)\ne\varnothing.
```

\mathbb QUse in source · 3 passages

Search aliases: Q mathbb

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Use in source · Read passage · q_lean.tex, line 177
The exact phase arithmetic. The structure Cyclo12 represents (a,b,c,d)⟼ a+bsqrt3+i(c+dsqrt3), a,b,c,d∈ Q. Its multiplication explicitly incorporates (sqrt3)^2=3 and i^2=-1. It is a concrete arithmetic representation. The development does not need or install a global field instance to certify the finite example.
Exact mathematical passage
```
 (a,b,c,d)\longmapsto a+b\sqrt3+i(c+d\sqrt3),\qquad a,b,c,d\in\mathbb Q.
```

q'Use in source · 5 passages

Search aliases: q'

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Use in source · Read passage · q_main.tex, line 149
Proof. The equation (ζ_q^a)^t=1 holds exactly when q divides at. Write g=gcd(a,q), q=gq', and a=ga', with gcd(a',q')=1. The divisibility condition is q'mid a't, which is equivalent to q'mid t. Its smallest positive solution is t=q'. This includes a=0: then g=q and the order is one. square
Exact mathematical passage
```
q=gq'
```

Q^{-1}Use in source · 7 passages

Search aliases: Q^-1

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Use in source · Read passage · c_main.tex, line 95
Exact cut comparison and safe discrepancies. claimC01 [Cut-comparison identity and complete coverage] Let a restored oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with target z, correctness of the full …
Exact mathematical passage
```
R=Q^{-1}
```

q^{-1}Use in source · 4 passages

Search aliases: q^-1

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Use in source · Read passage · c_lean.tex, line 120
pageheadFrom the cut to a numerical lower boundEXISTENCE, BALANCE, AND COMPLETE COVERAGE Write c=pmathbin++q. At this cut, Block 01 defines H_i(x)=run(p)(x,0^a,0)_ix run(q^-1)(x,0^a,0)_i and proves H_i≤ f, f(x)=1iff∃ i H_i(x)=1. Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives deg(H_i)≤max{TC(p),TC(q)}+1.
Exact mathematical passage
```
H_i(x)=\operatorname{run}(p)(x,0^a,0)_i\x
       \operatorname{run}(q^{-1})(x,0^a,0)_i
```

q^{-1/2}Use in source · 1 passage

Search aliases: q^-1/2

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Use in source · Read passage · q_main.tex, line 129
(F_q)_na=q^-1/2ζ_q^na.
Exact mathematical passage
```
(F_q)_{na}=q^{-1/2}\zeta_q^{na}.
```

q^{-d}Use in source · 4 passages

Search aliases: q^-d

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Use in source · Read passage · g_main.tex, line 600
Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estimate can be sharpened by orbit counting, stabilizers, and the geometry of word-map fibers.
Exact mathematical passage
```
\Prob[\Delta\ge d]
=O_{N,M}(q^{-d}).
```

q^2Use in source · 1 passage

Search aliases: q^2

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Use in source · Read passage · q_main.tex, line 319
c_q ^2 =frac1q∑_k|widehatc_q(k)|^2 =frac1qφ(q)q^2. square
Exact mathematical passage
```
\|c_q\|^2
=\frac1q\sum_k|\widehat{c_q}(k)|^2
=\frac1q\varphi(q)q^2.
```

\mathbb Q^4Use in source · 1 passage

Search aliases: Q^4 mathbb

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Use in source · Read passage · q_main.tex, line 1172
(a,b,c,d)∈ Q^4
Exact mathematical passage
```
(a,b,c,d)\in\mathbb Q^4
```

q^NUse in source · 3 passages

Search aliases: q^N

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Use in source · Read passage · g_main.tex, line 599
Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estimate can be sharpened by orbit counting, stabilizers, and the geometry of word-map fibers.
Exact mathematical passage
```
q^N
```

q^rUse in source · 2 passages

Search aliases: q^r

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Use in source · Read passage · g_main.tex, line 5149
Template VII: finite-field sparsity. Suppose there are at most B gate directions and at most q^r parameter choices per gate over F_q. Then the number of words of length at most L is at most ∑_j=0^L(Bq^r)^j. If the target family has q^N points and defect d forces L≤ (N-d)/r, this yields a density bound of order q^-d up to a fixed combinatorial factor.
Exact mathematical passage
```
q^r
```

Q_0Use in source · 1 passage

Search aliases: Q_0

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Use in source · Read passage · g_main.tex, line 811
Quiver gate models. Let Q=(Q_0,Q_1) be a quiver. Associate a vector space V_v to every vertex. Internal gates change bases inside V_v, while edge gates transmit low-rank maps along arrows. A target is a representation of the path algebra or an automorphism of a filtered quiver module.
Exact mathematical passage
```
Q=(Q_0,Q_1)
```

Q_1Use in source · 2 passages

Search aliases: Q_1

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Use in source · Read passage · g_main.tex, line 811
Quiver gate models. Let Q=(Q_0,Q_1) be a quiver. Associate a vector space V_v to every vertex. Internal gates change bases inside V_v, while edge gates transmit low-rank maps along arrows. A target is a representation of the path algebra or an automorphism of a filtered quiver module.
Exact mathematical passage
```
Q=(Q_0,Q_1)
```

q_1Use in source · 4 passages

Search aliases: q_1

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Use in source · Read passage · c_visual.tex, line 169
guidepageDegree bounds must retain shared register products Take four independent input bits. Compute q_1=x_1x_2, q_2=x_3x_4, q_3=q_1q_2. After two ANDs, each nonlinear register has degree two, but their product has degree four. After the third AND, a coordinate of degree four appears.
Exact mathematical passage
```
 q_1=x_1x_2,\qquad q_2=x_3x_4,\qquad q_3=q_1q_2.
```

q_2Use in source · 2 passages

Search aliases: q_2

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Use in source · Read passage · c_visual.tex, line 169
guidepageDegree bounds must retain shared register products Take four independent input bits. Compute q_1=x_1x_2, q_2=x_3x_4, q_3=q_1q_2. After two ANDs, each nonlinear register has degree two, but their product has degree four. After the third AND, a coordinate of degree four appears.
Exact mathematical passage
```
 q_1=x_1x_2,\qquad q_2=x_3x_4,\qquad q_3=q_1q_2.
```

Q_3Use in source · 4 passages

Search aliases: Q_3

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Use in source · Read passage · c_main.tex, line 462
d not be supported on the positive set of P_m. Take m=3, identify the columns with the integers 1,…,6 using their three-bit expansions, and set x_1=x_3=x_4=1 with all other inputs zero. Their column sum is 1+3+4=6 under bitwise XOR, not a=7, so P_3=0. Among the complementary pairs (1,6),(2,5),(3,4), exactly (3,4) has both inputs one. Thus Q_3=1. In the source ordering this is the logical assignment encoded by integer 13.
Exact mathematical passage
```
Q_3=1
```

q_3Use in source · 1 passage

Search aliases: q_3

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Use in source · Read passage · c_visual.tex, line 169
guidepageDegree bounds must retain shared register products Take four independent input bits. Compute q_1=x_1x_2, q_2=x_3x_4, q_3=q_1q_2. After two ANDs, each nonlinear register has degree two, but their product has degree four. After the third AND, a coordinate of degree four appears.
Exact mathematical passage
```
 q_1=x_1x_2,\qquad q_2=x_3x_4,\qquad q_3=q_1q_2.
```

Q_{>r}Use in source · 3 passages

Search aliases: Q_>r

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Use in source · Read passage · g_main.tex, line 780
C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
Q_{>r}\in U_t
```

Q_CUse in source · 2 passages

Search aliases: Q_C

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Use in source · Read passage · g_main.tex, line 4567
Define Q_R=fracI+Rn, Q_C=fracI+Cn, Q_S=fracI+Sn, B_0=fracJn^2. Then the five joint projectors are B_0, Q_R-B_0, Q_C-B_0, Q_S-B_0, I-Q_R-Q_C-Q_S+2B_0. Their multiplicities are 1, n-1, n-1, n-1, (n-1)(n-2). The Latin-square graph R+C+S has universal spectrum 3(n-1)^1, (n-3)^3(n-1), (-3)^(n-1)(n-2). At n=6 this becomes 15^1,qquad3^15, (-3)^20. Thus the graph spectrum cannot distinguish the twelve main classes of order six.
Exact mathematical passage
```
Q_R=\frac{I+R}{n},
\qquad
Q_C=\frac{I+C}{n},
\qquad
Q_S=\frac{I+S}{n},
\qquad
B_0=\frac{J}{n^2}.
```

Q_{\mathrm{final}}Use in source · 1 passage

Search aliases: Q_final mathrm

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Use in source · Read passage · g_main.tex, line 791
endpmatrix. A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
P_{\mathrm{final}}=I_s,
\qquad
Q_{\mathrm{final}}=I_t.
```

q_kUse in source · 1 passage

Search aliases: q_k

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Use in source · Read passage · s_main.tex, line 103
Conserved observables. Let Q=∑_k q_kP_k be a self-adjoint observable with spectral projectors P_k.
Exact mathematical passage
```
Q=\sum_k q_kP_k
```

q_nUse in source · 2 passages

Search aliases: q_n

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Use in source · Read passage · g_appendix.tex, line 167
Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).
Exact mathematical passage
```
(q_1,\ldots,q_n,p_1,\ldots,p_n)
```

Q_RUse in source · 2 passages

Search aliases: Q_R

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Use in source · Read passage · g_main.tex, line 4567
Define Q_R=fracI+Rn, Q_C=fracI+Cn, Q_S=fracI+Sn, B_0=fracJn^2. Then the five joint projectors are B_0, Q_R-B_0, Q_C-B_0, Q_S-B_0, I-Q_R-Q_C-Q_S+2B_0. Their multiplicities are 1, n-1, n-1, n-1, (n-1)(n-2). The Latin-square graph R+C+S has universal spectrum 3(n-1)^1, (n-3)^3(n-1), (-3)^(n-1)(n-2). At n=6 this becomes 15^1,qquad3^15, (-3)^20. Thus the graph spectrum cannot distinguish the twelve main classes of order six.
Exact mathematical passage
```
Q_R=\frac{I+R}{n},
\qquad
Q_C=\frac{I+C}{n},
\qquad
Q_S=\frac{I+S}{n},
\qquad
B_0=\frac{J}{n^2}.
```

Q_SUse in source · 2 passages

Search aliases: Q_S

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Use in source · Read passage · g_main.tex, line 4567
Define Q_R=fracI+Rn, Q_C=fracI+Cn, Q_S=fracI+Sn, B_0=fracJn^2. Then the five joint projectors are B_0, Q_R-B_0, Q_C-B_0, Q_S-B_0, I-Q_R-Q_C-Q_S+2B_0. Their multiplicities are 1, n-1, n-1, n-1, (n-1)(n-2). The Latin-square graph R+C+S has universal spectrum 3(n-1)^1, (n-3)^3(n-1), (-3)^(n-1)(n-2). At n=6 this becomes 15^1,qquad3^15, (-3)^20. Thus the graph spectrum cannot distinguish the twelve main classes of order six.
Exact mathematical passage
```
Q_R=\frac{I+R}{n},
\qquad
Q_C=\frac{I+C}{n},
\qquad
Q_S=\frac{I+S}{n},
\qquad
B_0=\frac{J}{n^2}.
```

Q_sUse in source · 1 passage

Search aliases: Q_s

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Use in source · Read passage · s_main.tex, line 181
Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential preprocessing merely relocates the complexity.
Exact mathematical passage
```
Q_1,\ldots,Q_s
```

\mathbf rUse in source · 2 passages

Search aliases: r mathbf

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Use in source · Read passage · g_main.tex, line 4094
For n bosonic modes, a Gaussian unitary acts on quadratures r=(q_1,…,q_n,p_1,…,p_n)^T by an affine symplectic transformation r⟼ S r+d, S∈Sp(2n,R). For n fermionic modes, a fermionic Gaussian unitary acts on 2n Majorana operators by γ⟼ Oγ, O∈ SO(2n). Nearest-neighbor matchgates give a circuit presentation of a fermionic Gaussian subgroup on qubits.
Exact mathematical passage
```
\mathbf r=(q_1,\ldots,q_n,p_1,\ldots,p_n)^T
```

\mathcal RUse in source · 16 passages

Search aliases: R mathcal

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Use in source · Read passage · c_main.tex, line 5
Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies alor b=a+b+ab. A statement h≤ f is pointwise and means that h has no positive value at a negative input of f.
Exact mathematical passage
```
\mathcal R_n=\F[x_1,\ldots,x_n]/(x_i^2-x_i:1\le i\le n).
```

\mathcal R^*_{n,\le L}Use in source · 3 passages

Search aliases: R^*_n,≤ L mathcal le

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Use in source · Read passage · g_main.tex, line 2418
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\mathcal R^*_{n,\le L}
=
\U_n\cap
\bigcup_{\sigma\in\{(t,s):t\ne s\}^L}
\operatorname{im}\Phi_\sigma
```

\mathcal R^*_{n,\le N-d}Use in source · 1 passage

Search aliases: R^*_n,≤ N-d mathcal le

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Use in source · Read passage · g_main.tex, line 2527
If DeltaDef(A)≥ d, then ellAll(A) ≤ ellDown(A)-d ≤ N-d. Thus D_n,≥ d⊆ R^*_n,≤ N-d, and [source reference] gives the first assertion. On the intrinsic-length-L stratum one instead has ellAll(A)≤ L-d, proving the refinement.
Exact mathematical passage
```
\mathcal D_{n,\ge d}\subseteq\mathcal R^*_{n,\le N-d}
```

R^6Use in source · 1 passage

Search aliases: R^6

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Use in source · Read passage · g_main.tex, line 1421
Project the rows to the quotient module R^6/W. Rows 0,1,2 have zero image, while rows 3,4,5 begin and end at the three distinct quotient basis vectors of the free quotient module. An output row targeted exactly once must therefore use a source lying in W, by [source reference].
Exact mathematical passage
```
R^6/W
```

\mathcal R^\downarrow_{n,\le L}Use in source · 2 passages

Search aliases: R^downarrow_n,≤ L mathcal downarrow le

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Use in source · Read passage · g_main.tex, line 2425
mial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\mathcal R^\downarrow_{n,\le L}
```

R^nUse in source · 1 passage

Search aliases: R^n

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1293
[One-target source condition] Let W≤ R^n. Suppose row t starts at e_t, is targeted exactly once, and finishes at e_t+w, w∈ W, where at least one coordinate of w is a unit of R. At the moment of its unique targeting, the gate coefficient is a unit and the source row is a unit multiple of w.
Exact mathematical passage
```
W\le R^n
```

r^qUse in source · 2 passages

Search aliases: r^q

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Use in source · Read passage · q_main.tex, line 167
Proof. If qmid h, every summand is one. Otherwise r=ζ_q^h≠1 and r^q=1. Multiplying the sum by 1-r gives 1-r^q=0, so the sum is zero. This argument also covers q=1, where only the first case occurs. square
Exact mathematical passage
```
r^q=1
```

\R_+^kUse in source · 2 passages

Search aliases: R_+^k R

Link to this entry ·

Use in source · Read passage · g_main.tex, line 5221
Template XII: vector costs. Let gates carry costs in R_+^k. A lower bound may be a vector inequality F(U)preceq∑_g∈ Wc(g) or a family of scalar inequalities indexed by dual weights w∈R_+^k: w· F(U)≤ w· c(W). A complete Pareto proof may require enough supporting weights to separate the restricted and ambient convex lower hulls.
Exact mathematical passage
```
\R_+^k
```

R_0Use in source · 6 passages

Search aliases: R_0

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Use in source · Read passage · g_main.tex, line 1228
For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0⊂ V_1⊂⋯⊂ V_n-1. An upper-directed gate leaves this fiber of the flag map. A strict defect means tha…
Exact mathematical passage
```
(R_0,\ldots,R_{n-1})
```

r_0Use in source · 2 passages

Search aliases: r_0

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Use in source · Read passage · g_main.tex, line 4836
State models. Binary matrices. An m× m matrix over F_2 is stored as (r_0,…,r_m-1), where bit j of r_i is the (i,j) entry. Matrix multiplication is exact XOR accumulation.
Exact mathematical passage
```
(r_0,\ldots,r_{m-1}),
```

R_1Use in source · 5 passages

Search aliases: R_1

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Use in source · Read passage · g_main.tex, line 1318
Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.
Exact mathematical passage
```
R_0&=e_0,
& R_1&=e_1+e_0,
& R_2&=e_2,\\
R_3&=e_3+v,
& R_4&=e_4+w,
& R_5&=e_5+w.
```

R_2Use in source · 5 passages

Search aliases: R_2

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1318
Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.
Exact mathematical passage
```
R_0&=e_0,
& R_1&=e_1+e_0,
& R_2&=e_2,\\
R_3&=e_3+v,
& R_4&=e_4+w,
& R_5&=e_5+w.
```

R_3Use in source · 5 passages

Search aliases: R_3

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1318
Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.
Exact mathematical passage
```
R_0&=e_0,
& R_1&=e_1+e_0,
& R_2&=e_2,\\
R_3&=e_3+v,
& R_4&=e_4+w,
& R_5&=e_5+w.
```

R_4Use in source · 5 passages

Search aliases: R_4

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1318
Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.
Exact mathematical passage
```
R_0&=e_0,
& R_1&=e_1+e_0,
& R_2&=e_2,\\
R_3&=e_3+v,
& R_4&=e_4+w,
& R_5&=e_5+w.
```

R_5Use in source · 5 passages

Search aliases: R_5

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1318
Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.
Exact mathematical passage
```
R_0&=e_0,
& R_1&=e_1+e_0,
& R_2&=e_2,\\
R_3&=e_3+v,
& R_4&=e_4+w,
& R_5&=e_5+w.
```

R_{CP_jC^\dagger}Use in source · 1 passage

Search aliases: R_CP_jC^† dagger

Link to this entry ·

Use in source · Read passage · g_main.tex, line 3932
The simplest symbolic sandwich is C(∏_j=1^m R_P_j(θ_j))C^† = ∏_j=1^m R_CP_jC^†(θ_j). If the P_j commute, their conjugates commute. A catalyst can therefore change the geometry of the support without changing the algebra of the angles.
Exact mathematical passage
```
C\left(\prod_{j=1}^m R_{P_j}(\theta_j)\right)C^\dagger
=
\prod_{j=1}^m R_{CP_jC^\dagger}(\theta_j).
```

R_{CPC^\dagger}Use in source · 2 passages

Search aliases: R_CPC^† dagger

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 175
Pauli rotations. For a Hermitian Pauli P, R_P(θ)=e^-iθ P/2. Clifford conjugation gives C R_P(θ) C^†=R_CPC^†(θ). The sign of CPC^† can be absorbed by changing θ to -θ.
Exact mathematical passage
```
C R_P(\theta) C^\dagger=R_{CPC^\dagger}(\theta).
```

R_iUse in source · 1 passage

Search aliases: R_i

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Use in source · Read passage · g_main.tex, line 1236
For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0⊂ V_1⊂⋯⊂ V_n-1. An upper-directed gate leaves this fiber of the flag map. A strict defect means tha…
Exact mathematical passage
```
R_i\in e_i+V_{i-1}
```

r_iUse in source · 7 passages

Search aliases: r_i

Link to this entry ·

Use in source · Read passage · g_main.tex, line 305
Many consumers and catalytic excess. Suppose the middle word is k=h_m⋯ h_1, h_i∈ H. The ambient word gh_m⋯ h_1g^-1 has cost at most 2c(g)+∑_i c(h_i). If the restricted images gh_ig^-1 require costs r_i and a lower-bound invariant proves that their obligations cannot be shared, then ℓ_H(gkg^-1)≥∑_i r_i. A sufficient inequality is ∑_i(r_i-c(h_i))>2c(g). The left side is the gross catalytic benefit. The right side is the restoration cost.
Exact mathematical passage
```
r_i
```

r_{m-1}Use in source · 1 passage

Search aliases: r_m-1

Link to this entry ·

Use in source · Read passage · g_main.tex, line 4836
State models. Binary matrices. An m× m matrix over F_2 is stored as (r_0,…,r_m-1), where bit j of r_i is the (i,j) entry. Matrix multiplication is exact XOR accumulation.
Exact mathematical passage
```
(r_0,\ldots,r_{m-1}),
```

\mathcal R_nUse in source · 7 passages

Search aliases: R_n mathcal

Link to this entry ·

Use in source · Read passage · c_main.tex, line 5
Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies alor b=a+b+ab. A statement h≤ f is pointwise and means that h has no positive value at a negative input of f.
Exact mathematical passage
```
\mathcal R_n=\F[x_1,\ldots,x_n]/(x_i^2-x_i:1\le i\le n).
```

r_nUse in source · 1 passage

Search aliases: r_n

Link to this entry ·

Use in source · Read passage · q_lean.tex, line 103
The rational signal and its filter. The first layer uses literal rational lists. The chosen perturbation is r_n=3cos(π n/2)+2, whose displayed rational entries can be certified without numerical trigonometric evaluation. Its sum with c is the test signal.
Exact mathematical passage
```
r_n=3\cos(\pi n/2)+2
```

R_{n-1}Use in source · 1 passage

Search aliases: R_n-1

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1228
For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0⊂ V_1⊂⋯⊂ V_n-1. An upper-directed gate leaves this fiber of the flag map. A strict defect means tha…
Exact mathematical passage
```
(R_0,\ldots,R_{n-1})
```

R_PUse in source · 4 passages

Search aliases: R_P

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 171
Pauli rotations. For a Hermitian Pauli P, R_P(θ)=e^-iθ P/2. Clifford conjugation gives C R_P(θ) C^†=R_CPC^†(θ). The sign of CPC^† can be absorbed by changing θ to -θ.
Exact mathematical passage
```
R_P(\theta)=e^{-i\theta P/2}.
```

R_pUse in source · 5 passages

Search aliases: R_p

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1509
Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y;…
Exact mathematical passage
```
R_p\leftarrow R_p+aR_q;
\quad
[R_q\leftarrow R_q+b_xR_x]_{x\in X};
\quad
[R_y\leftarrow R_y+c_yR_p]_{y\in Y};
\quad
R_p\leftarrow R_p-aR_q.
```

R_{P_j}Use in source · 1 passage

Search aliases: R_P_j

Link to this entry ·

Use in source · Read passage · g_main.tex, line 3932
The simplest symbolic sandwich is C(∏_j=1^m R_P_j(θ_j))C^† = ∏_j=1^m R_CP_jC^†(θ_j). If the P_j commute, their conjugates commute. A catalyst can therefore change the geometry of the support without changing the algebra of the angles.
Exact mathematical passage
```
C\left(\prod_{j=1}^m R_{P_j}(\theta_j)\right)C^\dagger
=
\prod_{j=1}^m R_{CP_jC^\dagger}(\theta_j).
```

R_qUse in source · 6 passages

Search aliases: R_q

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1509
Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y;…
Exact mathematical passage
```
R_p\leftarrow R_p+aR_q;
\quad
[R_q\leftarrow R_q+b_xR_x]_{x\in X};
\quad
[R_y\leftarrow R_y+c_yR_p]_{y\in Y};
\quad
R_p\leftarrow R_p-aR_q.
```

R_rUse in source · 3 passages

Search aliases: R_r

Link to this entry ·

Use in source · Read passage · q_main.tex, line 534
E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).
Exact mathematical passage
```
E_rf(t)=\sum_{m\ge0}a_{r+Lm}t^m,
\qquad
R_rf(z)=z^rE_rf(z^L).
```

R_sUse in source · 11 passages

Search aliases: R_s

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 12
Row-transvection convention. Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection τ_t,s(a)=I+aE_t,s performs R_t← R_t+aR_s. In the binary CNOT interpretation, s is the control and t is the target.
Exact mathematical passage
```
R_t\leftarrow R_t+aR_s.
```

R_tUse in source · 5 passages

Search aliases: R_t

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 12
Row-transvection convention. Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection τ_t,s(a)=I+aE_t,s performs R_t← R_t+aR_s. In the binary CNOT interpretation, s is the control and t is the target.
Exact mathematical passage
```
R_t\leftarrow R_t+aR_s.
```

R_{t_j}Use in source · 1 passage

Search aliases: R_t_j

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2021
[ 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.
Exact mathematical passage
```
\label{g:eq:snapshot-output}
R_{t_j}=e_{t_j}+e_q+w_{s_j}
\qquad(1\le j\le v).
```

R_{v_0}Use in source · 2 passages

Search aliases: R_v_0

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1963
Suppose the gate is s→ t with s<t. Let v_0 be the lowest node of T whose two children separate s and t. The order of the leaves forces s∈ L_v_0 and t∈ R_v_0. At this one node, a single row of the cross block X[R_v_0,L_v_0] changes, so its rank changes by at most one.
Exact mathematical passage
```
t\in R_{v_0}
```

R_xUse in source · 2 passages

Search aliases: R_x

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1509
Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y;…
Exact mathematical passage
```
R_p\leftarrow R_p+aR_q;
\quad
[R_q\leftarrow R_q+b_xR_x]_{x\in X};
\quad
[R_y\leftarrow R_y+c_yR_p]_{y\in Y};
\quad
R_p\leftarrow R_p-aR_q.
```

R_{x_i}Use in source · 1 passage

Search aliases: R_x_i

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2013
integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.
Exact mathematical passage
```
R_{x_i}=e_{x_i},
\qquad
R_p=w_u,
\qquad
R_q=e_q,
```

R_yUse in source · 3 passages

Search aliases: R_y

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1509
Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y;…
Exact mathematical passage
```
R_p\leftarrow R_p+aR_q;
\quad
[R_q\leftarrow R_q+b_xR_x]_{x\in X};
\quad
[R_y\leftarrow R_y+c_yR_p]_{y\in Y};
\quad
R_p\leftarrow R_p-aR_q.
```

R_ZUse in source · 2 passages

Search aliases: R_Z

Link to this entry ·

Use in source · Read passage · s_main.tex, line 747
Number-preserving gates. Single-qubit R_Z gates commute with widehat N. A two-qubit exchange rotation acts in the ordered basis ket00,ket01,ket10,ket11 as U_XY(θ,ϕ) = 1 · 0 · 0 · 0
Exact mathematical passage
```
R_Z
```

r_\alphaUse in source · 1 passage

Search aliases: r_α alpha

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1857
Hybrid transfer graphs. Suppose packets are nodes of a sparse graph and each node carries an internal representation. A gate may update a node internally or transfer a compact component to neighboring nodes. The simulation cost depends on ∑_α∈ A_tsize(r_α) + #{nonzero transfers at time t}. This model combines symmetry selection rules with compact state classes.
Exact mathematical passage
```
\sum_{\alpha\in A_t}\operatorname{size}(r_\alpha)
+
\#\{\text{nonzero transfers at time }t\}.
```

\operatorname{rank}Use in source · 2 passages

Search aliases: rank operatorname

Link to this entry ·

Use in source · Read passage · q_main.tex, line 363

(P_d^(N))^† · =P_d^(N), P_d^(N)P_e^(N) · =δ_deP_d^(N), ∑_dmid NP_d^(N) · =Id_N, rankP_d^(N) · =trP_d^(N)=φ(d).

Exact mathematical passage


\begin{aligned}
(P_d^{(N)})^\dagger&=P_d^{(N)},\\
P_d^{(N)}P_e^{(N)}&=\delta_{de}P_d^{(N)},\\
\sum_{d\mid N}P_d^{(N)}&=\operatorname{Id}_N,\\
\operatorname{rank}P_d^{(N)}&=\operatorname{tr}P_d^{(N)}=\varphi(d).
\end{aligned}

\rceilUse in source · 9 passages

Search aliases: rceil

Link to this entry ·

Use in source · Read passage · c_lean.tex, line 146
The balanced cut is constructed. Because each gate has cost zero or one, a prefix can realize every Toffoli count from zero to T. The source proves this fact and extracts a cut with max{TC(p),TC(q)}≤lceil T/2rceil. The complete safe cut cover then proves β(f)≤lceil T/2rceil+1, 2β(f)≤ T+3. Natural-number arithmetic retains the floor/ceiling and small-T cases. No balance assumption is added to the final theorem.
Exact mathematical passage
```
\max\{\TC(p),\TC(q)\}\le\left\lceil T/2\right\rceil.
```

\operatorname{Re}Use in source · 1 passage

Search aliases: Re operatorname

Link to this entry ·

Use in source · Read passage · q_visual.tex, line 199
A real three-dimensional view of one amplitude. addcontentslinetocsectionA real three-dimensional view of one amplitude leadThe plot follows an actual complex amplitude as the relative phase advances. At sample n=6, the phase-family amplitude is χ_θ(6)=frace^iθsqrt3. fig16_complex_amplitude_trajectory0.84Coordinates are (θ,Reχ_θ(6),Imχ_θ(6)). In the real-imaginary plane the amplitude traces a radius-1/sqrt3 circle. Advancing the phase coordinate produces the three-dimensional trajectory. Its mag…
Exact mathematical passage
```
(\theta,\operatorname{Re}\chi_\theta(6),\operatorname{Im}\chi_\theta(6))
```

\operatorname{Real}_{\mathrm{entangled}}Use in source · 1 passage

Search aliases: Real_entangled operatorname mathrm

Link to this entry ·

Use in source · Read passage · g_main.tex, line 4698
This is naturally expressed as Real_product(P_6)=∅, Real_entangled(P_6)≠∅. It is an existence defect, not a gate-count defect.
Exact mathematical passage
```
\operatorname{Real}_{\mathrm{product}}(P_6)=\varnothing,
\qquad
\operatorname{Real}_{\mathrm{entangled}}(P_6)\ne\varnothing.
```

\operatorname{Real}_{\mathrm{product}}Use in source · 1 passage

Search aliases: Real_product operatorname mathrm

Link to this entry ·

Use in source · Read passage · g_main.tex, line 4698
This is naturally expressed as Real_product(P_6)=∅, Real_entangled(P_6)≠∅. It is an existence defect, not a gate-count defect.
Exact mathematical passage
```
\operatorname{Real}_{\mathrm{product}}(P_6)=\varnothing,
\qquad
\operatorname{Real}_{\mathrm{entangled}}(P_6)\ne\varnothing.
```

\mathrm{rest}Use in source · 1 passage

Search aliases: rest mathrm

Link to this entry ·

Use in source · Read passage · c_visual.tex, line 451
guidepageExact nonlinear projection costs versus actual input length The source proves the all-size formulas M(P_m)=m-1, T_rest(P_m)=2m-3, M(Q_m)=T_rest(Q_m)=n/2. The chart uses actual input length n=2^m-2 on its horizontal axis.
Exact mathematical passage
```
 M(P_m)=m-1,\quad T_{\mathrm{rest}}(P_m)=2m-3,\qquad
 M(Q_m)=T_{\mathrm{rest}}(Q_m)=n/2.
```

\operatorname{rk}Use in source · 1 passage

Search aliases: rk operatorname

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Use in source · Read passage · g_main.tex, line 694
Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but tensor-regularized defect positive, item ambient and restricted border ranks both smaller than exact ranks but by different amounts. Each requires a se…
Exact mathematical passage
```
\operatorname{rk}_{X}(U)
=
\min\{L:U\in X^L\},
```

\operatorname{rk}_{X}Use in source · 1 passage

Search aliases: rk_X operatorname

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Use in source · Read passage · g_main.tex, line 694
Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but tensor-regularized defect positive, item ambient and restricted border ranks both smaller than exact ranks but by different amounts. Each requires a se…
Exact mathematical passage
```
\operatorname{rk}_{X}(U)
=
\min\{L:U\in X^L\},
```

\rmUse in source · 19 passages

Search aliases: rm

Link to this entry ·

Use in source · Read passage · s_main.tex, line 828
Repeated application versus one shot. Let C_rm compile be the cost of constructing the packet basis and restricted gate plans, and let C_rm apply be one packet application. Reusing the compiled plan R times costs C_rm packet(R) = C_rm compile+RC_rm apply. If the full path costs RC_rm full, the packet method breaks even when R> fracC_rm compile C_rm full-C_rm apply. This is why time evolution, parameter sweeps, variational loops, and repeated likelihood evaluations are natural applications.
Exact mathematical passage
```
C_{\rm compile}
```

\mathrm{run}Use in source · 1 passage

Search aliases: run mathrm

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Use in source · Read passage · c_lean.tex, line 222
The restoration predicate is ∀ x b, run(c)(encode(x,b)) =encode(x,b+f(x)). It is a hypothesis of the conditional theorem. The code does not assert that every gate list realizes a restored oracle. The canonical encoding sets each input wire to its input bit, each ancillary wire to zero, and the target to b. Its wire type is Sum (Fin n) (Sum (Fin a) Unit) which has exactly n+a+1 wires.
Exact mathematical passage
```
 \forall x\,b,\quad \mathrm{run}(c)(\mathrm{encode}(x,b))
 =\mathrm{encode}(x,b+f(x)).
```

S^{-1}Use in source · 2 passages

Search aliases: S^-1

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Use in source · Read passage · g_main.tex, line 456
Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.
Exact mathematical passage
```
S=S^{-1}
```

S^TUse in source · 2 passages

Search aliases: S^T

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Use in source · Read passage · g_main.tex, line 4217
Covariance-matrix lower bounds. For a Gaussian state, the covariance matrix Γ transforms as Γ⟼ SΓ S^T in the bosonic case and Γ⟼ OΓ O^T in the fermionic case. Possible restricted lower bounds include: item rank change of off-diagonal covariance blocks, item symplectic singular-value obligations, item growth of mode support, item graph bandwidth and cut rank, item number of nontrivial canonical angles between invariant subspaces. One elementary gate changes only a controlled portion of these data…
Exact mathematical passage
```
\Gamma\longmapsto S\Gamma S^T
```

S_0Use in source · 5 passages

Search aliases: S_0

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Use in source · Read passage · g_appendix.tex, line 121
1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.
Exact mathematical passage
```
H_1,\ \CX_{1\to0},\ \CX_{0\to1},\ H_0,\ S_1,\ S_0,
```

S_0^{d_0}Use in source · 1 passage

Search aliases: S_0^d_0

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Use in source · Read passage · g_main.tex, line 3607
I_2 · J_2). Applying S_0^d_0S_1^d_1 afterwards left multiplies by (I_2 · 0
Exact mathematical passage
```
S_0^{d_0}S_1^{d_1}
```

S_1Use in source · 9 passages

Search aliases: S_1

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Use in source · Read passage · g_appendix.tex, line 117
1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.
Exact mathematical passage
```
\CX_{0\to1},\ S_1,\ \CX_{1\to0},\ \CX_{0\to1},
```

S_1^{d_1}Use in source · 1 passage

Search aliases: S_1^d_1

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Use in source · Read passage · g_main.tex, line 3607
I_2 · J_2). Applying S_0^d_0S_1^d_1 afterwards left multiplies by (I_2 · 0
Exact mathematical passage
```
S_0^{d_0}S_1^{d_1}
```

s_2Use in source · 2 passages

Search aliases: s_2

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Use in source · Read passage · g_main.tex, line 242
[Holonomy-compression identity] Let every generator have unit cost. For h∈ H, ℓ_G(h) is the minimum length of a based loop in G/H whose lifted word has holonomy h. The intrinsic length ℓ_H(h) is the minimum length among loops that remain in the base fiber. An ambient word s_k⋯ s_1=h projects to the loop H, s_1H, s_2s_1H, …, hH=H. Conversely, a based Schreier loop has product in H. If every prefix lies in H, each edge label is the product of two elements of H and belongs to S_G∩ H= S_H. Conversel…
Exact mathematical passage
```
H,
s_1H,
s_2s_1H,
\ldots,
hH=H.
```

S_8Use in source · 4 passages

Search aliases: S_8

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Use in source · Read passage · g_appendix.tex, line 160
Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric group.
Exact mathematical passage
```
S_8
```

S_AUse in source · 3 passages

Search aliases: S_A

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Use in source · Read passage · g_main.tex, line 4862
[Breadth-first defect census] Given restricted generators S_R, ambient generators S_A, and identity e: item compute d_R on ⟨ S_R⟩; item compute d_A on ⟨ S_A⟩; item for every restricted state x, compare d_R(x) and d_A(x); item record the pair (d_R(x),d_A(x)) and reconstruct witnesses for strict inequalities.
Exact mathematical passage
```
S_A
```

S_dUse in source · 8 passages

Search aliases: S_d

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Use in source · Read passage · q_main.tex, line 337
S_d={a∈ I_N:N/gcd(a,N)=d}.
Exact mathematical passage
```
S_d=\{a\in I_N:N/\gcd(a,N)=d\}.
```

\mathcal S_GUse in source · 5 passages

Search aliases: S_G mathcal

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Use in source · Read passage · g_main.tex, line 224
Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry is the left Schreier graph G/H, with edges gH⟼ sgH. For the base-fiber assertion below, assume also S_H= S_G∩ H. This last condition excludes addition…
Exact mathematical passage
```
H=\langle\mathcal S_H\rangle
\le
G=\langle\mathcal S_G\rangle.
```

\mathcal S_HUse in source · 4 passages

Search aliases: S_H mathcal

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Use in source · Read passage · g_main.tex, line 224
Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry is the left Schreier graph G/H, with edges gH⟼ sgH. For the base-fiber assertion below, assume also S_H= S_G∩ H. This last condition excludes addition…
Exact mathematical passage
```
H=\langle\mathcal S_H\rangle
\le
G=\langle\mathcal S_G\rangle.
```

s_iUse in source · 10 passages

Search aliases: s_i

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Use in source · Read passage · c_lean.tex, line 87
pageheadWhy sharing does not break the degree boundA STRONGER INVARIANT THAN SINGLE-REGISTER DEGREE For a current state s(x) after t Toffolis, the invariant is ∀ L, Deg (t+|L|, x↦∏_i∈ Ls_i(x)). L may be any finite list, including repeated registers. Initially each coordinate is a variable or a constant, so a product of r coordinates has degree at most r.
Exact mathematical passage
```
\forall L,\qquad
\Deg\!\left(t+|L|,\;x\mapsto\prod_{i\in L}s_i(x)\right).
```

s_i'Use in source · 1 passage

Search aliases: s_i'

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Use in source · Read passage · c_visual.tex, line 197
guidepageThe invariant handles duplicates, NOT, and CNOT Suppose a gate toggles target u by a product of c controls. In a product list L that contains u, remove all occurrences of u to obtain R. Boolean idempotence leaves one updated target factor, and ∏_i∈ Ls_i'= s_u∏_i∈ Rs_i+ (∏_j∈controlss_j)∏_i∈ Rs_i. The old invariant bounds these terms by t+1+|R| and t+c+|R|. Since |R|+1≤ |L|, the budget rises by at most max{c-1,0}.
Exact mathematical passage
```
 \prod_{i\in L}s_i'=
 s_u\prod_{i\in R}s_i+
 \left(\prod_{j\in\mathrm{controls}}s_j\right)\prod_{i\in R}s_i.
```

S_jUse in source · 1 passage

Search aliases: S_j

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Use in source · Read passage · s_main.tex, line 1513
Syndrome projectors. Let S_1,…,S_m be commuting Pauli stabilizers with eigenvalues ±1. A syndrome s∈{0,1}^m defines P_s = ∏_j=1^m fracI+(-1)^s_jS_j2. These projectors are orthogonal and sum to the identity. Syndrome measurement selects one packet, while recovery applies a syndrome-dependent transfer toward the code packet.
Exact mathematical passage
```
P_s
=
\prod_{j=1}^m
\frac{I+(-1)^{s_j}S_j}{2}.
```

s_kUse in source · 2 passages

Search aliases: s_k

Link to this entry ·

Use in source · Read passage · g_main.tex, line 241
[Holonomy-compression identity] Let every generator have unit cost. For h∈ H, ℓ_G(h) is the minimum length of a based loop in G/H whose lifted word has holonomy h. The intrinsic length ℓ_H(h) is the minimum length among loops that remain in the base fiber. An ambient word s_k⋯ s_1=h projects to the loop H, s_1H, s_2s_1H, …, hH=H. Conversely, a based Schreier loop has product in H. If every prefix lies in H, each edge label is the product of two elements of H and belongs to S_G∩ H= S_H. Conversel…
Exact mathematical passage
```
s_k\cdots s_1=h
```

S_mUse in source · 1 passage

Search aliases: S_m

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Use in source · Read passage · s_main.tex, line 1512
Syndrome projectors. Let S_1,…,S_m be commuting Pauli stabilizers with eigenvalues ±1. A syndrome s∈{0,1}^m defines P_s = ∏_j=1^m fracI+(-1)^s_jS_j2. These projectors are orthogonal and sum to the identity. Syndrome measurement selects one packet, while recovery applies a syndrome-dependent transfer toward the code packet.
Exact mathematical passage
```
S_1,\ldots,S_m
```

s_mUse in source · 2 passages

Search aliases: s_m

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Use in source · Read passage · g_main.tex, line 1170
Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_1(a_1).
Exact mathematical passage
```
w=(\tau_{t_m,s_m}(a_m),\ldots,\tau_{t_1,s_1}(a_1))
```

S_nUse in source · 5 passages

Search aliases: S_n

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Use in source · Read passage · s_main.tex, line 904
The symmetric subspace. Let the symmetric group S_n act by permuting the n tensor factors. The fully symmetric qubit subspace is spanned by the Dicke states ketD_n^k = binom nk^-1/2 ∑_|x|=kket x, 0≤ k≤ n. Its dimension is n+1.
Exact mathematical passage
```
S_n
```

S_RUse in source · 3 passages

Search aliases: S_R

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Use in source · Read passage · g_main.tex, line 4862
[Breadth-first defect census] Given restricted generators S_R, ambient generators S_A, and identity e: item compute d_R on ⟨ S_R⟩; item compute d_A on ⟨ S_A⟩; item for every restricted state x, compare d_R(x) and d_A(x); item record the pair (d_R(x),d_A(x)) and reconstruct witnesses for strict inequalities.
Exact mathematical passage
```
S_R
```

S_{\rm transfer}Use in source · 1 passage

Search aliases: S_rm transfer rm

Link to this entry ·

Use in source · Read passage · s_main.tex, line 2273
The core research invariant is the complete classical representation record ( discovery cost, N_rm packets, D_rm internal, S_rm transfer, L_rm leakage, measurement cost ). A future compiler that computes and certifies this record can decide when the full state vector is unnecessary, when a hybrid representation is preferable, and when symmetry alone is insufficient.
Exact mathematical passage
```
\bigl(
\text{discovery cost},
N_{\rm packets},
D_{\rm internal},
S_{\rm transfer},
L_{\rm leakage},
\text{measurement cost}
\bigr).
```

S_tUse in source · 5 passages

Search aliases: S_t

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Use in source · Read passage · s_main.tex, line 544
Sparse packet-transfer graphs. Let I=∑_α∈ΛP_α be an orthogonal packet decomposition. Write the gate blocks as G_t;βα=P_β G_tP_α. Starting from an active packet set S_0, define recursively S_t = { β:∃α∈ S_t-1 with G_t;βα≠0 }.
Exact mathematical passage
```
S_t
=
\left\{
\beta:\exists\alpha\in S_{t-1}
\text{ with }G_{t;\beta\alpha}\ne0
\right\}.
\label{s:eq:reachable-packet-set}
```

S_{t-1}Use in source · 1 passage

Search aliases: S_t-1

Link to this entry ·

Use in source · Read passage · s_main.tex, line 544
Sparse packet-transfer graphs. Let I=∑_α∈ΛP_α be an orthogonal packet decomposition. Write the gate blocks as G_t;βα=P_β G_tP_α. Starting from an active packet set S_0, define recursively S_t = { β:∃α∈ S_t-1 with G_t;βα≠0 }.
Exact mathematical passage
```
S_t
=
\left\{
\beta:\exists\alpha\in S_{t-1}
\text{ with }G_{t;\beta\alpha}\ne0
\right\}.
\label{s:eq:reachable-packet-set}
```

S_{\mathrm{transfer}}Use in source · 2 passages

Search aliases: S_transfer mathrm

Link to this entry ·

Use in source · Read passage · s_main.tex, line 33
The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:
Exact mathematical passage
```
\mathfrak C_n=
\bigl(
N_{\mathrm{packets}},
D_{\mathrm{internal}},
S_{\mathrm{transfer}},
L_{\mathrm{leakage}}
\bigr).
```

s_uUse in source · 3 passages

Search aliases: s_u

Link to this entry ·

Use in source · Read passage · c_lean.tex, line 94
Suppose the next gate toggles target u by the product of a control list of length q. A product that omits u is unchanged. If u occurs, idempotence removes all its repetitions. With R the remaining list, the updated product is (s_u∏_i∈ Rs_i) x (∏_j∈controlss_j∏_i∈ Rs_i). The two old-state products have certified degrees at most t+1+|R| and t+q+|R|. Since |R|+1≤ |L|, the invariant's budget rises by at most max{q-1,0}: zero for NOT and CNOT, one for Toffoli.
Exact mathematical passage
```
\left(s_u\prod_{i\in R}s_i\right)
\x
\left(\prod_{j\in\mathrm{controls}}s_j\prod_{i\in R}s_i\right).
```

\sinUse in source · 6 passages

Search aliases: sin

Link to this entry ·

Use in source · Read passage · q_lean.tex, line 764
QA19 · Relative-phase family · Prose extension sin^2(θ/2); not an inherited Lean theorem.
Exact mathematical passage
```
\sin^2(\theta/2)
```

\sin^2Use in source · 5 passages

Search aliases: sin^2 sin

Link to this entry ·

Use in source · Read passage · q_lean.tex, line 764
QA19 · Relative-phase family · Prose extension sin^2(θ/2); not an inherited Lean theorem.
Exact mathematical passage
```
\sin^2(\theta/2)
```

\operatorname{size}Use in source · 1 passage

Search aliases: size operatorname

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1857
Hybrid transfer graphs. Suppose packets are nodes of a sparse graph and each node carries an internal representation. A gate may update a node internally or transfer a compact component to neighboring nodes. The simulation cost depends on ∑_α∈ A_tsize(r_α) + #{nonzero transfers at time t}. This model combines symmetry selection rules with compact state classes.
Exact mathematical passage
```
\sum_{\alpha\in A_t}\operatorname{size}(r_\alpha)
+
\#\{\text{nonzero transfers at time }t\}.
```

\SLUse in source · 8 passages

Search aliases: SL

Link to this entry ·

Use in source · Read passage · g_main.tex, line 910
A CNOT circuit on n labeled wires represents an element of GL_n(F_2)=SL_n(F_2), and minimizing its gate count is the word-metric problem for transvection generators. This formulation supports asymptotically efficient synthesis, architecture-aware elimination, computable lower bounds, and exact small-width optimization [source reference]. Recent work has also produced stronger explicit lower-bound families and established NP-completeness of exact all-to-all CNOT distance [source reference]. The p…
Exact mathematical passage
```
\GL_n(\F_2)=\SL_n(\F_2)
```

\SL_nUse in source · 8 passages

Search aliases: SL_n SL

Link to this entry ·

Use in source · Read passage · g_main.tex, line 910
A CNOT circuit on n labeled wires represents an element of GL_n(F_2)=SL_n(F_2), and minimizing its gate count is the word-metric problem for transvection generators. This formulation supports asymptotically efficient synthesis, architecture-aware elimination, computable lower bounds, and exact small-width optimization [source reference]. Recent work has also produced stronger explicit lower-bound families and established NP-completeness of exact all-to-all CNOT distance [source reference]. The p…
Exact mathematical passage
```
\GL_n(\F_2)=\SL_n(\F_2)
```

\mathrm{small}Use in source · 1 passage

Search aliases: small mathrm

Link to this entry ·

Use in source · Read passage · g_main.tex, line 5165
a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). This transfers a catalytic atom to architectures containing it as a retract.
Exact mathematical passage
```
\Delta_{\mathrm{large}}(\iota U)
\ge
\Delta_{\mathrm{small}}(U).
```

\mathrm{small,res}Use in source · 1 passage

Search aliases: small,res mathrm

Link to this entry ·

Use in source · Read passage · g_main.tex, line 5159
Template VIII: block retraction. Let a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). This transfers a catalytic atom to architectures containing it as a retract.
Exact mathematical passage
```
\ell_{\mathrm{large,res}}(\iota U)
\ge
\ell_{\mathrm{small,res}}(U).
```

\mathrm{space-time}Use in source · 1 passage

Search aliases: space-time mathrm

Link to this entry ·

Use in source · Read passage · g_main.tex, line 3662
Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its total count is large. Conversely, a two-gate count saving can increase depth under locality constraints.
Exact mathematical passage
```
\Delta_{\mathrm{depth}}(U),
\qquad
\Delta_{\mathrm{2q-depth}}(U),
\qquad
\Delta_{\mathrm{space-time}}(U).
```

\SpanUse in source · 7 passages

Search aliases: Span

Link to this entry ·

Use in source · Read passage · c_main.tex, line 215
[Dual certificate for failure of safe detection] For every positive input v, β_v(f)>d Longleftrightarrow v_d(v)∈Span_F{v_d(u):u∈ U}. A degree-at-most-d polynomial is a coefficient covector on the evaluation space. It is safe exactly when it annihilates every negative evaluation vector, equivalently their span. Such a covector can equal one on v_d(v) exactly when v_d(v) lies outside that span. This is finite-dimensional linear duality over F.
Exact mathematical passage
```
\label{c:eq:dual}
 \beta_v(f)>d
 \quad\Longleftrightarrow\quad
 v_d(v)\in\Span_{\F}\{v_d(u):u\in U\}.
```

\Span_{\F}Use in source · 1 passage

Search aliases: Span_F Span F

Link to this entry ·

Use in source · Read passage · c_main.tex, line 215
[Dual certificate for failure of safe detection] For every positive input v, β_v(f)>d Longleftrightarrow v_d(v)∈Span_F{v_d(u):u∈ U}. A degree-at-most-d polynomial is a coefficient covector on the evaluation space. It is safe exactly when it annihilates every negative evaluation vector, equivalently their span. Such a covector can equal one on v_d(v) exactly when v_d(v) lies outside that span. This is finite-dimensional linear duality over F.
Exact mathematical passage
```
\label{c:eq:dual}
 \beta_v(f)>d
 \quad\Longleftrightarrow\quad
 v_d(v)\in\Span_{\F}\{v_d(u):u\in U\}.
```

\Span_{\F_2}Use in source · 1 passage

Search aliases: Span_F_2 Span F

Link to this entry ·

Use in source · Read passage · c_visual.tex, line 355
guidepageThe finite evaluation-span dual in full Fix a positive v and integer d≥0. Let v_d(x)=(x^S)_S⊆[n], |S|≤ d, U=f^-1(0), N=Span_F_2{v_d(u):u∈ U}. The constant monomial S=∅ is included. Degree-d coefficient covectors form the finite dual of this evaluation space. A covector is safe exactly when it annihilates N.
Exact mathematical passage
```
 v_d(x)=(x^S)_{S\subseteq[n],\ |S|\le d},\qquad
 U=f^{-1}(0),\qquad N=\Span_{\F_2}\{v_d(u):u\in U\}.
```

\squareUse in source · 20 passages

Search aliases: square

Link to this entry ·

Use in source · Read passage · q_main.tex, line 153
Proof. The equation (ζ_q^a)^t=1 holds exactly when q divides at. Write g=gcd(a,q), q=gq', and a=ga', with gcd(a',q')=1. The divisibility condition is q'mid a't, which is equivalent to q'mid t. Its smallest positive solution is t=q'. This includes a=0: then g=q and the order is one. square
Exact mathematical passage
```
\square
```

\operatorname{Stretch}Use in source · 2 passages

Search aliases: Stretch operatorname

Link to this entry ·

Use in source · Read passage · g_main.tex, line 276
The basic conjugation inequality. Let g∈ G and k∈ H satisfy gkg^-1∈ H. Then ℓ_G(gkg^-1) ≤ 2ℓ_G(g)+ℓ_H(k). Define the intrinsic conjugation stretch Stretch_g(k) = ℓ_H(gkg^-1)-ℓ_H(k).
Exact mathematical passage
```
\operatorname{Stretch}_g(k)
=
\ell_H(gkg^{-1})-\ell_H(k).
```

\operatorname{Stretch}_gUse in source · 2 passages

Search aliases: Stretch_g operatorname

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Use in source · Read passage · g_main.tex, line 276
The basic conjugation inequality. Let g∈ G and k∈ H satisfy gkg^-1∈ H. Then ℓ_G(gkg^-1) ≤ 2ℓ_G(g)+ℓ_H(k). Define the intrinsic conjugation stretch Stretch_g(k) = ℓ_H(gkg^-1)-ℓ_H(k).
Exact mathematical passage
```
\operatorname{Stretch}_g(k)
=
\ell_H(gkg^{-1})-\ell_H(k).
```

\SUMUse in source · 3 passages

Search aliases: SUM

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 51
Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.
Exact mathematical passage
```
\SUM^{(a)}_{s\to t}
\ket{x_s,x_t}
=
\ket{x_s,x_t+a x_s\bmod d}.
```

\SUM^{(-1)}_{q\to p}Use in source · 1 passage

Search aliases: SUM^(-1)_q→ p SUM to

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Use in source · Read passage · g_main.tex, line 3253
t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.
Exact mathematical passage
```
\SUM_{q\to p},
\quad
\SUM_{p\to t_0},
\quad
\SUM_{x\to p},
\quad
\SUM_{p\to t_1},
\quad
\SUM_{p\to t_2},
\quad
\SUM^{(-1)}_{q\to p}.
```

\SUM^{(a)}_{s\to t}Use in source · 2 passages

Search aliases: SUM^(a)_s→ t SUM to

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Use in source · Read passage · g_appendix.tex, line 51
Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.
Exact mathematical passage
```
\SUM^{(a)}_{s\to t}
\ket{x_s,x_t}
=
\ket{x_s,x_t+a x_s\bmod d}.
```

\SUM_{p\to t_0}Use in source · 1 passage

Search aliases: SUM_p→ t_0 SUM to

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Use in source · Read passage · g_main.tex, line 3253
t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.
Exact mathematical passage
```
\SUM_{q\to p},
\quad
\SUM_{p\to t_0},
\quad
\SUM_{x\to p},
\quad
\SUM_{p\to t_1},
\quad
\SUM_{p\to t_2},
\quad
\SUM^{(-1)}_{q\to p}.
```

\SUM_{p\to t_1}Use in source · 1 passage

Search aliases: SUM_p→ t_1 SUM to

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Use in source · Read passage · g_main.tex, line 3253
t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.
Exact mathematical passage
```
\SUM_{q\to p},
\quad
\SUM_{p\to t_0},
\quad
\SUM_{x\to p},
\quad
\SUM_{p\to t_1},
\quad
\SUM_{p\to t_2},
\quad
\SUM^{(-1)}_{q\to p}.
```

\SUM_{p\to t_2}Use in source · 1 passage

Search aliases: SUM_p→ t_2 SUM to

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Use in source · Read passage · g_main.tex, line 3253
t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.
Exact mathematical passage
```
\SUM_{q\to p},
\quad
\SUM_{p\to t_0},
\quad
\SUM_{x\to p},
\quad
\SUM_{p\to t_1},
\quad
\SUM_{p\to t_2},
\quad
\SUM^{(-1)}_{q\to p}.
```

\SUM_{q\to p}Use in source · 1 passage

Search aliases: SUM_q→ p SUM to

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Use in source · Read passage · g_main.tex, line 3253
t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.
Exact mathematical passage
```
\SUM_{q\to p},
\quad
\SUM_{p\to t_0},
\quad
\SUM_{x\to p},
\quad
\SUM_{p\to t_1},
\quad
\SUM_{p\to t_2},
\quad
\SUM^{(-1)}_{q\to p}.
```

\SUM_{x\to p}Use in source · 1 passage

Search aliases: SUM_x→ p SUM to

Link to this entry ·

Use in source · Read passage · g_main.tex, line 3253
t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.
Exact mathematical passage
```
\SUM_{q\to p},
\quad
\SUM_{p\to t_0},
\quad
\SUM_{x\to p},
\quad
\SUM_{p\to t_1},
\quad
\SUM_{p\to t_2},
\quad
\SUM^{(-1)}_{q\to p}.
```

\sup_{A\in\U_n(\F)}Use in source · 2 passages

Search aliases: sup_A∈U_n(F) sup in U F

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Use in source · Read passage · g_main.tex, line 1116
[Explicit nlog n distortion] For D_n(F) = sup_A∈U_n(F) (ellDown(A)-ellAll(A)), one has, for every field and every n, D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n. The coefficient is attained within the snapshot family by the twenty-one-dimensional seed A(9;0,1,…,9).
Exact mathematical passage
```
D_n(\F)
=
\sup_{A\in\U_n(\F)}
\bigl(\ellDown(A)-\ellAll(A)\bigr),
```

\sup_{A\in\U_n(\F),\,A\ne I}Use in source · 1 passage

Search aliases: sup_A∈U_n(F), A≠ I sup in U F ne

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Use in source · Read passage · g_main.tex, line 2193
[Asymptotic multiplicative distortion] For every field, limsup_n→∞ sup_A∈U_n(F), A≠ I fracellDown(A)ellAll(A) ≥frac32.
Exact mathematical passage
```
\limsup_{n\to\infty}
\sup_{A\in\U_n(\F),\,A\ne I}
\frac{\ellDown(A)}{\ellAll(A)}
\ge\frac32.
```

\sup_{n,A\in\U_n(\F)}Use in source · 1 passage

Search aliases: sup_n,A∈U_n(F) sup in U F

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Use in source · Read passage · g_main.tex, line 3042
[Multiplicative distortion] Is sup_n,A∈U_n(F) fracellDown(A)ellAll(A) bounded? The connected snapshot family gives an asymptotic lower bound of 3/2. Exceeding 3/2 appears to require a mechanism beyond one clean carrier exporting independent snapshots.
Exact mathematical passage
```
\sup_{n,A\in\U_n(\F)}
\frac{\ellDown(A)}{\ellAll(A)}
```

\suppUse in source · 2 passages

Search aliases: supp

Link to this entry ·

Use in source · Read passage · c_main.tex, line 241
Monotone specialization. [Safe monomial ideal for a monotone function] If f is monotone, I_f has a monomial basis indexed by the positive supports of f. For each positive v, β_v(f)=min{|S|: S⊆supp(v), f(1_S)=1}. Consequently, for nonzero f, α_1(f)=minimum minterm size, β(f)=maximum minterm size. For any Boolean polynomial h, its coefficient on x^S is [x^S]h=∑_T⊆ Sh(1_T) over F. If S is negative for monotone f, every subset T is negative. A safe h vanishes at each 1_T, so its x^S coefficient is z…
Exact mathematical passage
```
 \beta_v(f)=\min\{|S|: S\subseteq\supp(v),\ f(1_S)=1\}.
```

\SWAPUse in source · 1 passage

Search aliases: SWAP

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Use in source · Read passage · g_main.tex, line 3517
SWAP sensitivity. Add SWAP_ij to both alphabets as a unit-cost primitive. The target groups remain unchanged, but the metric changes.
Exact mathematical passage
```
\SWAP_{ij}
```

\SWAP_{ij}Use in source · 1 passage

Search aliases: SWAP_ij SWAP

Link to this entry ·

Use in source · Read passage · g_main.tex, line 3517
SWAP sensitivity. Add SWAP_ij to both alphabets as a unit-cost primitive. The target groups remain unchanged, but the metric changes.
Exact mathematical passage
```
\SWAP_{ij}
```

\SymUse in source · 1 passage

Search aliases: Sym

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Use in source · Read passage · s_main.tex, line 961
Symmetric qudits. For local dimension d, the symmetric subspace has occupation-number basis (n_1,…,n_d), n_1+⋯+n_d=n, and dimension dimSym^n(C^d) = binomn+d-1d-1. For fixed d, this is polynomial in n.
Exact mathematical passage
```
\dim\Sym^n(\C^d)
=
\binom{n+d-1}{d-1}.
\label{s:eq:symmetric-qudit-dim}
```

\Sym^nUse in source · 1 passage

Search aliases: Sym^n Sym

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Use in source · Read passage · s_main.tex, line 961
Symmetric qudits. For local dimension d, the symmetric subspace has occupation-number basis (n_1,…,n_d), n_1+⋯+n_d=n, and dimension dimSym^n(C^d) = binomn+d-1d-1. For fixed d, this is polynomial in n.
Exact mathematical passage
```
\dim\Sym^n(\C^d)
=
\binom{n+d-1}{d-1}.
\label{s:eq:symmetric-qudit-dim}
```

T^{-1}Use in source · 2 passages

Search aliases: T^-1

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Use in source · Read passage · g_main.tex, line 3776
A formal nonlinear catalyst. A clean one-workspace template is T h_m⋯ h_1 T^-1, where T is Toffoli and the middle gates use the quadratic term created in the workspace. Since Toffoli is self-inverse, the displayed circuit uses two nonlinear gates.
Exact mathematical passage
```
T\;h_m\cdots h_1\;T^{-1},
```

t^4Use in source · 2 passages

Search aliases: t^4

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Use in source · Read passage · c_main.tex, line 620
The exact tests use GF(16)=F[t]/(t^4+t+1). They evaluate every specified verifier through three witness bits at every field point, while the independent top-coefficient check uses all field points through two witness bits and the declared smaller coordinate grid at three bits. These domains must not be conflated.
Exact mathematical passage
```
\mathrm{GF}(16)=\F[t]/(t^4+t+1)
```

T^kUse in source · 1 passage

Search aliases: T^k

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Use in source · Read passage · g_main.tex, line 3944
Phase polynomials and parity carriers. A CNOT network computes linear parities λ(x)=a· x∈F_2. Applying T^k to a wire carrying λ(x) contributes ω^kλ(x) to the phase polynomial. A parity network must create each required parity, expose it to a phase gate, and restore the linear map required at the endpoint.
Exact mathematical passage
```
T^k
```

t^mUse in source · 1 passage

Search aliases: t^m

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Use in source · Read passage · q_main.tex, line 534
E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).
Exact mathematical passage
```
E_rf(t)=\sum_{m\ge0}a_{r+Lm}t^m,
\qquad
R_rf(z)=z^rE_rf(z^L).
```

t_0Use in source · 3 passages

Search aliases: t_0

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Use in source · Read passage · g_main.tex, line 3242
The universal six-qudit circuit. Use the coordinate names x,p,q,t_0,t_1,t_2. The target acts on basis labels by p · ↦ p+x,
Exact mathematical passage
```
x,p,q,t_0,t_1,t_2.
```

T_1Use in source · 1 passage

Search aliases: T_1

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Use in source · Read passage · g_main.tex, line 4545
Spherical packets. Let T_1,…,T_m be commuting normal operators on a finite-dimensional Hilbert space. Their unique coarsest joint spectral resolution is T_j=∑_λt_j(λ)B_λ, ∑_λ B_λ=I. The tuple λ=(t_1(λ),…,t_m(λ)) is the spherical label. The abstract spectral packet records these labels and the ranks of B_λ.
Exact mathematical passage
```
T_1,\ldots,T_m
```

t_1Use in source · 11 passages

Search aliases: t_1

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Use in source · Read passage · g_main.tex, line 846
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
\ell_\Psi(u)
=
\min\{L:u=x_{\beta_L}(t_L)\cdots x_{\beta_1}(t_1),\ \beta_i\in\Psi\},
```

t_2Use in source · 3 passages

Search aliases: t_2

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Use in source · Read passage · g_main.tex, line 3242
The universal six-qudit circuit. Use the coordinate names x,p,q,t_0,t_1,t_2. The target acts on basis labels by p · ↦ p+x,
Exact mathematical passage
```
x,p,q,t_0,t_1,t_2.
```

t_iUse in source · 1 passage

Search aliases: t_i

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Use in source · Read passage · g_main.tex, line 2405
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\sigma=((t_1,s_1),\ldots,(t_L,s_L)),
\qquad t_i\ne s_i,
```

T_jUse in source · 3 passages

Search aliases: T_j

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Use in source · Read passage · g_main.tex, line 4546
Spherical packets. Let T_1,…,T_m be commuting normal operators on a finite-dimensional Hilbert space. Their unique coarsest joint spectral resolution is T_j=∑_λt_j(λ)B_λ, ∑_λ B_λ=I. The tuple λ=(t_1(λ),…,t_m(λ)) is the spherical label. The abstract spectral packet records these labels and the ranks of B_λ.
Exact mathematical passage
```
T_j=\sum_{\lambda}t_j(\lambda)B_\lambda,
\qquad
\sum_\lambda B_\lambda=I.
```

t_jUse in source · 4 passages

Search aliases: t_j

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Use in source · Read passage · g_main.tex, line 2021
[ 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.
Exact mathematical passage
```
\label{g:eq:snapshot-output}
R_{t_j}=e_{t_j}+e_q+w_{s_j}
\qquad(1\le j\le v).
```

t_j^{\mathrm{close}}Use in source · 1 passage

Search aliases: t_j^close mathrm

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Use in source · Read passage · g_main.tex, line 343
Catalytic interval geometry. Associate to every matched opening and closing operation an interval I_j=[t_j^open,t_j^close]. The family of intervals can be disjoint, nested, laminar, or crossing.
Exact mathematical passage
```
I_j=[t_j^{\mathrm{open}},t_j^{\mathrm{close}}].
```

t_j^{\mathrm{open}}Use in source · 1 passage

Search aliases: t_j^open mathrm

Link to this entry ·

Use in source · Read passage · g_main.tex, line 343
Catalytic interval geometry. Associate to every matched opening and closing operation an interval I_j=[t_j^open,t_j^close]. The family of intervals can be disjoint, nested, laminar, or crossing.
Exact mathematical passage
```
I_j=[t_j^{\mathrm{open}},t_j^{\mathrm{close}}].
```

t_LUse in source · 3 passages

Search aliases: t_L

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Use in source · Read passage · g_main.tex, line 846
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
\ell_\Psi(u)
=
\min\{L:u=x_{\beta_L}(t_L)\cdots x_{\beta_1}(t_1),\ \beta_i\in\Psi\},
```

T_mUse in source · 1 passage

Search aliases: T_m

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Use in source · Read passage · g_main.tex, line 4545
Spherical packets. Let T_1,…,T_m be commuting normal operators on a finite-dimensional Hilbert space. Their unique coarsest joint spectral resolution is T_j=∑_λt_j(λ)B_λ, ∑_λ B_λ=I. The tuple λ=(t_1(λ),…,t_m(λ)) is the spherical label. The abstract spectral packet records these labels and the ranks of B_λ.
Exact mathematical passage
```
T_1,\ldots,T_m
```

t_mUse in source · 3 passages

Search aliases: t_m

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1170
Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_1(a_1).
Exact mathematical passage
```
w=(\tau_{t_m,s_m}(a_m),\ldots,\tau_{t_1,s_1}(a_1))
```

T_{q,r}Use in source · 3 passages

Search aliases: T_q,r

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Use in source · Read passage · g_main.tex, line 4344
Sector-flow invariants. Let P_q be the projector onto sector q. For an intermediate unitary U_t, define transfer blocks T_q,r(t)=P_q U_t P_r. A restricted path has T_q,r=0 for q≠ r. Ambient paths may populate off-diagonal blocks. Possible excursion measures include ∑_q≠ rrank T_q,r, max_q≠ r T_q,r , bandwidth in |q-r|. These are analogues of strict-upper rank and bandwidth in the unitriangular theory.
Exact mathematical passage
```
T_{q,r}(t)=P_q U_t P_r.
```

T_{qr}Use in source · 1 passage

Search aliases: T_qr

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Use in source · Read passage · g_main.tex, line 5245
Template XIV: symmetry-sector escape. Given projectors P_q, define off-sector transfer blocks T_qr=P_qUP_r. A restricted gate preserves all diagonal sectors. An ambient gate changes a controlled collection of blocks. A candidate potential is a weighted sum of ranks or norms of obligations relative to the final target. The proof must charge both creation and useful consumption of off-sector coherence.
Exact mathematical passage
```
T_{qr}=P_qUP_r.
```

T_{\mathrm{rest}}Use in source · 1 passage

Search aliases: T_rest mathrm

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Use in source · Read passage · c_visual.tex, line 451
guidepageExact nonlinear projection costs versus actual input length The source proves the all-size formulas M(P_m)=m-1, T_rest(P_m)=2m-3, M(Q_m)=T_rest(Q_m)=n/2. The chart uses actual input length n=2^m-2 on its horizontal axis.
Exact mathematical passage
```
 M(P_m)=m-1,\quad T_{\mathrm{rest}}(P_m)=2m-3,\qquad
 M(Q_m)=T_{\mathrm{rest}}(Q_m)=n/2.
```

t_vUse in source · 2 passages

Search aliases: t_v

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2001
Fix integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.
Exact mathematical passage
```
x_1,\ldots,x_u,\ p,\ q,\ t_1,\ldots,t_v.
```

\mathrm{target}Use in source · 1 passage

Search aliases: target mathrm

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Use in source · Read passage · c_lean.tex, line 213
pageheadReading the formal model literallyQUANTIFIERS AND ENCODING CARRY THE HYPOTHESES The generic cut theory uses a logical-input type X, a wire type W with decidable equality, and a boundary structure. Its essential fields are an encoding encode:X→Bool→(W→Bool), a target wire, and the proof encode(x,b)(target)=b. This target equation separates the two target branches for each fixed input. It excludes an encoding that silently ignores its supplied target bit. The source proves codeBoundary.tar…
Exact mathematical passage
```
 \mathrm{encode}(x,b)(\mathrm{target})=b.
```

\operatorname{TC}Use in source · 1 passage

Search aliases: TC operatorname

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1942
A representation-relative invariant. For a declared classical representation class R, define TC^ R_ε(C) = min{ cost(R): R∈ R, |R(C)-p_C|≤ε }, where p_C is the target acceptance probability. This permits a disciplined ladder:
Exact mathematical passage
```
\operatorname{TC}^{\mathcal R}_\varepsilon(C)
=
\min\left\{
\operatorname{cost}(R):
R\in\mathcal R,
\ |R(C)-p_C|\le\varepsilon
\right\},
```

\operatorname{TC}^{\mathcal R}_\varepsilonUse in source · 1 passage

Search aliases: TC^ R_ε operatorname mathcal varepsilon

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Use in source · Read passage · s_main.tex, line 1942
A representation-relative invariant. For a declared classical representation class R, define TC^ R_ε(C) = min{ cost(R): R∈ R, |R(C)-p_C|≤ε }, where p_C is the target acceptance probability. This permits a disciplined ladder:
Exact mathematical passage
```
\operatorname{TC}^{\mathcal R}_\varepsilon(C)
=
\min\left\{
\operatorname{cost}(R):
R\in\mathcal R,
\ |R(C)-p_C|\le\varepsilon
\right\},
```

\TOFUse in source · 1 passage

Search aliases: TOF

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Use in source · Read passage · g_appendix.tex, line 157
Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric group.
Exact mathematical passage
```
\TOF_{a,b\to t}:x_t\leftarrow x_t+x_ax_b.
```

\TOF_{a,b\to t}Use in source · 1 passage

Search aliases: TOF_a,b→ t TOF to

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Use in source · Read passage · g_appendix.tex, line 157
Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric group.
Exact mathematical passage
```
\TOF_{a,b\to t}:x_t\leftarrow x_t+x_ax_b.
```

\TrUse in source · 5 passages

Search aliases: Tr

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Use in source · Read passage · s_main.tex, line 1067
[Invariant observables see only the twirl] If A commutes with every U_π, then Tr(Aρ) = Tr(A T_n(ρ)) for every density operator ρ.
Exact mathematical passage
```
\Tr(A\rho)
=
\Tr\bigl(A\mathcal T_n(\rho)\bigr)
```

\operatorname{tr}Use in source · 6 passages

Search aliases: tr operatorname

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Use in source · Read passage · g_main.tex, line 519
Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint series after Fourier inversion.
Exact mathematical passage
```
\mu_k(h)
=
\frac1{|H|}
\sum_{\tau\in\widehat H}
\dim(\tau)
\operatorname{tr}
\bigl(
\tau(h^{-1})\widehat\mu_k(\tau)
\bigr).
```

\mathrm{transfer}Use in source · 2 passages

Search aliases: transfer mathrm

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Use in source · Read passage · s_main.tex, line 33
The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:
Exact mathematical passage
```
\mathfrak C_n=
\bigl(
N_{\mathrm{packets}},
D_{\mathrm{internal}},
S_{\mathrm{transfer}},
L_{\mathrm{leakage}}
\bigr).
```

\mathcal UUse in source · 5 passages

Search aliases: U mathcal

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Use in source · Read passage · s_main.tex, line 988
Decomposition. For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.
Exact mathematical passage
```
(\C^d)^{\otimes n}
\cong
\bigoplus_{\lambda\vdash n,\,\ell(\lambda)\le d}
\mathcal U_\lambda\otimes\mathcal V_\lambda,
```

U^*Use in source · 2 passages

Search aliases: U^*

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Use in source · Read passage · s_main.tex, line 1090
If a unitary U and acceptance operator M are permutation invariant, then U^*MU is invariant. Therefore the final probability depends on the input only through its twirl. This permits an observable-specific collapse even when the original pure input is not itself symmetric.
Exact mathematical passage
```
U^*MU
```

U^\daggerUse in source · 1 passage

Search aliases: U^† dagger

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Use in source · Read passage · s_main.tex, line 1617
Symmetry-preserving unitary dynamics. If U commutes with every packet projector, then P_α Uρ U^† P_β = U(P_αρ P_β)U^†. The left and right packet labels are preserved. Populations and coherences evolve independently by their declared labels.
Exact mathematical passage
```
P_\alpha U\rho U^\dagger P_\beta
=
U(P_\alpha\rho P_\beta)U^\dagger.
```

U_1Use in source · 1 passage

Search aliases: U_1

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Use in source · Read passage · q_main.tex, line 287
At q=1, U_1={0} because gcd(0,1)=1. The phase sum and the divisor formula both give c_1(n)=1. This boundary case is included in the proofs even though the inherited arithmetic regression range begins at modulus two.
Exact mathematical passage
```
U_1=\{0\}
```

U_{12}Use in source · 14 passages

Search aliases: U_12

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Use in source · Read passage · q_main.tex, line 680
U_12={1,5,7,11}, φ(12)=4, ζ=e^π i/6=fracsqrt32+frac i2.
Exact mathematical passage
```
U_{12}=\{1,5,7,11\},
\qquad \varphi(12)=4,
\qquad \zeta=e^{\pi i/6}=\frac{\sqrt3}{2}+\frac i2.
```

\U_{2m}Use in source · 1 passage

Search aliases: U_2m U

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Use in source · Read passage · g_main.tex, line 2660
For A∈U_m(F), define Z(A) = A · 0 A · A ∈U_2m(F).
Exact mathematical passage
```
\label{g:eq:zeta-lift}
\mathcal Z(A)
=
\begin{pmatrix}
A&0\\
A&A
\end{pmatrix}
\in\U_{2m}(\F).
```

U_6Use in source · 1 passage

Search aliases: U_6

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Use in source · Read passage · g_appendix.tex, line 245
Core binary six-dimensional census. For U_6(F_2) there are 2^15=32,768 targets. The complete comparison is: @rrr@ Restricted distance · Ambient distance · Number of targets
Exact mathematical passage
```
U_6(\F_2)
```

\U_6Use in source · 11 passages

Search aliases: U_6 U

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Use in source · Read passage · g_main.tex, line 943
[Universal catalytic atom over rings] For every nonzero commutative ring R with identity, let C= 1 · 0 · 0 · 0 · 0 · 0 1 · 1 · 0 · 0 · 0 · 0 0 · 0 · 1 · 0 · 0 · 0 0 · 1 · 1 · 1 · 0 · 0 1 · 1 · 1 · 0 · 1 · 0 1 · 1 · 1 · 0 · 0 · 1 ∈U_6(R). Then ellDown(C)=7, ellAll(C)=6.
Exact mathematical passage
```
\label{g:eq:canonical-C}
C=
\begin{pmatrix}
1&0&0&0&0&0\\
1&1&0&0&0&0\\
0&0&1&0&0&0\\
0&1&1&1&0&0\\
1&1&1&0&1&0\\
1&1&1&0&0&1
\end{pmatrix}
\in\U_6(R).
```

U_DUse in source · 1 passage

Search aliases: U_D

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Use in source · Read passage · g_main.tex, line 5233
Template XIII: ancilla restoration. Let W act on data D and ancilla A. To certify a dirty catalyst, prove the operator identity W=U_D⊗ I_A on the full joint space, not only on one ancilla input. For a clean catalyst, prove W(ketψ⊗ket0) =(Uketψ)⊗ket0 for all ketψ. The distinction changes both the target equations and lower bounds.
Exact mathematical passage
```
W=U_D\otimes I_A
```

U_dUse in source · 1 passage

Search aliases: U_d

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Use in source · Read passage · q_main.tex, line 381
(F_ND_dF_N^†)_nm · =frac1N∑_a∈ S_dζ_N^a(n-m) · =frac1N∑_b∈ U_dζ_d^b(n-m) =frac1N c_d(n-m).
Exact mathematical passage
```
\begin{aligned}
(F_ND_dF_N^\dagger)_{nm}
&=\frac1N\sum_{a\in S_d}\zeta_N^{a(n-m)}\\
&=\frac1N\sum_{b\in U_d}\zeta_d^{b(n-m)}
=\frac1N c_d(n-m).
\end{aligned}
```

U_gUse in source · 3 passages

Search aliases: U_g

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Use in source · Read passage · s_main.tex, line 157
Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).
Exact mathematical passage
```
U_g
```

U_iUse in source · 1 passage

Search aliases: U_i

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Use in source · Read passage · s_main.tex, line 115
[Invariant quantum packets] If a unitary U satisfies [U,Q]=0, then P_iUP_j=0 (i≠ j), and U=⊕_i U_i, U_i=P_iUP_i|_P_iHh. If P_kketψ=ketψ, then P_kUketψ=Uketψ.
Exact mathematical passage
```
U=\bigoplus_i U_i,
\qquad
U_i=P_iUP_i\big|_{P_i\Hh}.
```

u_jUse in source · 1 passage

Search aliases: u_j

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Use in source · Read passage · s_main.tex, line 1193
Qubit corollary. For qubits, Schur labels are total spins j. The carrier dimension is u_j=2j+1≤ n+1, and there are at most lfloor n/2rfloor+1 labels. Hence ∑_j(2j+1)^2=O(n^3).
Exact mathematical passage
```
u_j=2j+1\le n+1,
```

U_{ji}Use in source · 1 passage

Search aliases: U_ji

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Use in source · Read passage · s_main.tex, line 1324
Gate updates. A number-preserving one-body unitary acts on mode creation operators by a_i^† ⟼ ∑_j U_jia_j^†. A generic dense mode transformation can induce many occupation-basis transitions, but for fixed N the basis dimension remains polynomial. Local hopping and phase gates are sparse. Few-body interactions remain polynomial to apply for fixed N, although the exponent can grow with N.
Exact mathematical passage
```
a_i^\dagger
\longmapsto
\sum_j U_{ji}a_j^\dagger.
```

u_kUse in source · 1 passage

Search aliases: u_k

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Use in source · Read passage · q_visual.tex, line 150
The arithmetic signal is a coherent amplitude vector. addcontentslinetocsectionThe arithmetic signal is a coherent amplitude vector leadNormalization makes the signal a unit vector without changing its relative phases. Define u_k=1/2 for k∈ U_12 and zero otherwise. Then u =1, and ψ=Fu, ψ_n=fracc_12(n)sqrt48, F^†ψ=u. fig11_same_state_two_bases1.0Separate panels preserve the different basis labels: k for the frequency-basis coordinates and n for the sample-basis coordinates. Heights are amplitudes…
Exact mathematical passage
```
u_k=1/2
```

\U_mUse in source · 4 passages

Search aliases: U_m U

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Use in source · Read passage · g_main.tex, line 2659
For A∈U_m(F), define Z(A) = A · 0
Exact mathematical passage
```
A\in\U_m(\F)
```

\U_{n_1+\cdots+n_k}Use in source · 1 passage

Search aliases: U_n_1+⋯+n_k U cdots

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Use in source · Read passage · g_main.tex, line 2611
For the lower bound, view an element of U_n_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in every factor. If m_i gates survive in factor i, then the total word length is at least ∑_i m_i ≥ ∑_iellDow…
Exact mathematical passage
```
\U_{n_1+\cdots+n_k}(\F)
```

\U_{n_i}Use in source · 1 passage

Search aliases: U_n_i U

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Use in source · Read passage · g_main.tex, line 2600
[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).
Exact mathematical passage
```
A_i\in\U_{n_i}(\F)
```

U_PUse in source · 2 passages

Search aliases: U_P

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Use in source · Read passage · g_main.tex, line 830
Pattern groups and architecture constraints. For a poset P, define the pattern group U_P(K) = {I+A:A_ij=0 unless j<_P i}. When the allowed positions are transitively closed, U_P is a group. Compare the intrinsic metric generated by P-allowed transvections with the metric induced by all transvections.
Exact mathematical passage
```
U_P(K)
=
\{I+A:A_{ij}=0\text{ unless }j<_P i\}.
```

U_qUse in source · 5 passages

Search aliases: U_q

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Use in source · Read passage · q_main.tex, line 222
U_q={a∈ I_q:gcd(a,q)=1}, φ(q)=|U_q|.
Exact mathematical passage
```
U_q=\{a\in I_q:\gcd(a,q)=1\},
\qquad \varphi(q)=|U_q|.
```

U_sUse in source · 1 passage

Search aliases: U_s

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Use in source · Read passage · g_main.tex, line 780
C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
P_r\in U_s
```

U_tUse in source · 3 passages

Search aliases: U_t

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Use in source · Read passage · g_main.tex, line 780
C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
Q_{>r}\in U_t
```

\U_{u+v+2}Use in source · 2 passages

Search aliases: U_u+v+2 U

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Use in source · Read passage · g_main.tex, line 1014
[Connected snapshot family] Let u≥1, let 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<⋯<s_v≤ u. There is a connected target A(u;s_1,…,s_v)∈U_u+v+2(F) with ellDown(A(u;s_1,…,s_v))=u+2v and ellAll(A(u;s_1,…,s_v))≤ u+v+2. Hence its certified defect is at least v-2.
Exact mathematical passage
```
A(u;s_1,\ldots,s_v)\in\U_{u+v+2}(\F)
```

U_{XY}Use in source · 1 passage

Search aliases: U_XY

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Use in source · Read passage · s_main.tex, line 748
Number-preserving gates. Single-qubit R_Z gates commute with widehat N. A two-qubit exchange rotation acts in the ordered basis ket00,ket01,ket10,ket11 as U_XY(θ,ϕ) = 1 · 0 · 0 · 0 0 · cosθ · -ie^-iϕsinθ · 0 0 · -ie^iϕsinθ · cosθ · 0 0 · 0 · 0 · 1 . It rotates ket01 and ket10 while fixing ket00 and ket11, so it preserves exact Hamming weight.
Exact mathematical passage
```
U_{XY}(\theta,\phi)
=
\begin{pmatrix}
1&0&0&0\\
0&\cos\theta&-ie^{-i\phi}\sin\theta&0\\
0&-ie^{i\phi}\sin\theta&\cos\theta&0\\
0&0&0&1
\end{pmatrix}.
```

U_\lambdaUse in source · 2 passages

Search aliases: U_λ lambda

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Use in source · Read passage · s_main.tex, line 1124
A permutation-invariant unitary has the form U = ⊕_λ U_λ⊗ I_ V_λ, and a permutation-invariant measurement operator has the analogous form M = ⊕_λ M_λ⊗ I_ V_λ. The output probability is Tr(MUρ U^*) = ∑_λ p_λ Tr(M_λ U_λρ_λ U_λ^*).
Exact mathematical passage
```
U
=
\bigoplus_\lambda
U_\lambda\otimes I_{\mathcal V_\lambda},
\label{s:eq:commutant-unitary-form}
```

\mathcal U_\lambdaUse in source · 5 passages

Search aliases: U_λ mathcal lambda

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Use in source · Read passage · s_main.tex, line 988
Decomposition. For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.
Exact mathematical passage
```
(\C^d)^{\otimes n}
\cong
\bigoplus_{\lambda\vdash n,\,\ell(\lambda)\le d}
\mathcal U_\lambda\otimes\mathcal V_\lambda,
```

u_\lambdaUse in source · 2 passages

Search aliases: u_λ lambda

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Use in source · Read passage · s_main.tex, line 1094
Exact commutant record. Let u_λ=dim U_λ, f_λ=dim V_λ. Schur's lemma gives the following normal form.
Exact mathematical passage
```
u_\lambda=\dim\mathcal U_\lambda,
\qquad
f_\lambda=\dim\mathcal V_\lambda.
```

U_\lambda^*Use in source · 1 passage

Search aliases: U_λ^* lambda

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Use in source · Read passage · s_main.tex, line 1139
A permutation-invariant unitary has the form U = ⊕_λ U_λ⊗ I_ V_λ, and a permutation-invariant measurement operator has the analogous form M = ⊕_λ M_λ⊗ I_ V_λ. The output probability is Tr(MUρ U^*) = ∑_λ p_λ Tr(M_λ U_λρ_λ U_λ^*).
Exact mathematical passage
```
\Tr(MU\rho U^*)
=
\sum_\lambda
p_\lambda
\Tr\bigl(M_\lambda U_\lambda\rho_\lambda U_\lambda^*\bigr).
\label{s:eq:commutant-output}
```

u_\lambda^2Use in source · 1 passage

Search aliases: u_λ^2 lambda

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1166
umber of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).
Exact mathematical passage
```
\sum_\lambda u_\lambda^2
=O_d\left(n^{d^2-1}\right).
\label{s:eq:commutant-parameter-bound}
```

U_\piUse in source · 4 passages

Search aliases: U_π pi

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Use in source · Read passage · s_main.tex, line 1054
The permutation twirl. Define the twirling channel T_n(X) = frac1n! ∑_π∈ S_n U_π XU_π^*. Its image is the commutant of the permutation action.
Exact mathematical passage
```
\mathcal T_n(X)
=
\frac1{n!}
\sum_{\pi\in S_n}
U_\pi XU_\pi^*.
\label{s:eq:permutation-twirl}
```

U_\pi^*Use in source · 3 passages

Search aliases: U_π^* pi

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Use in source · Read passage · s_main.tex, line 1054
The permutation twirl. Define the twirling channel T_n(X) = frac1n! ∑_π∈ S_n U_π XU_π^*. Its image is the commutant of the permutation action.
Exact mathematical passage
```
\mathcal T_n(X)
=
\frac1{n!}
\sum_{\pi\in S_n}
U_\pi XU_\pi^*.
\label{s:eq:permutation-twirl}
```

U_\PsiUse in source · 1 passage

Search aliases: U_Ψ Psi

Link to this entry ·

Use in source · Read passage · g_main.tex, line 845
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
U_\Psi
```

\underline{\operatorname{rk}}Use in source · 1 passage

Search aliases: underlinerk underline operatorname

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Use in source · Read passage · g_main.tex, line 700
Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but tensor-regularized defect positive, item ambient and restricted border ranks both smaller than exact ranks but by different amounts. Each requires a se…
Exact mathematical passage
```
\underline{\operatorname{rk}}_{X}(U)
=
\min\{L:U\in\overline{X^L}\}.
```

\underline{\operatorname{rk}}_{X}Use in source · 1 passage

Search aliases: underlinerk_X underline operatorname

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Use in source · Read passage · g_main.tex, line 700
Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but tensor-regularized defect positive, item ambient and restricted border ranks both smaller than exact ranks but by different amounts. Each requires a se…
Exact mathematical passage
```
\underline{\operatorname{rk}}_{X}(U)
=
\min\{L:U\in\overline{X^L}\}.
```

\underline\ell_XUse in source · 1 passage

Search aliases: underlineℓ_X underline ell

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2582
For an algebraic generating set X, define the border product length underlineℓ_X(A) = min{L:A∈overlineX^L}. This is analogous to the passage from tensor rank to border rank [source reference]. The generic value is still N by [source reference], but exact and border defects can differ on exceptional strata. It is presently open whether the canonical atom C belongs to the closure of the six-factor lower transvection locus. A negative answer would show that its one-gate defect is stable under algeb…
Exact mathematical passage
```
\underline\ell_X(A)
=
\min\{L:A\in\overline{X^L}\}.
```

\mathcal VUse in source · 12 passages

Search aliases: V mathcal

Link to this entry ·

Use in source · Read passage · g_main.tex, line 744
Intermediate states as a representable matroid. A linear circuit visits vectors that satisfy exact dependence relations. If V is the set of all row states appearing during a computation, then V determines a representable matroid. A transvection u⟼ u+a v creates a three-point projective line [u], [v], [u+av]. The ambient circuit chooses a tour through this projective configuration. A flag-preserving circuit must lift the same dependencies through prescribed pivot positions.
Exact mathematical passage
```
\mathcal V
```

v^\daggerUse in source · 1 passage

Search aliases: v^† dagger

Link to this entry ·

Use in source · Read passage · q_main.tex, line 1378
For a normalized vector v, its density matrix is vv^†. Complete dephasing in the sample basis keeps only its diagonal entries. In the present example, dephasing either ψ or χ gives the same diagonal matrix
Exact mathematical passage
```
vv^\dagger
```

v_+Use in source · 7 passages

Search aliases: v_+

Link to this entry ·

Use in source · Read passage · q_main.tex, line 1304
The two endpoint states are especially simple, but the mechanism is continuous. Define the nonnegative vectors v_+ and v_- by placing |c(n)|/sqrt48 respectively on the positive and negative entries of c, with zero elsewhere. Thus
Exact mathematical passage
```
v_+
```

v_-Use in source · 7 passages

Search aliases: v_-

Link to this entry ·

Use in source · Read passage · q_main.tex, line 1304
The two endpoint states are especially simple, but the mechanism is continuous. Define the nonnegative vectors v_+ and v_- by placing |c(n)|/sqrt48 respectively on the positive and negative entries of c, with zero elsewhere. Thus
Exact mathematical passage
```
v_-
```

V_0Use in source · 1 passage

Search aliases: V_0

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1240
, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0⊂ V_1⊂⋯⊂ V_n-1. An upper-directed gate leaves this fiber of the flag map. A strict defect means that the flag-compatible frame subgraph is not geodesi…
Exact mathematical passage
```
V_0\subset V_1\subset\cdots\subset V_{n-1}.
```

v_0Use in source · 9 passages

Search aliases: v_0

Link to this entry ·

Use in source · Read passage · c_main.tex, line 532
[Scalar unital summaries are evaluations] Every unital K-linear multiplicative map Φ:K^W→ K is evaluation at exactly one witness. An idempotent scalar in a field is either zero or one. Orthogonality allows at most one Φ(δ_v) to be one, and preservation of the unit requires exactly one. If it occurs at v_0, linearity gives Φ(g)=Φ(∑_v g(v)δ_v)=g(v_0). For m≥1, such a scalar summary loses at least one singleton witness other than v_0. At m=0, the unique empty witness is preserved by the identity ma…
Exact mathematical passage
```
v_0
```

V_1Use in source · 4 passages

Search aliases: V_1

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Use in source · Read passage · g_main.tex, line 1240
, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0⊂ V_1⊂⋯⊂ V_n-1. An upper-directed gate leaves this fiber of the flag map. A strict defect means that the flag-compatible frame subgraph is not geodesi…
Exact mathematical passage
```
V_0\subset V_1\subset\cdots\subset V_{n-1}.
```

v_1Use in source · 1 passage

Search aliases: v_1

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Use in source · Read passage · c_visual.tex, line 330
guidepageA degree-one dual certificate for two-input AND Let f(a,b)=ab, with unique positive point (1,1). For degree at most one, the evaluation vector is v_1(a,b)=(1,a,b). The three negative points already span the positive vector.
Exact mathematical passage
```
v_1(a,b)=(1,a,b)
```

v_dUse in source · 13 passages

Search aliases: v_d

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Use in source · Read passage · c_main.tex, line 208
Exact evaluation-span certificates. For an integer d≥0, form the evaluation vector v_d(x)=(x^S)_S⊆[n], |S|≤ d, and write U=f^-1(0).
Exact mathematical passage
```
 v_d(x)=\bigl(x^S\bigr)_{S\subseteq[n],\ |S|\le d},
```

V_iUse in source · 1 passage

Search aliases: V_i

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Use in source · Read passage · g_main.tex, line 1232
For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0⊂ V_1⊂⋯⊂ V_n-1. An upper-directed gate leaves this fiber of the flag map. A strict defect means tha…
Exact mathematical passage
```
V_i=\Span(e_0,\ldots,e_i).
```

v_iUse in source · 2 passages

Search aliases: v_i

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Use in source · Read passage · c_lean.tex, line 135
The least safe degree exists. For a positive v, take the point indicator δ_v(x)=∏_i=1^n x_i, · v_i=1, 1+x_i, · v_i=0. It is safe, detects v, and has a degree-n certificate. Hence complete safe coverage exists at degree n for every function. Strong induction proves the existence of a least admissible natural degree. The definition pathrefbeta chooses this bound and its specification and minimality are proved.
Exact mathematical passage
```
\delta_v(x)=\prod_{i=1}^{n}
\begin{cases}
x_i,&v_i=1,\\
1+x_i,&v_i=0.
\end{cases}
```

V_{i-1}Use in source · 1 passage

Search aliases: V_i-1

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Use in source · Read passage · g_main.tex, line 1236
For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0⊂ V_1⊂⋯⊂ V_n-1. An upper-directed gate leaves this fiber of the flag map. A strict defect means tha…
Exact mathematical passage
```
R_i\in e_i+V_{i-1}
```

V_jUse in source · 3 passages

Search aliases: V_j

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Use in source · Read passage · s_appendix.tex, line 18
Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].
Exact mathematical passage
```
V_j=PG_jP
```

V_kUse in source · 3 passages

Search aliases: V_k

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Use in source · Read passage · s_appendix.tex, line 19
Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].
Exact mathematical passage
```
G_m\cdots G_1P-V_m\cdots V_1P
=
\sum_{k=1}^m
G_m\cdots G_{k+1}(G_k-V_k)V_{k-1}\cdots V_1P.
```

v_kUse in source · 1 passage

Search aliases: v_k

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Use in source · Read passage · c_main.tex, line 353
An explicit moment certificate. On the face b=1, every point except v=(a=0,z=(1,…,1)) is negative for G_k. A polynomial of degree at most k sums to zero over the (k+1)-dimensional cube: each monomial omits some coordinate and contributes an even number of ones. Therefore v_k(v)=∑_u≠ v on the facev_k(u). This is the certificate from Theorem~[source reference], proving that no safe degree-k polynomial detects v.
Exact mathematical passage
```
 v_k(v)=\sum_{u\ne v\text{ on the face}}v_k(u).
```

V_{k-1}Use in source · 2 passages

Search aliases: V_k-1

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Use in source · Read passage · s_appendix.tex, line 19
Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].
Exact mathematical passage
```
G_m\cdots G_1P-V_m\cdots V_1P
=
\sum_{k=1}^m
G_m\cdots G_{k+1}(G_k-V_k)V_{k-1}\cdots V_1P.
```

V_MUse in source · 1 passage

Search aliases: V_M

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Use in source · Read passage · s_main.tex, line 141
Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis and charge sectors in quantum theory.
Exact mathematical passage
```
V_M=\exp\left(\frac{2\pi i}{M}Q\right),
\qquad
\omega_M=e^{2\pi i/M}.
```

V_mUse in source · 3 passages

Search aliases: V_m

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Use in source · Read passage · s_appendix.tex, line 19
Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].
Exact mathematical passage
```
G_m\cdots G_1P-V_m\cdots V_1P
=
\sum_{k=1}^m
G_m\cdots G_{k+1}(G_k-V_k)V_{k-1}\cdots V_1P.
```

V_M^\ellUse in source · 1 passage

Search aliases: V_M^ℓ ell

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Use in source · Read passage · s_main.tex, line 147
Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis and charge sectors in quantum theory.
Exact mathematical passage
```
B_r^{(M)}
=
\frac1M\sum_{\ell=0}^{M-1}
\omega_M^{-r\ell}V_M^\ell.
\label{s:eq:residue-projector}
```

V_{n-1}Use in source · 1 passage

Search aliases: V_n-1

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Use in source · Read passage · g_main.tex, line 1240
, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0⊂ V_1⊂⋯⊂ V_n-1. An upper-directed gate leaves this fiber of the flag map. A strict defect means that the flag-compatible frame subgraph is not geodesi…
Exact mathematical passage
```
V_0\subset V_1\subset\cdots\subset V_{n-1}.
```

V_vUse in source · 2 passages

Search aliases: V_v

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Use in source · Read passage · g_main.tex, line 811
Quiver gate models. Let Q=(Q_0,Q_1) be a quiver. Associate a vector space V_v to every vertex. Internal gates change bases inside V_v, while edge gates transmit low-rank maps along arrows. A target is a representation of the path algebra or an automorphism of a filtered quiver module.
Exact mathematical passage
```
V_v
```

\mathcal V_\lambdaUse in source · 10 passages

Search aliases: V_λ mathcal lambda

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Use in source · Read passage · s_main.tex, line 988
Decomposition. For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.
Exact mathematical passage
```
(\C^d)^{\otimes n}
\cong
\bigoplus_{\lambda\vdash n,\,\ell(\lambda)\le d}
\mathcal U_\lambda\otimes\mathcal V_\lambda,
```

\vdashUse in source · 2 passages

Search aliases: vdash

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Use in source · Read passage · s_main.tex, line 988
Decomposition. For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.
Exact mathematical passage
```
(\C^d)^{\otimes n}
\cong
\bigoplus_{\lambda\vdash n,\,\ell(\lambda)\le d}
\mathcal U_\lambda\otimes\mathcal V_\lambda,
```

WUse in source · 55 passages

Search aliases: W

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Use in source · Read passage · c_lean.tex, line 59
pageheadModel and definitionsTHE ASSUMPTIONS ARE PART OF THE THEOREM A general register state is a Boolean function W→Bool on any wire type with decidable equality. C01 does not assume that W is finite. The canonical degree model has a finite wire set with n input wires, a ancillas, and one target. A NOT complements a bit. A CNOT adds one control to a distinct target. A Toffoli adds the product of two distinct controls to a third distinct wire. Gate constructors carry the required inequalities.
Exact mathematical passage
```
W\to\mathrm{Bool}
```

\mathcal WUse in source · 1 passage

Search aliases: W mathcal

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Use in source · Read passage · g_main.tex, line 3994
A vector-valued defect. For a target U in a restricted phase model, define C_res(U) = { c( W): W is a restricted circuit for U} and similarly C_amb(U). Their Pareto frontiers are denoted Pareto_res(U) and Pareto_amb(U).
Exact mathematical passage
```
\mathcal C_{\mathrm{res}}(U)
=
\{\mathbf c(\mathcal W):\mathcal W\text{ is a restricted circuit for }U\}
```

w^2Use in source · 5 passages

Search aliases: w^2

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Use in source · Read passage · c_main.tex, line 653
[Shared-variable multiplication fails] For a single witness bit, define KTr_z(A)=A(0)+zA(1). Then in the Boolean w algebra, KTr_z(w)=z, KTr_z(1+w)=1, but KTr_z(w(1+w))=0 ≠ z=KTr_z(w)KTr_z(1+w) whenever z≠0. Also, KTr_z(w^2)=z, KTr_z(w)^2=z^2, which differ for field elements z with z^2≠ z. Evaluate w and 1+w at zero and one. In the Boolean quotient w(1+w)=w+w^2=0 and w^2=w. The displayed identities follow directly. In GF(16), any element outside F supplies the second inequality.
Exact mathematical passage
```
 \KTr_z(w^2)=z,\qquad\KTr_z(w)^2=z^2,
```

W_0Use in source · 1 passage

Search aliases: W_0

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Use in source · Read passage · s_main.tex, line 519
[Moving-frame collapse] Suppose a polynomial-time uniform circuit admits efficiently described frames W_t for which the transformed gates in [source reference] preserve one fixed polynomial-dimensional packet, the initial state has efficiently computable coordinates in W_0PHh, and their restricted matrices and the transformed final measurement are efficiently computable. Then the circuit family is classically simulable in polynomial time.
Exact mathematical passage
```
W_0P\Hh
```

w_1Use in source · 4 passages

Search aliases: w_1

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Use in source · Read passage · c_main.tex, line 505
and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
 K[w_1,\ldots,w_m]/(w_i^2-w_i:1\le i\le m).
```

w_2Use in source · 1 passage

Search aliases: w_2

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Use in source · Read passage · c_visual.tex, line 667
For A=w_1 and B=w_2, τ_z_1(A)=z_1, τ_z_2(B)=z_2, τ_z_1,z_2(AB)=z_1z_2. Tracing both factors over the full two-bit ambient domain introduces unused variable sums: τ_z_1,z_2(A)=z_1(1+z_2), τ_z_1,z_2(B)=z_2(1+z_1). Their product generally differs from the union trace. At z_1=z_2=1, each full ambient factor is zero, while the trace of AB is one. Disjoint dependence alone does not justify multiplying traces with mismatched domains.
Exact mathematical passage
```
B=w_2
```

w_{\mathrm{cat}}Use in source · 1 passage

Search aliases: w_cat mathrm

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Use in source · Read passage · g_main.tex, line 353
Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.
Exact mathematical passage
```
w_{\mathrm{cat}}=
\max_t
\#\{j:t\in I_j\}.
```

w_iUse in source · 9 passages

Search aliases: w_i

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Use in source · Read passage · c_main.tex, line 505
and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
 K[w_1,\ldots,w_m]/(w_i^2-w_i:1\le i\le m).
```

w_i^2Use in source · 1 passage

Search aliases: w_i^2

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Use in source · Read passage · c_main.tex, line 505
and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
 K[w_1,\ldots,w_m]/(w_i^2-w_i:1\le i\le m).
```

w_jUse in source · 2 passages

Search aliases: w_j

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Use in source · Read passage · g_main.tex, line 1302
If the unique gate is R_t← R_t+aR_s, then aR_s=w. Choose a coordinate j for which w_j is a unit. Since a(R_s)_j=w_j, the factor a is a unit in the commutative ring R. Hence R_s=a^-1w.
Exact mathematical passage
```
w_j
```

w_mUse in source · 3 passages

Search aliases: w_m

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Use in source · Read passage · c_main.tex, line 505
and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
 K[w_1,\ldots,w_m]/(w_i^2-w_i:1\le i\le m).
```

w_sUse in source · 2 passages

Search aliases: w_s

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Use in source · Read passage · g_main.tex, line 2005
Fix integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.
Exact mathematical passage
```
\label{g:eq:prefix-state}
w_s=e_p+\sum_{i=1}^{s}e_{x_i}.
```

w_{s_1}Use in source · 1 passage

Search aliases: w_s_1

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Use in source · Read passage · g_main.tex, line 2063
It remains to prove optimality in the lower model. At the root, split {x_1,…,x_u,p} | {q,t_1,…,t_v}. The nonzero rows of the root cross block are w_s_1,…,w_s_v. Distinct prefix vectors are linearly independent: in a linear relation, inspect successively a coordinate occurring in w_s_j but in none of the earlier prefixes, starting from the largest phase. Hence the root contributes v.
Exact mathematical passage
```
w_{s_1},\ldots,w_{s_v}
```

w_{s_j}Use in source · 2 passages

Search aliases: w_s_j

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Use in source · Read passage · g_main.tex, line 2021
[ 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.
Exact mathematical passage
```
\label{g:eq:snapshot-output}
R_{t_j}=e_{t_j}+e_q+w_{s_j}
\qquad(1\le j\le v).
```

w_{s_v}Use in source · 1 passage

Search aliases: w_s_v

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Use in source · Read passage · g_main.tex, line 2063
It remains to prove optimality in the lower model. At the root, split {x_1,…,x_u,p} | {q,t_1,…,t_v}. The nonzero rows of the root cross block are w_s_1,…,w_s_v. Distinct prefix vectors are linearly independent: in a linear relation, inspect successively a coordinate occurring in w_s_j but in none of the earlier prefixes, starting from the largest phase. Hence the root contributes v.
Exact mathematical passage
```
w_{s_1},\ldots,w_{s_v}
```

W_tUse in source · 5 passages

Search aliases: W_t

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Use in source · Read passage · s_main.tex, line 502
Moving frames and hidden symmetry. The packet schedule has a useful coordinate form. Let P be one fixed projector and let W_t be unitaries with P_t=W_tPW_t^*. In the moving frame, the transformed gate is widetilde G_t =W_t^*G_tW_t-1. The schedule condition is equivalent to widetilde G_tP=Pwidetilde G_tP.
Exact mathematical passage
```
W_t
```

W_{t-1}Use in source · 1 passage

Search aliases: W_t-1

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Use in source · Read passage · s_main.tex, line 507
Moving frames and hidden symmetry. The packet schedule has a useful coordinate form. Let P be one fixed projector and let W_t be unitaries with P_t=W_tPW_t^*. In the moving frame, the transformed gate is widetilde G_t =W_t^*G_tW_t-1. The schedule condition is equivalent to widetilde G_tP=Pwidetilde G_tP.
Exact mathematical passage
```
\widetilde G_t
=W_t^*G_tW_{t-1}.
\label{s:eq:moving-frame-gate}
```

W_t^*Use in source · 2 passages

Search aliases: W_t^*

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Use in source · Read passage · s_main.tex, line 503
Moving frames and hidden symmetry. The packet schedule has a useful coordinate form. Let P be one fixed projector and let W_t be unitaries with P_t=W_tPW_t^*. In the moving frame, the transformed gate is widetilde G_t =W_t^*G_tW_t-1. The schedule condition is equivalent to widetilde G_tP=Pwidetilde G_tP.
Exact mathematical passage
```
P_t=W_tPW_t^*.
```

w_uUse in source · 1 passage

Search aliases: w_u

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Use in source · Read passage · g_main.tex, line 2013
integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.
Exact mathematical passage
```
R_{x_i}=e_{x_i},
\qquad
R_p=w_u,
\qquad
R_q=e_q,
```

\widehat aUse in source · 1 passage

Search aliases: widehat a widehat

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Use in source · Read passage · q_main.tex, line 1030
widehat a=(16,0,4,0,4,0,16,0,4,0,4,0).
Exact mathematical passage
```
\widehat a=(16,0,4,0,4,0,16,0,4,0,4,0).
```

\widehat bUse in source · 1 passage

Search aliases: widehat b widehat

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Use in source · Read passage · q_main.tex, line 857
Thus widehat b equals twenty-four at zero, eighteen at three and nine, and zero elsewhere. Multiplication by D fixes the spectrum of c and kills the spectrum of b, proving the filter identities. Their supports are disjoint, so Parseval for inner products gives ⟨ c,b⟩=0. QA05 gives c ^2=12·4=48, and
Exact mathematical passage
```
\widehat b
```

\widehat cUse in source · 5 passages

Search aliases: widehat c widehat

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Use in source · Read passage · q_main.tex, line 709
widehat c=(0,12,0,0,0,12,0,12,0,0,0,12).
Exact mathematical passage
```
\widehat c=(0,12,0,0,0,12,0,12,0,0,0,12).
```

\widehat c_0Use in source · 1 passage

Search aliases: widehat c_0 widehat

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Use in source · Read passage · q_main.tex, line 781
widehat c_0=(0,8,0,8), widehat c_1=(0,4i,0,-4i), widehat c_2=(0,4,0,4).
Exact mathematical passage
```
\widehat c_0=(0,8,0,8),
\quad \widehat c_1=(0,4i,0,-4i),
\quad \widehat c_2=(0,4,0,4).
```

\widehat c_1Use in source · 1 passage

Search aliases: widehat c_1 widehat

Link to this entry ·

Use in source · Read passage · q_main.tex, line 781
widehat c_0=(0,8,0,8), widehat c_1=(0,4i,0,-4i), widehat c_2=(0,4,0,4).
Exact mathematical passage
```
\widehat c_0=(0,8,0,8),
\quad \widehat c_1=(0,4i,0,-4i),
\quad \widehat c_2=(0,4,0,4).
```

\widehat c_2Use in source · 1 passage

Search aliases: widehat c_2 widehat

Link to this entry ·

Use in source · Read passage · q_main.tex, line 781
widehat c_0=(0,8,0,8), widehat c_1=(0,4i,0,-4i), widehat c_2=(0,4,0,4).
Exact mathematical passage
```
\widehat c_0=(0,8,0,8),
\quad \widehat c_1=(0,4i,0,-4i),
\quad \widehat c_2=(0,4,0,4).
```

\widehat GUse in source · 3 passages

Search aliases: widehat G widehat

Link to this entry ·

Use in source · Read passage · s_main.tex, line 612
Approximate dynamic packets. Define the one-step leakage from the scheduled packet by δ_t = (I-P_t)G_tP_t-1 . Let widehat G_t=P_tG_tP_t-1.
Exact mathematical passage
```
\widehat G_t=P_tG_tP_{t-1}.
```

\widehat{\mathcal G}Use in source · 2 passages

Search aliases: widehat G widehat mathcal

Link to this entry ·

Use in source · Read passage · g_main.tex, line 531
Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint series after Fourier inversion.
Exact mathematical passage
```
\widehat{\mathcal G}_\tau(z)
=
\sum_{k\ge0}\widehat\mu_k(\tau)z^k
```

\widehat G_1Use in source · 1 passage

Search aliases: widehat G_1 widehat

Link to this entry ·

Use in source · Read passage · s_main.tex, line 619
[Dynamic sum-of-leakages bound] For an initial unit vector in P_0Hh, G_m⋯ G_1ketψ_0 - widehat G_m⋯widehat G_1ketψ_0 ≤ ∑_t=1^mδ_t. For any acceptance operator 0≤ M≤ I, the corresponding acceptance probabilities differ by at most 2∑_t=1^mδ_t.
Exact mathematical passage
```
\left\|
G_m\cdots G_1\ket{\psi_0}
-
\widehat G_m\cdots\widehat G_1\ket{\psi_0}
\right\|
\le
\sum_{t=1}^{m}\delta_t.
```

\widehat G_mUse in source · 1 passage

Search aliases: widehat G_m widehat

Link to this entry ·

Use in source · Read passage · s_main.tex, line 619
[Dynamic sum-of-leakages bound] For an initial unit vector in P_0Hh, G_m⋯ G_1ketψ_0 - widehat G_m⋯widehat G_1ketψ_0 ≤ ∑_t=1^mδ_t. For any acceptance operator 0≤ M≤ I, the corresponding acceptance probabilities differ by at most 2∑_t=1^mδ_t.
Exact mathematical passage
```
\left\|
G_m\cdots G_1\ket{\psi_0}
-
\widehat G_m\cdots\widehat G_1\ket{\psi_0}
\right\|
\le
\sum_{t=1}^{m}\delta_t.
```

\widehat G_tUse in source · 2 passages

Search aliases: widehat G_t widehat

Link to this entry ·

Use in source · Read passage · s_main.tex, line 612
Approximate dynamic packets. Define the one-step leakage from the scheduled packet by δ_t = (I-P_t)G_tP_t-1 . Let widehat G_t=P_tG_tP_t-1.
Exact mathematical passage
```
\widehat G_t=P_tG_tP_{t-1}.
```

\widehat{\mathcal G}_\tauUse in source · 2 passages

Search aliases: widehat G_τ widehat mathcal tau

Link to this entry ·

Use in source · Read passage · g_main.tex, line 531
Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint series after Fourier inversion.
Exact mathematical passage
```
\widehat{\mathcal G}_\tau(z)
=
\sum_{k\ge0}\widehat\mu_k(\tau)z^k
```

\widehat HUse in source · 3 passages

Search aliases: widehat H widehat

Link to this entry ·

Use in source · Read passage · g_main.tex, line 512
Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint series after Fourier inversion.
Exact mathematical passage
```
\tau\in\widehat H
```

\widehat NUse in source · 2 passages

Search aliases: widehat N widehat

Link to this entry ·

Use in source · Read passage · s_main.tex, line 718
Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.
Exact mathematical passage
```
\widehat N
=
\sum_{q=1}^n\frac{I-Z_q}{2}.
```

\widehat xUse in source · 3 passages

Search aliases: widehat x widehat

Link to this entry ·

Use in source · Read passage · q_visual.tex, line 77
The spectrum recovers the selector. addcontentslinetocsectionThe spectrum recovers the selector leadThe Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform widehat x(k)=∑_n=0^11x(n)e^-2π ikn/12. fig04_exact_spectrum1.0The coefficient is exactly 12 at k=1,5,7,11 and exactly zero elsewhere. This is an algebraic identity, not a spectral thresholding decision. widehatc_12(k)=∑_a∈ U_12∑_n=0^11ζ^(a-k)n =12 1_k∈ U_12.…
Exact mathematical passage
```
 \widehat x(k)=\sum_{n=0}^{11}x(n)e^{-2\pi ikn/12}.
```

\widehat{c_{12}}Use in source · 1 passage

Search aliases: widehatc_12 widehat

Link to this entry ·

Use in source · Read passage · q_visual.tex, line 81
s the selector leadThe Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform widehat x(k)=∑_n=0^11x(n)e^-2π ikn/12. fig04_exact_spectrum1.0The coefficient is exactly 12 at k=1,5,7,11 and exactly zero elsewhere. This is an algebraic identity, not a spectral thresholding decision. widehatc_12(k)=∑_a∈ U_12∑_n=0^11ζ^(a-k)n =12 1_k∈ U_12. The inner geometric sum is twelve when a=k modulo twelve and zero otherwise. The…
Exact mathematical passage
```
 \widehat{c_{12}}(k)=\sum_{a\in U_{12}}\sum_{n=0}^{11}\zeta^{(a-k)n}
 =12\,\mathbf1_{k\in U_{12}}.
```

\widehat{c_q}Use in source · 4 passages

Search aliases: widehatc_q widehat

Link to this entry ·

Use in source · Read passage · q_main.tex, line 298
widehatc_q(k)=q 1_k∈ U_q, k∈ I_q.
Exact mathematical passage
```
\boxed{\widehat{c_q}(k)=q\,\mathbf1_{k\in U_q},\qquad k\in I_q.}
```

\widehat{E_rx}Use in source · 1 passage

Search aliases: widehatE_rx widehat

Link to this entry ·

Use in source · Read passage · q_visual.tex, line 211
Phase factors connect the lane spectra. addcontentslinetocsectionPhase factors connect the lane spectra leadEach local Fourier transform contributes to every appropriate global frequency. With four-point negative-sign DFTs in the lanes, widehat x(k)=∑_r=0^2e^-2π ikr/12 widehatE_rx(kbmod4). fig18_polyphase_alias_assembly1.0Left: magnitudes of the three local transforms. Right: the recovered global spectrum. Assembly uses the complex coefficients and phase factors; adding the displayed magnitudes …
Exact mathematical passage
```
 \widehat x(k)=\sum_{r=0}^{2}e^{-2\pi ikr/12}
 \widehat{E_rx}(k\bmod4).
```

\widehat{x}Use in source · 9 passages

Search aliases: widehatx widehat

Link to this entry ·

Use in source · Read passage · q_main.tex, line 123
widehatx(k)=∑_n=0^q-1x(n)ζ_q^-kn.
Exact mathematical passage
```
\widehat{x}(k)=\sum_{n=0}^{q-1}x(n)\zeta_q^{-kn}.
```

\widehat\muUse in source · 5 passages

Search aliases: widehatμ widehat mu

Link to this entry ·

Use in source · Read passage · g_main.tex, line 513
Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint series after Fourier inversion.
Exact mathematical passage
```
\widehat\mu_k(\tau)
=
\sum_{h\in H}\mu_k(h)\tau(h).
```

\widehat\mu_kUse in source · 5 passages

Search aliases: widehatμ_k widehat mu

Link to this entry ·

Use in source · Read passage · g_main.tex, line 513
Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint series after Fourier inversion.
Exact mathematical passage
```
\widehat\mu_k(\tau)
=
\sum_{h\in H}\mu_k(h)\tau(h).
```

\widetilde GUse in source · 2 passages

Search aliases: widetilde G widetilde

Link to this entry ·

Use in source · Read passage · s_main.tex, line 507
Moving frames and hidden symmetry. The packet schedule has a useful coordinate form. Let P be one fixed projector and let W_t be unitaries with P_t=W_tPW_t^*. In the moving frame, the transformed gate is widetilde G_t =W_t^*G_tW_t-1. The schedule condition is equivalent to widetilde G_tP=Pwidetilde G_tP.
Exact mathematical passage
```
\widetilde G_t
=W_t^*G_tW_{t-1}.
\label{s:eq:moving-frame-gate}
```

\widetilde G_tUse in source · 2 passages

Search aliases: widetilde G_t widetilde

Link to this entry ·

Use in source · Read passage · s_main.tex, line 507
Moving frames and hidden symmetry. The packet schedule has a useful coordinate form. Let P be one fixed projector and let W_t be unitaries with P_t=W_tPW_t^*. In the moving frame, the transformed gate is widetilde G_t =W_t^*G_tW_t-1. The schedule condition is equivalent to widetilde G_tP=Pwidetilde G_tP.
Exact mathematical passage
```
\widetilde G_t
=W_t^*G_tW_{t-1}.
\label{s:eq:moving-frame-gate}
```

\widetilde HUse in source · 2 passages

Search aliases: widetilde H widetilde

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1236
The quotient matrix is widetilde H_ab = bra O_aHket O_b.
Exact mathematical passage
```
\widetilde H_{ab}
=
\bra{\mathcal O_a}H\ket{\mathcal O_b}.
```

\widetilde H_{ab}Use in source · 1 passage

Search aliases: widetilde H_ab widetilde

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1236
The quotient matrix is widetilde H_ab = bra O_aHket O_b.
Exact mathematical passage
```
\widetilde H_{ab}
=
\bra{\mathcal O_a}H\ket{\mathcal O_b}.
```

X'Use in source · 2 passages

Search aliases: X'

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1956
[One-cut locality] Let X∈U_n(F) and let X' be obtained from X by one lower-directed transvection. Then |Φ_T(X')-Φ_T(X)|≤1.
Exact mathematical passage
```
X'
```

X^{(L)}Use in source · 2 passages

Search aliases: X^(L)

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2458
[Dimension per gate] Let G be an algebraic group, let V⊆ G be an irreducible subvariety of dimension D, and let X⊆ G be a finite union of locally closed subvarieties, each of dimension at most r, with the identity in X. Write X^(L) = {x_L⋯ x_1:x_1,…,x_L∈ X}. Then dimoverlineV∩ X^(L)≤ rL. In particular, if rL<D, a Zariski-generic point of V has X-product length greater than L.
Exact mathematical passage
```
X^{(L)}
=
\{x_L\cdots x_1:x_1,\ldots,x_L\in X\}.
```

X^LUse in source · 4 passages

Search aliases: X^L

Link to this entry ·

Use in source · Read passage · g_main.tex, line 549
Word maps and product loci. Let K be a field and let X⊆ G be an algebraic gate family. For a fixed length L, multiplication defines μ_L:X^L→ G, (x_L,…,x_1)↦ x_L⋯ x_1. The exact length-L locus is μ_L(X^L), and the length-at-most-L locus is a finite union of such images when the gate family has finitely many components.
Exact mathematical passage
```
\mu_L:X^L\to G,
\qquad
(x_L,\ldots,x_1)\mapsto x_L\cdots x_1.
```

x^SUse in source · 6 passages

Search aliases: x^S

Link to this entry ·

Use in source · Read passage · c_main.tex, line 12
For S⊆[n], write x^S=∏_i∈ Sx_i and let 1_S be the input with support S. A minterm of a monotone function means an inclusion-minimal positive support, not an arbitrary positive input.
Exact mathematical passage
```
x^S=\prod_{i\in S}x_i
```

X^{\times L}Use in source · 1 passage

Search aliases: X^× L times

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2471
Every irreducible component of the Cartesian power X^× L has dimension at most rL. Multiplication G^L→ G is a morphism, so the closure of the image of each component has dimension at most rL. There are finitely many components, and intersecting the resulting finite union with V cannot increase dimension. Since the identity belongs to X, products of length at most L are already contained in X^(L).
Exact mathematical passage
```
X^{\times L}
```

x^\daggerUse in source · 2 passages

Search aliases: x^† dagger

Link to this entry ·

Use in source · Read passage · q_main.tex, line 202
F_q^† x ^2 =x^† F_qF_q^† x =x^† x.
Exact mathematical passage
```
\|F_q^\dagger x\|^2
=x^\dagger F_qF_q^\dagger x
=x^\dagger x.
```

X_*Use in source · 1 passage

Search aliases: X_*

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2428
Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.
Exact mathematical passage
```
X_*
=
\bigcup_{t\ne s}\{I+aE_{t,s}:a\in K\},
\qquad
X_\downarrow
=
\bigcup_{s<t}\{I+aE_{t,s}:a\in K\}.
```

X_*^LUse in source · 1 passage

Search aliases: X_*^L

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2437
Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.
Exact mathematical passage
```
A\in X_*^L
```

x_0Use in source · 5 passages

Search aliases: x_0

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 61
Binary symplectic form. Pauli exponents are ordered as (x_0,…,x_n-1,z_0,…,z_n-1). The symplectic form is J= 0 · I_n
Exact mathematical passage
```
(x_0,\ldots,x_{n-1},z_0,\ldots,z_{n-1}).
```

x_1Use in source · 17 passages

Search aliases: x_1

Link to this entry ·

Use in source · Read passage · c_main.tex, line 5
Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies alor b=a+b+ab. A statement h≤ f is pointwise and means that h has no positive value at a negative input of f.
Exact mathematical passage
```
\mathcal R_n=\F[x_1,\ldots,x_n]/(x_i^2-x_i:1\le i\le n).
```

X_{11}Use in source · 1 passage

Search aliases: X_11

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2612
For the lower bound, view an element of U_n_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in every factor. If m_i gates survive in factor i, then the total word length is at least ∑_i m_i ≥ ∑_iellDow…
Exact mathematical passage
```
\pi(X)=(X_{11},\ldots,X_{kk}).
```

x_2Use in source · 9 passages

Search aliases: x_2

Link to this entry ·

Use in source · Read passage · c_main.tex, line 262
[Exact monomial oracle cost] For k≥2, KTrest(x_1x_2⋯ x_k)=2k-3. The unique positive input of the k-variable monomial needs a safe polynomial of degree k. The complete-cover lower bound gives 2k-3. For the matching construction, successively compute x_1x_2, then x_1x_2x_3, and so on up to a (k-1)-fold product on clean work wires. Toggle the output using that product and x_k, then uncompute the k-2 work products. The count is (k-2)+1+(k-2)=2k-3. For k=2, a single Toffoli suffices.
Exact mathematical passage
```
 \KTrest(x_1x_2\cdots x_k)=2k-3.
```

x_3Use in source · 7 passages

Search aliases: x_3

Link to this entry ·

Use in source · Read passage · c_main.tex, line 267
[Exact monomial oracle cost] For k≥2, KTrest(x_1x_2⋯ x_k)=2k-3. The unique positive input of the k-variable monomial needs a safe polynomial of degree k. The complete-cover lower bound gives 2k-3. For the matching construction, successively compute x_1x_2, then x_1x_2x_3, and so on up to a (k-1)-fold product on clean work wires. Toggle the output using that product and x_k, then uncompute the k-2 work products. The count is (k-2)+1+(k-2)=2k-3. For k=2, a single Toffoli suffices.
Exact mathematical passage
```
x_1x_2x_3
```

x_4Use in source · 5 passages

Search aliases: x_4

Link to this entry ·

Use in source · Read passage · c_main.tex, line 400
[Exact cost of a nondegenerate paired quadratic] For even n, let Q(x)=x_1x_2+x_3x_4+⋯+x_n-1x_n. Then M(Q)=KTrest(Q)=n/2. The polar bilinear form of Q is the full-rank alternating form made of n/2 disjoint 2×2 blocks. If Q is affine on an affine subspace with direction space U, its second differences vanish on U, so U is totally isotropic. Nondegeneracy gives dim U≤ n/2.
Exact mathematical passage
```
 Q(x)=x_1x_2+x_3x_4+\cdots+x_{n-1}x_n.
```

x_5Use in source · 2 passages

Search aliases: x_5

Link to this entry ·

Use in source · Read passage · c_visual.tex, line 504
guidepageQuadratic projection can destroy safety At m=3, variables are ordered by source labels v=1,2,3,4,5,6. Assignment index 13 is read least-significant-bit first: (x_1,x_2,x_3,x_4,x_5,x_6)=(1,0,1,1,0,0). The active labels have syndrome 1+3+4=6, not a=7.
Exact mathematical passage
```
 (x_1,x_2,x_3,x_4,x_5,x_6)=(1,0,1,1,0,0).
```

x_6Use in source · 2 passages

Search aliases: x_6

Link to this entry ·

Use in source · Read passage · c_visual.tex, line 504
guidepageQuadratic projection can destroy safety At m=3, variables are ordered by source labels v=1,2,3,4,5,6. Assignment index 13 is read least-significant-bit first: (x_1,x_2,x_3,x_4,x_5,x_6)=(1,0,1,1,0,0). The active labels have syndrome 1+3+4=6, not a=7.
Exact mathematical passage
```
 (x_1,x_2,x_3,x_4,x_5,x_6)=(1,0,1,1,0,0).
```

x_aUse in source · 2 passages

Search aliases: x_a

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 157
Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric group.
Exact mathematical passage
```
\TOF_{a,b\to t}:x_t\leftarrow x_t+x_ax_b.
```

X_{\mathrm{amb}}Use in source · 1 passage

Search aliases: X_amb mathrm

Link to this entry ·

Use in source · Read passage · g_main.tex, line 693
Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but tensor-regularized defect positive, item ambient and restricted border ranks both smaller than exact ranks but by different amounts. Each requires a se…
Exact mathematical passage
```
X_{\mathrm{amb}}
```

x_bUse in source · 2 passages

Search aliases: x_b

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 157
Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric group.
Exact mathematical passage
```
\TOF_{a,b\to t}:x_t\leftarrow x_t+x_ax_b.
```

x_cUse in source · 2 passages

Search aliases: x_c

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 87
Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.
Exact mathematical passage
```
x_t\leftarrow x_t+x_c,
\qquad
z_c\leftarrow z_c+z_t.
```

X_\downarrowUse in source · 1 passage

Search aliases: X_downarrow downarrow

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2428
Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.
Exact mathematical passage
```
X_*
=
\bigcup_{t\ne s}\{I+aE_{t,s}:a\in K\},
\qquad
X_\downarrow
=
\bigcup_{s<t}\{I+aE_{t,s}:a\in K\}.
```

X_\downarrow^LUse in source · 1 passage

Search aliases: X_downarrow^L downarrow

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Use in source · Read passage · g_main.tex, line 2437
Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.
Exact mathematical passage
```
A\in X_\downarrow^L
```

X_iUse in source · 1 passage

Search aliases: X_i

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Use in source · Read passage · g_appendix.tex, line 151
Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric group.
Exact mathematical passage
```
X_i:x_i\leftarrow x_i+1,
```

x_iUse in source · 17 passages

Search aliases: x_i

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Use in source · Read passage · c_lean.tex, line 75
Polynomial certificates are not postulated degree oracles. The inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator polyEval(p,x)=⊕_L∈ p∏_i∈ Lx_i by a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a certificate.
Exact mathematical passage
```
\operatorname{polyEval}(p,x)=\bigoplus_{L\in p}\prod_{i\in L}x_i
```

x_i^2Use in source · 2 passages

Search aliases: x_i^2

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Use in source · Read passage · c_main.tex, line 5
Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies alor b=a+b+ab. A statement h≤ f is pointwise and means that h has no positive value at a negative input of f.
Exact mathematical passage
```
\mathcal R_n=\F[x_1,\ldots,x_n]/(x_i^2-x_i:1\le i\le n).
```

x_jUse in source · 5 passages

Search aliases: x_j

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Use in source · Read passage · c_main.tex, line 373
A cubic on three variables has the form x_1x_2x_3+q(x)+ℓ(x), with q quadratic and ℓ affine. Translating x_i↦ x_i+c_i adjusts independently the coefficients of x_jx_k, so all quadratic terms can be cancelled. The resulting function is a three-variable monomial plus an affine function. Affine input changes and affine output additions cost no Toffolis. Corollary~[source reference] gives both the lower bound three and a matching restored construction.
Exact mathematical passage
```
x_jx_k
```

x_kUse in source · 3 passages

Search aliases: x_k

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Use in source · Read passage · c_main.tex, line 262
[Exact monomial oracle cost] For k≥2, KTrest(x_1x_2⋯ x_k)=2k-3. The unique positive input of the k-variable monomial needs a safe polynomial of degree k. The complete-cover lower bound gives 2k-3. For the matching construction, successively compute x_1x_2, then x_1x_2x_3, and so on up to a (k-1)-fold product on clean work wires. Toggle the output using that product and x_k, then uncompute the k-2 work products. The count is (k-2)+1+(k-2)=2k-3. For k=2, a single Toffoli suffices.
Exact mathematical passage
```
 \KTrest(x_1x_2\cdots x_k)=2k-3.
```

X_{kk}Use in source · 1 passage

Search aliases: X_kk

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Use in source · Read passage · g_main.tex, line 2612
For the lower bound, view an element of U_n_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in every factor. If m_i gates survive in factor i, then the total word length is at least ∑_i m_i ≥ ∑_iellDow…
Exact mathematical passage
```
\pi(X)=(X_{11},\ldots,X_{kk}).
```

x_LUse in source · 2 passages

Search aliases: x_L

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Use in source · Read passage · g_main.tex, line 549
Word maps and product loci. Let K be a field and let X⊆ G be an algebraic gate family. For a fixed length L, multiplication defines μ_L:X^L→ G, (x_L,…,x_1)↦ x_L⋯ x_1. The exact length-L locus is μ_L(X^L), and the length-at-most-L locus is a finite union of such images when the gate family has finitely many components.
Exact mathematical passage
```
\mu_L:X^L\to G,
\qquad
(x_L,\ldots,x_1)\mapsto x_L\cdots x_1.
```

x_{L-1}Use in source · 1 passage

Search aliases: x_L-1

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Use in source · Read passage · q_main.tex, line 587
The addressing map T:x↦(x_0,…,x_L-1) is bijective. Interleaving the lanes reconstructs x, and
Exact mathematical passage
```
T:x\mapsto(x_0,\ldots,x_{L-1})
```

x_nUse in source · 2 passages

Search aliases: x_n

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Use in source · Read passage · c_main.tex, line 5
Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies alor b=a+b+ab. A statement h≤ f is pointwise and means that h has no positive value at a negative input of f.
Exact mathematical passage
```
\mathcal R_n=\F[x_1,\ldots,x_n]/(x_i^2-x_i:1\le i\le n).
```

x_{n-1}Use in source · 4 passages

Search aliases: x_n-1

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Use in source · Read passage · c_main.tex, line 400
[Exact cost of a nondegenerate paired quadratic] For even n, let Q(x)=x_1x_2+x_3x_4+⋯+x_n-1x_n. Then M(Q)=KTrest(Q)=n/2. The polar bilinear form of Q is the full-rank alternating form made of n/2 disjoint 2×2 blocks. If Q is affine on an affine subspace with direction space U, its second differences vanish on U, so U is totally isotropic. Nondegeneracy gives dim U≤ n/2.
Exact mathematical passage
```
 Q(x)=x_1x_2+x_3x_4+\cdots+x_{n-1}x_n.
```

x_rUse in source · 3 passages

Search aliases: x_r

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Use in source · Read passage · q_main.tex, line 582
x_r(m)=x(r+Lm), 0≤ r<L, 0≤ m<M.
Exact mathematical passage
```
x_r(m)=x(r+Lm),
\qquad 0\le r<L,\quad 0\le m<M.
```

X_{\mathrm{res}}Use in source · 1 passage

Search aliases: X_res mathrm

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Use in source · Read passage · g_main.tex, line 693
Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but tensor-regularized defect positive, item ambient and restricted border ranks both smaller than exact ranks but by different amounts. Each requires a se…
Exact mathematical passage
```
X_{\mathrm{res}}
```

x_sUse in source · 2 passages

Search aliases: x_s

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Use in source · Read passage · g_appendix.tex, line 51
Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.
Exact mathematical passage
```
\SUM^{(a)}_{s\to t}
\ket{x_s,x_t}
=
\ket{x_s,x_t+a x_s\bmod d}.
```

X_{s,s}Use in source · 1 passage

Search aliases: X_s,s

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Use in source · Read passage · g_main.tex, line 2301
A lower-directed gate preserves lower triangularity. Conversely, if X is lower unitriangular and an upper-directed gate R_t← R_t+aR_s with t<s is applied, then the new (t,s) entry equals X_t,s+aX_s,s=a≠0. The next state is therefore not lower triangular. A circuit whose every prefix has zero strict upper part uses only lower-directed gates.
Exact mathematical passage
```
X_{t,s}+aX_{s,s}=a\ne0.
```

x_tUse in source · 6 passages

Search aliases: x_t

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Use in source · Read passage · g_appendix.tex, line 51
Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.
Exact mathematical passage
```
\SUM^{(a)}_{s\to t}
\ket{x_s,x_t}
=
\ket{x_s,x_t+a x_s\bmod d}.
```

X_{t,s}Use in source · 1 passage

Search aliases: X_t,s

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Use in source · Read passage · g_main.tex, line 2301
A lower-directed gate preserves lower triangularity. Conversely, if X is lower unitriangular and an upper-directed gate R_t← R_t+aR_s with t<s is applied, then the new (t,s) entry equals X_t,s+aX_s,s=a≠0. The next state is therefore not lower triangular. A circuit whose every prefix has zero strict upper part uses only lower-directed gates.
Exact mathematical passage
```
X_{t,s}+aX_{s,s}=a\ne0.
```

x_\alphaUse in source · 1 passage

Search aliases: x_α alpha

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Use in source · Read passage · g_main.tex, line 854
The Chevalley conjugation formula has the schematic form x_α(a)x_β(b)x_α(a)^-1 = x_β(b) ∏_k>0 x_β+kα(c_k a^k b). A forbidden root α can therefore create several allowed root effects before being removed.
Exact mathematical passage
```
x_\alpha(a)x_\beta(b)x_\alpha(a)^{-1}
=
x_\beta(b)
\prod_{k>0}
x_{\beta+k\alpha}(c_k a^k b).
```

x_\betaUse in source · 2 passages

Search aliases: x_β beta

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Use in source · Read passage · g_main.tex, line 845
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
x_\beta(t)
```

x_{\beta+k\alpha}Use in source · 1 passage

Search aliases: x_β+kα beta alpha

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Use in source · Read passage · g_main.tex, line 854
The Chevalley conjugation formula has the schematic form x_α(a)x_β(b)x_α(a)^-1 = x_β(b) ∏_k>0 x_β+kα(c_k a^k b). A forbidden root α can therefore create several allowed root effects before being removed.
Exact mathematical passage
```
x_\alpha(a)x_\beta(b)x_\alpha(a)^{-1}
=
x_\beta(b)
\prod_{k>0}
x_{\beta+k\alpha}(c_k a^k b).
```

x_{\beta_1}Use in source · 1 passage

Search aliases: x_β_1 beta

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Use in source · Read passage · g_main.tex, line 846
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
\ell_\Psi(u)
=
\min\{L:u=x_{\beta_L}(t_L)\cdots x_{\beta_1}(t_1),\ \beta_i\in\Psi\},
```

x_{\beta_L}Use in source · 1 passage

Search aliases: x_β_L beta

Link to this entry ·

Use in source · Read passage · g_main.tex, line 846
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
\ell_\Psi(u)
=
\min\{L:u=x_{\beta_L}(t_L)\cdots x_{\beta_1}(t_1),\ \beta_i\in\Psi\},
```

\xrightarrowUse in source · 2 passages

Search aliases: xrightarrow

Link to this entry ·

Use in source · Read passage · c_visual.tex, line 107
guidepageA dirty scratch wire can be fully restored Consider the four-wire oracle in the Lean examples. First compute ab into scratch s, copy s into the target, and uncompute s: (a,b,s,y)xrightarrowT(a,b;s)(a,b,s+ab,y) xrightarrowC(s;y)(a,b,s+ab,y+s+ab) xrightarrowT(a,b;s)(a,b,s,y+s+ab). The displayed final expression specializes to the required oracle when the boundary scratch is zero. On the positive input a=b=1, the scratch is dirty between the first and last gates.
Exact mathematical passage
```
 (a,b,s,y)\xrightarrow{T(a,b;s)}(a,b,s+ab,y)
 \xrightarrow{C(s;y)}(a,b,s+ab,y+s+ab)
 \xrightarrow{T(a,b;s)}(a,b,s,y+s+ab).
```

YUse in source · 11 passages

Search aliases: Y

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Use in source · Read passage · g_main.tex, line 383
Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.
Exact mathematical passage
```
e^{\varepsilon X}e^{\varepsilon Y}e^{-\varepsilon X}e^{-\varepsilon Y}
=
e^{\varepsilon^2[X,Y]+O(\varepsilon^3)}.
```

\mathbb ZUse in source · 8 passages

Search aliases: Z mathbb

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Use in source · Read passage · g_main.tex, line 3185
The program further_generalization_tests.py verifies the universal circuits over Z/4 Z, Z/6 Z, and Z/8 Z, audits the rank-one bandwidth-one excursion, checks the finite-field counting bounds symbolically, and verifies the stable tensor subadditivity identities.
Exact mathematical passage
```
\mathbb Z/4\mathbb Z
```

\mathcal ZUse in source · 5 passages

Search aliases: Z mathcal

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2660
For A∈U_m(F), define Z(A) = A · 0 A · A ∈U_2m(F).
Exact mathematical passage
```
\label{g:eq:zeta-lift}
\mathcal Z(A)
=
\begin{pmatrix}
A&0\\
A&A
\end{pmatrix}
\in\U_{2m}(\F).
```

z^{10}Use in source · 2 passages

Search aliases: z^10

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1894
For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.
Exact mathematical passage
```
\mathcal G_{\downarrow,C}(z)
&=
543z^7+25080z^8+510703z^9+10824940z^{10}+\cdots,\\
\mathcal G_{*,C}(z)
&=
2z^6+776z^7+39454z^8+\cdots.
```

z^2Use in source · 4 passages

Search aliases: z^2

Link to this entry ·

Use in source · Read passage · c_main.tex, line 653
[Shared-variable multiplication fails] For a single witness bit, define KTr_z(A)=A(0)+zA(1). Then in the Boolean w algebra, KTr_z(w)=z, KTr_z(1+w)=1, but KTr_z(w(1+w))=0 ≠ z=KTr_z(w)KTr_z(1+w) whenever z≠0. Also, KTr_z(w^2)=z, KTr_z(w)^2=z^2, which differ for field elements z with z^2≠ z. Evaluate w and 1+w at zero and one. In the Boolean quotient w(1+w)=w+w^2=0 and w^2=w. The displayed identities follow directly. In GF(16), any element outside F supplies the second inequality.
Exact mathematical passage
```
 \KTr_z(w^2)=z,\qquad\KTr_z(w)^2=z^2,
```

z^4Use in source · 1 passage

Search aliases: z^4

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Use in source · Read passage · q_visual.tex, line 72
bottomrule noteThe finite verifier also evaluates the twelve phase sums in the exact ring Z[z]/(z^4-z^2+1). The plots use floating coordinates for rendering, while the sample identities use exact arithmetic.
Exact mathematical passage
```
\mathbb Z[z]/(z^4-z^2+1)
```

z^6Use in source · 2 passages

Search aliases: z^6

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1894
For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.
Exact mathematical passage
```
\mathcal G_{\downarrow,C}(z)
&=
543z^7+25080z^8+510703z^9+10824940z^{10}+\cdots,\\
\mathcal G_{*,C}(z)
&=
2z^6+776z^7+39454z^8+\cdots.
```

z^7Use in source · 3 passages

Search aliases: z^7

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1894
For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.
Exact mathematical passage
```
\mathcal G_{\downarrow,C}(z)
&=
543z^7+25080z^8+510703z^9+10824940z^{10}+\cdots,\\
\mathcal G_{*,C}(z)
&=
2z^6+776z^7+39454z^8+\cdots.
```

z^8Use in source · 3 passages

Search aliases: z^8

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1894
For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.
Exact mathematical passage
```
\mathcal G_{\downarrow,C}(z)
&=
543z^7+25080z^8+510703z^9+10824940z^{10}+\cdots,\\
\mathcal G_{*,C}(z)
&=
2z^6+776z^7+39454z^8+\cdots.
```

z^9Use in source · 2 passages

Search aliases: z^9

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1894
For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.
Exact mathematical passage
```
\mathcal G_{\downarrow,C}(z)
&=
543z^7+25080z^8+510703z^9+10824940z^{10}+\cdots,\\
\mathcal G_{*,C}(z)
&=
2z^6+776z^7+39454z^8+\cdots.
```

z^kUse in source · 6 passages

Search aliases: z^k

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Use in source · Read passage · g_main.tex, line 479
[Endpoint Green valuation] The word distance is d_S(e,g) = ord_z=0 ⟨ δ_g,(I-zA_S)^-1δ_e ⟩. Expand the resolvent as a formal power series: (I-zA_S)^-1=∑_k≥0z^kA_S^k. The first nonzero endpoint coefficient occurs at the shortest word length.
Exact mathematical passage
```
(I-zA_S)^{-1}=\sum_{k\ge0}z^kA_S^k.
```

z^LUse in source · 1 passage

Search aliases: z^L

Link to this entry ·

Use in source · Read passage · q_main.tex, line 534
E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).
Exact mathematical passage
```
E_rf(t)=\sum_{m\ge0}a_{r+Lm}t^m,
\qquad
R_rf(z)=z^rE_rf(z^L).
```

z^nUse in source · 2 passages

Search aliases: z^n

Link to this entry ·

Use in source · Read passage · q_main.tex, line 530
Let L≥1 and let f(z)=∑_n≥0a_nz^n be a formal power series over a field containing a primitive Lth root ω and 1/L. For 0≤ r<L, define
Exact mathematical passage
```
f(z)=\sum_{n\ge0}a_nz^n
```

z^rUse in source · 1 passage

Search aliases: z^r

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Use in source · Read passage · q_main.tex, line 534
E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).
Exact mathematical passage
```
E_rf(t)=\sum_{m\ge0}a_{r+Lm}t^m,
\qquad
R_rf(z)=z^rE_rf(z^L).
```

z_0Use in source · 4 passages

Search aliases: z_0

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Use in source · Read passage · g_appendix.tex, line 61
Binary symplectic form. Pauli exponents are ordered as (x_0,…,x_n-1,z_0,…,z_n-1). The symplectic form is J= 0 · I_n
Exact mathematical passage
```
(x_0,\ldots,x_{n-1},z_0,\ldots,z_{n-1}).
```

z_1Use in source · 19 passages

Search aliases: z_1

Link to this entry ·

Use in source · Read passage · c_main.tex, line 274
Calibration families and exact small oracles. claimC03 and C04 An unbalanced disjunction. For k≥2 and n=k+2, define B=z_1⋯ z_k, F_k(a,b,z)=(ab)lor B=ab+B+abB, G_k=F_k+a. This is a deliberately easy family with two differently sized accepting regions. It separates the cost of finding one safe packet from the degree needed for complete coverage.
Exact mathematical passage
```
 B=z_1\cdots z_k,\qquad
 F_k(a,b,z)=(ab)\lor B=ab+B+abB,\qquad G_k=F_k+a.
```

\mathbb Z_2Use in source · 1 passage

Search aliases: Z_2 mathbb

Link to this entry ·

Use in source · references.tex, line 169
bibitemg:GongYu2026 S.~Gong and A.~Yu, Explicit matrices over Z_2 with CNOT and row complexity 4n-o(n) and local logic gates, arXiv:2607.28598, 2026.
Exact mathematical passage
```
\mathbb Z_2
```

z_2Use in source · 9 passages

Search aliases: z_2

Link to this entry ·

Use in source · Read passage · c_visual.tex, line 234
guidepageOne safe packet need not cover every accepting point For k=3 consider F_3(a,b,z_1,z_2,z_3)=ab OR z_1z_2z_3. The packet ab is safe and has degree two. It detects every teal point below. It misses three additional positives, which require the other packet z_1z_2z_3.
Exact mathematical passage
```
 F_3(a,b,z_1,z_2,z_3)=ab\ \mathrm{OR}\ z_1z_2z_3.
```

z_3Use in source · 6 passages

Search aliases: z_3

Link to this entry ·

Use in source · Read passage · c_visual.tex, line 234
guidepageOne safe packet need not cover every accepting point For k=3 consider F_3(a,b,z_1,z_2,z_3)=ab OR z_1z_2z_3. The packet ab is safe and has degree two. It detects every teal point below. It misses three additional positives, which require the other packet z_1z_2z_3.
Exact mathematical passage
```
 F_3(a,b,z_1,z_2,z_3)=ab\ \mathrm{OR}\ z_1z_2z_3.
```

\Z_8Use in source · 2 passages

Search aliases: Z_8 Z

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Use in source · Read passage · g_main.tex, line 3905
Let ω=e^iπ/4. A phase-polynomial circuit on n qubits has the form [source reference] Uketx=ω^P(x)ketAx+b, where A∈GL(n,2), b∈F_2^n, and P:F_2^n→Z_8 is represented as a sum of binary linear forms with coefficients in Z_8. The diagonal case has A=I and b=0.
Exact mathematical passage
```
P:\F_2^n\to\Z_8
```

z_cUse in source · 1 passage

Search aliases: z_c

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Use in source · Read passage · g_appendix.tex, line 87
Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.
Exact mathematical passage
```
x_t\leftarrow x_t+x_c,
\qquad
z_c\leftarrow z_c+z_t.
```

\Z_d^nUse in source · 1 passage

Search aliases: Z_d^n Z

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Use in source · Read passage · g_main.tex, line 3222
From transvections to quantum gates. Let d≥2. On n qudits with computational basis labeled by Z_d^n, define the scaled SUM gate SUM^(a)_s→ t ketx_0,…,x_n-1 = ketx_0,…,x_t+a x_s,…,x_n-1, a∈Z_d. It is a permutation unitary. Its action on basis labels is the row transvection I+aE_t,s. For d=2 and a=1, it is CNOT. For prime d, these gates lie in the standard qudit Clifford group. For composite d, they remain exact reversible modular-addition gates.
Exact mathematical passage
```
\Z_d^n
```

z_iUse in source · 11 passages

Search aliases: z_i

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Use in source · Read passage · c_main.tex, line 558
Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.
Exact mathematical passage
```
z_i
```

z_i^2Use in source · 1 passage

Search aliases: z_i^2

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Use in source · Read passage · c_main.tex, line 558
fingerprints claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.
Exact mathematical passage
```
z_i^2=z_i
```

z_i^{w_i}Use in source · 2 passages

Search aliases: z_i^w_i

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Use in source · Read passage · c_main.tex, line 554
Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.
Exact mathematical passage
```
 P_R(z_1,\ldots,z_m)=
 \sum_{w\in\B^m}R(w)\prod_{i=1}^m z_i^{w_i}\in K[z_1,\ldots,z_m].
```

z_jUse in source · 2 passages

Search aliases: z_j

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Use in source · Read passage · g_appendix.tex, line 93
Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.
Exact mathematical passage
```
z_i\leftarrow z_i+x_j,
\qquad
z_j\leftarrow z_j+x_i.
```

z_kUse in source · 3 passages

Search aliases: z_k

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Use in source · Read passage · c_main.tex, line 274
Calibration families and exact small oracles. claimC03 and C04 An unbalanced disjunction. For k≥2 and n=k+2, define B=z_1⋯ z_k, F_k(a,b,z)=(ab)lor B=ab+B+abB, G_k=F_k+a. This is a deliberately easy family with two differently sized accepting regions. It separates the cost of finding one safe packet from the degree needed for complete coverage.
Exact mathematical passage
```
 B=z_1\cdots z_k,\qquad
 F_k(a,b,z)=(ab)\lor B=ab+B+abB,\qquad G_k=F_k+a.
```

z_mUse in source · 3 passages

Search aliases: z_m

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Use in source · Read passage · c_main.tex, line 554
Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.
Exact mathematical passage
```
 P_R(z_1,\ldots,z_m)=
 \sum_{w\in\B^m}R(w)\prod_{i=1}^m z_i^{w_i}\in K[z_1,\ldots,z_m].
```

z_{m-1}Use in source · 1 passage

Search aliases: z_m-1

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Use in source · Read passage · c_main.tex, line 602
tional algorithmic consequence [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The leading coefficient has total degree at most d-r, so it vanishes at a random first (m-1)-tuple with probability at most (d-r)/|K|. Otherwise the remaini…
Exact mathematical passage
```
 P=\sum_{j=0}^r P_j(z_1,\ldots,z_{m-1})z_m^j,
 \qquad P_r\ne0.
```

z_m^jUse in source · 1 passage

Search aliases: z_m^j

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Use in source · Read passage · c_main.tex, line 602
tional algorithmic consequence [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The leading coefficient has total degree at most d-r, so it vanishes at a random first (m-1)-tuple with probability at most (d-r)/|K|. Otherwise the remaini…
Exact mathematical passage
```
 P=\sum_{j=0}^r P_j(z_1,\ldots,z_{m-1})z_m^j,
 \qquad P_r\ne0.
```

z_{n-1}Use in source · 2 passages

Search aliases: z_n-1

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Use in source · Read passage · g_appendix.tex, line 61
Binary symplectic form. Pauli exponents are ordered as (x_0,…,x_n-1,z_0,…,z_n-1). The symplectic form is J= 0 · I_n
Exact mathematical passage
```
(x_0,\ldots,x_{n-1},z_0,\ldots,z_{n-1}).
```

Z_qUse in source · 1 passage

Search aliases: Z_q

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Use in source · Read passage · s_main.tex, line 718
Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.
Exact mathematical passage
```
\widehat N
=
\sum_{q=1}^n\frac{I-Z_q}{2}.
```

z_tUse in source · 4 passages

Search aliases: z_t

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Use in source · Read passage · g_appendix.tex, line 87
Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.
Exact mathematical passage
```
x_t\leftarrow x_t+x_c,
\qquad
z_c\leftarrow z_c+z_t.
```

z_{t-1}Use in source · 3 passages

Search aliases: z_t-1

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Use in source · Read passage · s_main.tex, line 458
[Exact dynamic-packet theorem] Suppose P_0ketψ_0=ketψ_0 and [source reference] holds for every gate. Put z_0=B_0^*ketψ_0, z_t=M_tz_t-1. Then G_t⋯ G_1ketψ_0=B_tz_t for every t.
Exact mathematical passage
```
z_0=B_0^*\ket{\psi_0},
\qquad
z_t=M_tz_{t-1}.
```

z_uUse in source · 1 passage

Search aliases: z_u

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Use in source · Read passage · c_main.tex, line 628
[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independent and the factorization exact.
Exact mathematical passage
```
 P_{A\land B}(z_u,z_v)=P_A(z_u)P_B(z_v).
```

z_u^uUse in source · 1 passage

Search aliases: z_u^u

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Use in source · Read passage · c_main.tex, line 634
[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independent and the factorization exact.
Exact mathematical passage
```
 \sum_{u,v}A(u)B(v)z_u^uz_v^v
 =\left(\sum_uA(u)z_u^u\right)
  \left(\sum_vB(v)z_v^v\right).
```

z_vUse in source · 1 passage

Search aliases: z_v

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Use in source · Read passage · c_main.tex, line 628
[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independent and the factorization exact.
Exact mathematical passage
```
 P_{A\land B}(z_u,z_v)=P_A(z_u)P_B(z_v).
```

z_v^vUse in source · 1 passage

Search aliases: z_v^v

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Use in source · Read passage · c_main.tex, line 634
[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independent and the factorization exact.
Exact mathematical passage
```
 \sum_{u,v}A(u)B(v)z_u^uz_v^v
 =\left(\sum_uA(u)z_u^u\right)
  \left(\sum_vB(v)z_v^v\right).
```

|^2Use in source · 13 passages

Search aliases: |^2

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Use in source · Read passage · q_main.tex, line 184
x(n)=frac1q∑_k=0^q-1widehatx(k)ζ_q^kn, ∑_n|x(n)|^2=frac1q∑_k|widehatx(k)|^2.
Exact mathematical passage
```
x(n)=\frac1q\sum_{k=0}^{q-1}\widehat{x}(k)\zeta_q^{kn},
\qquad
\sum_n|x(n)|^2=\frac1q\sum_k|\widehat{x}(k)|^2.
```

|_{\mathcal S_G}Use in source · 1 passage

Search aliases: |_ S_G mathcal

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Use in source · Read passage · g_main.tex, line 230
Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry is the left Schreier graph G/H, with edges gH⟼ sgH. For the base-fiber assertion below, assume also S_H= S_G∩ H. This last condition excludes addition…
Exact mathematical passage
```
\ell_H(h)=|h|_{\mathcal S_H},
\qquad
\ell_G(h)=|h|_{\mathcal S_G}.
```

|_{\mathcal S_H}Use in source · 1 passage

Search aliases: |_ S_H mathcal

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Use in source · Read passage · g_main.tex, line 230
Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry is the left Schreier graph G/H, with edges gH⟼ sgH. For the base-fiber assertion below, assume also S_H= S_G∩ H. This last condition excludes addition…
Exact mathematical passage
```
\ell_H(h)=|h|_{\mathcal S_H},
\qquad
\ell_G(h)=|h|_{\mathcal S_G}.
```

|_{a=0}Use in source · 2 passages

Search aliases: |_a=0

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Use in source · Read passage · c_main.tex, line 303
For equality, take ℓ=a. The degree-three polynomial a(1+b)(1+z_1) is safe for G_k. Suppose a safe h of degree at most two existed. On a=0, acceptance requires B=1. For k≥3, the monotone-ideal argument forces h|_a=0=0, hence h=aL with L affine in the remaining variables. The negative face a=b=1 forces L(1,z)=0, so L=c(1+b). The negative point a=1,b=0,z=(1,…,1) then forces c=0.
Exact mathematical passage
```
h|_{a=0}=0
```

|_{b=1}Use in source · 2 passages

Search aliases: |_b=1

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Use in source · Read passage · c_main.tex, line 316
[Affine optimization of complete coverage] For every k≥2, β^*(F_k)=k+1. Restrict G_k=F_k+a to b=1. Then G_k|_b=1=(1+a)B. This is the indicator of one point on k+1 variables. At the positive input a=0,b=1,z=(1,…,1), a safe detecting polynomial must restrict to that singleton indicator and therefore have degree at least k+1. Thus β(G_k)≥ k+1.
Exact mathematical passage
```
 G_k|_{b=1}=(1+a)B.
```

|_HUse in source · 1 passage

Search aliases: |_H

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Use in source · Read passage · g_main.tex, line 399
When defects are impossible. A useful positive theorem is a criterion guaranteeing geodesic convexity. [One-Lipschitz retraction criterion] Suppose there is a group homomorphism π:G→ H with π|_H=id_H such that every ambient generator s∈ S_G satisfies ℓ_H(π(s))≤ c(s). Then ℓ_H(h)≤ℓ_G(h) for every h∈ H, hence the two metrics agree on H. For an ambient word h=s_L⋯ s_1, multiplicativity gives h=π(h)=π(s_L)⋯π(s_1). Subadditivity and the generator bounds give ℓ_H(h)≤∑_j c(s_j). Taking the infimum over…
Exact mathematical passage
```
\pi|_H=\operatorname{id}_H
```

|_{P_i\Hh}Use in source · 1 passage

Search aliases: |_P_iHh Hh

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Use in source · Read passage · s_main.tex, line 115
[Invariant quantum packets] If a unitary U satisfies [U,Q]=0, then P_iUP_j=0 (i≠ j), and U=⊕_i U_i, U_i=P_iUP_i|_P_iHh. If P_kketψ=ketψ, then P_kUketψ=Uketψ.
Exact mathematical passage
```
U=\bigoplus_i U_i,
\qquad
U_i=P_iUP_i\big|_{P_i\Hh}.
```

|_{z_1=0}Use in source · 1 passage

Search aliases: |_z_1=0

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Use in source · Read passage · c_main.tex, line 297
[Affine optimization of the one-packet parameter] For every k≥2, max_ℓ∈Aff_nα_1(F_k+ℓ)=3. First restrict to the face z_1=0. There F_k+ℓ becomes ab+ℓ|_z_1=0, a nonzero quadratic. If the restricted affine term contains a remaining z variable, set a=0 and set the remaining affine form to one. This gives an accepting affine subspace of codimension at most two inside the face. If it contains no remaining z variable, fix a,b to an accepting pair instead. The indicator of the resulting affine subspace …
Exact mathematical passage
```
ab+\ell|_{z_1=0}
```

\pmUse in source · 2 passages

Search aliases: ± pm

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Use in source · Read passage · g_main.tex, line 1811
[Regular-matroid core] Every square minor of B belongs to {0,±1}, so B is totally unimodular. Hence the column-dependence matroid is the same over every field. Under the edge labeling c|cccccc state · p · x · x+p · p+q · q · x+p+q
Exact mathematical passage
```
\{0,\pm1\}
```

\cdotUse in source · 11 passages

Search aliases: · cdot

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Use in source · Read passage · c_visual.tex, line 477
guidepageAt m=10, the total-gate accounting points the other way Here n=1,022. The full function uses 17 Toffolis; its quadratic grade needs 511. However, computing and uncomputing the m parities also uses 2m(2^m-1-1)=20·511=10,220 CNOTs. The explicit full construction has 10,237 total gates. The quadratic accumulation uses only 511 total gates.
Exact mathematical passage
```
 2m(2^{m-1}-1)=20\cdot511=10,220\quad\text{CNOTs}.
```

\timesUse in source · 14 passages

Search aliases: × times

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Use in source · Read passage · c_main.tex, line 409
[Exact cost of a nondegenerate paired quadratic] For even n, let Q(x)=x_1x_2+x_3x_4+⋯+x_n-1x_n. Then M(Q)=KTrest(Q)=n/2. The polar bilinear form of Q is the full-rank alternating form made of n/2 disjoint 2×2 blocks. If Q is affine on an affine subspace with direction space U, its second differences vanish on U, so U is totally isotropic. Nondegeneracy gives dim U≤ n/2.
Exact mathematical passage
```
2\times2
```

\bm\alphaUse in source · 1 passage

Search aliases: α bm alpha

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Use in source · Read passage · s_main.tex, line 182
Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential preprocessing merely relocates the complexity.
Exact mathematical passage
```
P_{\bm\alpha}
=
P^{(1)}_{\alpha_1}\cdots P^{(s)}_{\alpha_s}.
```

\alpha_sUse in source · 1 passage

Search aliases: α_s alpha

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Use in source · Read passage · s_main.tex, line 182
Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential preprocessing merely relocates the complexity.
Exact mathematical passage
```
P_{\bm\alpha}
=
P^{(1)}_{\alpha_1}\cdots P^{(s)}_{\alpha_s}.
```

\beta_1Use in source · 1 passage

Search aliases: β_1 beta

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Use in source · Read passage · g_main.tex, line 846
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
\ell_\Psi(u)
=
\min\{L:u=x_{\beta_L}(t_L)\cdots x_{\beta_1}(t_1),\ \beta_i\in\Psi\},
```

\beta_{11}Use in source · 1 passage

Search aliases: β_11 beta

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Use in source · Read passage · c_visual.tex, line 341
The degree-two polynomial ab does detect it, so β_11(f)=β(f)=2. Since β^*≤ n=2 and β^*≥β, β^*=2. The complete bound gives T≥1, attained by one direct Toffoli to the target.
Exact mathematical passage
```
\beta_{11}(f)=\beta(f)=2
```

\beta_iUse in source · 1 passage

Search aliases: β_i beta

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Use in source · Read passage · g_main.tex, line 846
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
\ell_\Psi(u)
=
\min\{L:u=x_{\beta_L}(t_L)\cdots x_{\beta_1}(t_1),\ \beta_i\in\Psi\},
```

\beta_LUse in source · 1 passage

Search aliases: β_L beta

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Use in source · Read passage · g_main.tex, line 846
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
\ell_\Psi(u)
=
\min\{L:u=x_{\beta_L}(t_L)\cdots x_{\beta_1}(t_1),\ \beta_i\in\Psi\},
```

\GammaUse in source · 10 passages

Search aliases: Γ Gamma

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Use in source · Read passage · g_main.tex, line 3986
texorpdfstringTT-depth and commuting layers. Two phase gadgets may share a T layer when a suitable Clifford network maps their parities to distinct wires or when ancilla resources permit parallel extraction. Define the conflict graph Γ_P whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the chosen ancilla and connectivity rules. Then T-depth(P)≥χ(Γ_P) for any conflict graph whose edge relation is sound. Catalysis can alter Γ_P by changing…
Exact mathematical passage
```
\Gamma_P
```

\gammaUse in source · 2 passages

Search aliases: γ gamma

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Use in source · Read passage · g_main.tex, line 4104
For n bosonic modes, a Gaussian unitary acts on quadratures r=(q_1,…,q_n,p_1,…,p_n)^T by an affine symplectic transformation r⟼ S r+d, S∈Sp(2n,R). For n fermionic modes, a fermionic Gaussian unitary acts on 2n Majorana operators by γ⟼ Oγ, O∈ SO(2n). Nearest-neighbor matchgates give a circuit presentation of a fermionic Gaussian subgroup on qubits.
Exact mathematical passage
```
\gamma\longmapsto O\gamma,
\qquad O\in SO(2n).
```

\Gamma'Use in source · 1 passage

Search aliases: Γ' Gamma

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Use in source · Read passage · g_main.tex, line 4423
For a target U, define ℓ_Γ(U) = minimum hardware-respecting cost, and compare it with an enlarged graph Γ'. A matched routing excursion has the form π W π^-1, where π is a logical permutation. The excursion is profitable when many gates in W become local in the permuted frame.
Exact mathematical passage
```
\Gamma'
```

\Gamma_PUse in source · 3 passages

Search aliases: Γ_P Gamma

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Use in source · Read passage · g_main.tex, line 3986
texorpdfstringTT-depth and commuting layers. Two phase gadgets may share a T layer when a suitable Clifford network maps their parities to distinct wires or when ancilla resources permit parallel extraction. Define the conflict graph Γ_P whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the chosen ancilla and connectivity rules. Then T-depth(P)≥χ(Γ_P) for any conflict graph whose edge relation is sound. Catalysis can alter Γ_P by changing…
Exact mathematical passage
```
\Gamma_P
```

\deltaUse in source · 39 passages

Search aliases: δ delta

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Use in source · Read passage · c_lean.tex, line 135
The least safe degree exists. For a positive v, take the point indicator δ_v(x)=∏_i=1^n x_i, · v_i=1, 1+x_i, · v_i=0. It is safe, detects v, and has a degree-n certificate. Hence complete safe coverage exists at degree n for every function. Strong induction proves the existence of a least admissible natural degree. The definition pathrefbeta chooses this bound and its specification and minimality are proved.
Exact mathematical passage
```
\delta_v(x)=\prod_{i=1}^{n}
\begin{cases}
x_i,&v_i=1,\\
1+x_i,&v_i=0.
\end{cases}
```

\Delta^\lambdaUse in source · 1 passage

Search aliases: Δ^λ Delta lambda

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Use in source · Read passage · g_main.tex, line 102
For a weight vector λ∈R_≥0^r, scalarization gives ℓ_ M^λ(U) = inf_x∈ P_ M(U)λ· x. The scalar defect Δ^λ(U) can vary with λ even when the target and gate alphabets are fixed.
Exact mathematical passage
```
\Delta^\lambda(U)
```

\Delta_{\mathrm{2q-depth}}Use in source · 1 passage

Search aliases: Δ_2q-depth Delta mathrm

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Use in source · Read passage · g_main.tex, line 3662
Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its total count is large. Conversely, a two-gate count saving can increase depth under locality constraints.
Exact mathematical passage
```
\Delta_{\mathrm{depth}}(U),
\qquad
\Delta_{\mathrm{2q-depth}}(U),
\qquad
\Delta_{\mathrm{space-time}}(U).
```

\Delta_{a,\mathrm{clean}}Use in source · 1 passage

Search aliases: Δ_a,clean Delta mathrm

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Use in source · Read passage · g_main.tex, line 4375
Ancillas define a family of defects Δ_a,clean(U), Δ_a,dirty(U), where a is the number of ancillary qubits or modes. Stable ancilla capacity asks whether a fixed number of reusable ancillas can yield an unbounded saving as the data size grows.
Exact mathematical passage
```
\Delta_{a,\mathrm{clean}}(U),
\qquad
\Delta_{a,\mathrm{dirty}}(U),
```

\Delta_{a,\mathrm{dirty}}Use in source · 1 passage

Search aliases: Δ_a,dirty Delta mathrm

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Use in source · Read passage · g_main.tex, line 4375
Ancillas define a family of defects Δ_a,clean(U), Δ_a,dirty(U), where a is the number of ancillary qubits or modes. Stable ancilla capacity asks whether a fixed number of reusable ancillas can yield an unbounded saving as the data size grows.
Exact mathematical passage
```
\Delta_{a,\mathrm{clean}}(U),
\qquad
\Delta_{a,\mathrm{dirty}}(U),
```

\delta_{de}Use in source · 3 passages

Search aliases: δ_de delta

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Use in source · Read passage · q_main.tex, line 363

(P_d^(N))^† · =P_d^(N), P_d^(N)P_e^(N) · =δ_deP_d^(N), ∑_dmid NP_d^(N) · =Id_N, rankP_d^(N) · =trP_d^(N)=φ(d).

Exact mathematical passage


\begin{aligned}
(P_d^{(N)})^\dagger&=P_d^{(N)},\\
P_d^{(N)}P_e^{(N)}&=\delta_{de}P_d^{(N)},\\
\sum_{d\mid N}P_d^{(N)}&=\operatorname{Id}_N,\\
\operatorname{rank}P_d^{(N)}&=\operatorname{tr}P_d^{(N)}=\varphi(d).
\end{aligned}

\Delta_{\mathrm{depth}}Use in source · 1 passage

Search aliases: Δ_depth Delta mathrm

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Use in source · Read passage · g_main.tex, line 3662
Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its total count is large. Conversely, a two-gate count saving can increase depth under locality constraints.
Exact mathematical passage
```
\Delta_{\mathrm{depth}}(U),
\qquad
\Delta_{\mathrm{2q-depth}}(U),
\qquad
\Delta_{\mathrm{space-time}}(U).
```

\delta_eUse in source · 10 passages

Search aliases: δ_e delta

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Use in source · Read passage · g_main.tex, line 461
Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.
Exact mathematical passage
```
\langle\delta_g,A_S^k\delta_e\rangle
```

\delta_gUse in source · 10 passages

Search aliases: δ_g delta

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Use in source · Read passage · g_main.tex, line 461
Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.
Exact mathematical passage
```
\langle\delta_g,A_S^k\delta_e\rangle
```

\delta_hUse in source · 1 passage

Search aliases: δ_h delta

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Use in source · Read passage · g_main.tex, line 1848
Let G be a finite group and S=S^-1 a symmetric generating set. On ℓ^2(G), let λ be the left regular representation, λ(s)δ_h=δ_sh, and define A_S=∑_s∈ Sλ(s). For g∈ G, define the endpoint Green series G_S,g(z) = ⟨ δ_g, (I-zA_S)^-1δ_e ⟩.
Exact mathematical passage
```
\lambda(s)\delta_h=\delta_{sh},
```

\Delta_{H\hookrightarrow G}Use in source · 1 passage

Search aliases: Δ_H↪ G Delta hookrightarrow

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Use in source · Read passage · g_main.tex, line 253
Thus Δ_H↪ G(h) = intrinsic fiber cost - shortest holonomy loop cost. This is the geometric content of a dirty-workspace shortcut.
Exact mathematical passage
```
\Delta_{H\hookrightarrow G}(h)
=
\text{intrinsic fiber cost}
-
\text{shortest holonomy loop cost}.
```

\delta_jUse in source · 3 passages

Search aliases: δ_j delta

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Use in source · Read passage · s_main.tex, line 360
Circuit leakage. For a gate G_j, define δ_j= (I-P)G_jP . Let V_j=PG_jP. A telescoping expansion gives G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Taking norms yields the following.
Exact mathematical passage
```
\delta_j=\|(I-P)G_jP\|.
```

\Delta_{\mathrm{large}}Use in source · 1 passage

Search aliases: Δ_large Delta mathrm

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Use in source · Read passage · g_main.tex, line 5165
a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). This transfers a catalytic atom to architectures containing it as a retract.
Exact mathematical passage
```
\Delta_{\mathrm{large}}(\iota U)
\ge
\Delta_{\mathrm{small}}(U).
```

\delta_{rs}Use in source · 1 passage

Search aliases: δ_rs delta

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Use in source · Read passage · q_main.tex, line 550
R_rf(z) · =frac1L∑_j=0^L-1ω^-rjf(ω^jz), R_rR_s · =δ_rsR_r, ∑_r=0^L-1R_rf · =f.
Exact mathematical passage
```
\begin{aligned}
R_rf(z)&=\frac1L\sum_{j=0}^{L-1}\omega^{-rj}f(\omega^jz),\\
R_rR_s&=\delta_{rs}R_r,\\
\sum_{r=0}^{L-1}R_rf&=f.
\end{aligned}
```

\delta_{sh}Use in source · 1 passage

Search aliases: δ_sh delta

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Use in source · Read passage · g_main.tex, line 1848
Let G be a finite group and S=S^-1 a symmetric generating set. On ℓ^2(G), let λ be the left regular representation, λ(s)δ_h=δ_sh, and define A_S=∑_s∈ Sλ(s). For g∈ G, define the endpoint Green series G_S,g(z) = ⟨ δ_g, (I-zA_S)^-1δ_e ⟩.
Exact mathematical passage
```
\lambda(s)\delta_h=\delta_{sh},
```

\Delta_{\mathrm{small}}Use in source · 1 passage

Search aliases: Δ_small Delta mathrm

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Use in source · Read passage · g_main.tex, line 5165
a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). This transfers a catalytic atom to architectures containing it as a retract.
Exact mathematical passage
```
\Delta_{\mathrm{large}}(\iota U)
\ge
\Delta_{\mathrm{small}}(U).
```

\Delta_{\mathrm{space-time}}Use in source · 1 passage

Search aliases: Δ_space-time Delta mathrm

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Use in source · Read passage · g_main.tex, line 3662
Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its total count is large. Conversely, a two-gate count saving can increase depth under locality constraints.
Exact mathematical passage
```
\Delta_{\mathrm{depth}}(U),
\qquad
\Delta_{\mathrm{2q-depth}}(U),
\qquad
\Delta_{\mathrm{space-time}}(U).
```

\delta_tUse in source · 4 passages

Search aliases: δ_t delta

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Use in source · Read passage · s_main.tex, line 606
Approximate dynamic packets. Define the one-step leakage from the scheduled packet by δ_t = (I-P_t)G_tP_t-1 . Let widehat G_t=P_tG_tP_t-1.
Exact mathematical passage
```
\delta_t
=
\|(I-P_t)G_tP_{t-1}\|.
```

\delta_uUse in source · 2 passages

Search aliases: δ_u delta

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Use in source · Read passage · c_main.tex, line 499
The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
 \delta_v^2=\delta_v,\qquad
 \delta_v\delta_u=0\quad(u\ne v),\qquad
 \sum_{v\in W}\delta_v=1.
```

\delta_vUse in source · 14 passages

Search aliases: δ_v delta

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Use in source · Read passage · c_lean.tex, line 135
The least safe degree exists. For a positive v, take the point indicator δ_v(x)=∏_i=1^n x_i, · v_i=1, 1+x_i, · v_i=0. It is safe, detects v, and has a degree-n certificate. Hence complete safe coverage exists at degree n for every function. Strong induction proves the existence of a least admissible natural degree. The definition pathrefbeta chooses this bound and its specification and minimality are proved.
Exact mathematical passage
```
\delta_v(x)=\prod_{i=1}^{n}
\begin{cases}
x_i,&v_i=1,\\
1+x_i,&v_i=0.
\end{cases}
```

\delta_v^2Use in source · 2 passages

Search aliases: δ_v^2 delta

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Use in source · Read passage · c_main.tex, line 499
The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
 \delta_v^2=\delta_v,\qquad
 \delta_v\delta_u=0\quad(u\ne v),\qquad
 \sum_{v\in W}\delta_v=1.
```

\Delta_wUse in source · 1 passage

Search aliases: Δ_w Delta

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Use in source · Read passage · g_main.tex, line 4006
Scalarization by positive weights w gives Δ_w(U)= min_c∈ C_res(U)w· c - min_c∈ C_amb(U)w· c. Different w can change the magnitude of the defect and whether it is zero or positive. It remains nonnegative because every restricted realization is also an ambient realization with the same resource vector.
Exact mathematical passage
```
\Delta_w(U)=
\min_{c\in\mathcal C_{\mathrm{res}}(U)}w\cdot c
-
\min_{c\in\mathcal C_{\mathrm{amb}}(U)}w\cdot c.
```

\delta_wUse in source · 1 passage

Search aliases: δ_w delta

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Use in source · Read passage · c_main.tex, line 167
If v and w are distinct positive inputs, δ_v+δ_w is safe and equals one at v. The degree-n coefficients of its two summands are both one, so they cancel. Its degree is at most n-1. Do this for each positive v. The last assertion applies the general bound to every affine adjustment, with the stated zero convention.
Exact mathematical passage
```
\delta_v+\delta_w
```

\delta_{\alpha\beta}Use in source · 1 passage

Search aliases: δ_αβ delta alpha beta

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Use in source · Read passage · s_main.tex, line 82
Orthogonal packet decompositions. Let Hh be a finite-dimensional Hilbert space and let {P_α}_α∈ A be orthogonal projectors satisfying P_α P_β=δ_αβP_α, ∑_α∈ AP_α=I. Every state and operator decompose as ketψ=∑_α P_αketψ, X=∑_α,βP_α X P_β. The diagonal blocks act within packets. The off-diagonal blocks transfer amplitude or coherence between packets.
Exact mathematical passage
```
P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,
\qquad
\sum_{\alpha\in A}P_\alpha=I.
```

\delta_{\chi\psi}Use in source · 1 passage

Search aliases: δ_χψ delta chi psi

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Use in source · Read passage · s_main.tex, line 167
Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).
Exact mathematical passage
```
P_\chi P_\psi=\delta_{\chi\psi}P_\chi,
\qquad
\sum_\chi P_\chi=I
```

\varepsilon^2Use in source · 1 passage

Search aliases: ε^2 varepsilon

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Use in source · Read passage · g_main.tex, line 383
Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.
Exact mathematical passage
```
e^{\varepsilon X}e^{\varepsilon Y}e^{-\varepsilon X}e^{-\varepsilon Y}
=
e^{\varepsilon^2[X,Y]+O(\varepsilon^3)}.
```

\varepsilon^3Use in source · 1 passage

Search aliases: ε^3 varepsilon

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Use in source · Read passage · g_main.tex, line 383
Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.
Exact mathematical passage
```
e^{\varepsilon X}e^{\varepsilon Y}e^{-\varepsilon X}e^{-\varepsilon Y}
=
e^{\varepsilon^2[X,Y]+O(\varepsilon^3)}.
```

\zetaUse in source · 33 passages

Search aliases: ζ zeta

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Use in source · Read passage · q_lean.tex, line 227
The phase ζ=(0,1/2,1/2,0) has complex meaning sqrt3/2+i/2. Exact finite certificates establish its twelfth power, inverse, and absence of a smaller positive power below twelve. Later the convention module identifies it with exp(π i/6).
Exact mathematical passage
```
\zeta=(0,1/2,1/2,0)
```

\zeta^{(a-k)n}Use in source · 1 passage

Search aliases: ζ^(a-k)n zeta

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Use in source · Read passage · q_visual.tex, line 81
s the selector leadThe Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform widehat x(k)=∑_n=0^11x(n)e^-2π ikn/12. fig04_exact_spectrum1.0The coefficient is exactly 12 at k=1,5,7,11 and exactly zero elsewhere. This is an algebraic identity, not a spectral thresholding decision. widehatc_12(k)=∑_a∈ U_12∑_n=0^11ζ^(a-k)n =12 1_k∈ U_12. The inner geometric sum is twelve when a=k modulo twelve and zero otherwise. The…
Exact mathematical passage
```
 \widehat{c_{12}}(k)=\sum_{a\in U_{12}}\sum_{n=0}^{11}\zeta^{(a-k)n}
 =12\,\mathbf1_{k\in U_{12}}.
```

\zeta^{-1}Use in source · 1 passage

Search aliases: ζ^-1 zeta

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Use in source · Read passage · q_main.tex, line 789
widehat c(1)=8+4iζ^-1+4ζ^-2=12.
Exact mathematical passage
```
\widehat c(1)=8+4i\zeta^{-1}+4\zeta^{-2}=12.
```

\zeta^{-18}Use in source · 1 passage

Search aliases: ζ^-18 zeta

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Use in source · Read passage · q_main.tex, line 795
widehat c(9)=8+4iζ^-9+4ζ^-18=8-4-4=0.
Exact mathematical passage
```
\widehat c(9)=8+4i\zeta^{-9}+4\zeta^{-18}=8-4-4=0.
```

\zeta^{-2}Use in source · 1 passage

Search aliases: ζ^-2 zeta

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Use in source · Read passage · q_main.tex, line 789
widehat c(1)=8+4iζ^-1+4ζ^-2=12.
Exact mathematical passage
```
\widehat c(1)=8+4i\zeta^{-1}+4\zeta^{-2}=12.
```

\zeta^{-9}Use in source · 1 passage

Search aliases: ζ^-9 zeta

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Use in source · Read passage · q_main.tex, line 795
widehat c(9)=8+4iζ^-9+4ζ^-18=8-4-4=0.
Exact mathematical passage
```
\widehat c(9)=8+4i\zeta^{-9}+4\zeta^{-18}=8-4-4=0.
```

\zeta^{11n}Use in source · 1 passage

Search aliases: ζ^11n zeta

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Use in source · Read passage · q_main.tex, line 688
c(n)=ζ^n+ζ^5n+ζ^7n+ζ^11n =2cos(π n/6)+2cos(5π n/6).
Exact mathematical passage
```
c(n)=\zeta^n+\zeta^{5n}+\zeta^{7n}+\zeta^{11n}
=2\cos(\pi n/6)+2\cos(5\pi n/6).
```

\zeta^{3n}Use in source · 1 passage

Search aliases: ζ^3n zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 853
3cos(π n/2)=frac32(ζ^3n+ζ^9n).
Exact mathematical passage
```
3\cos(\pi n/2)=\frac32(\zeta^{3n}+\zeta^{9n}).
```

\zeta^{5n}Use in source · 1 passage

Search aliases: ζ^5n zeta

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Use in source · Read passage · q_main.tex, line 688
c(n)=ζ^n+ζ^5n+ζ^7n+ζ^11n =2cos(π n/6)+2cos(5π n/6).
Exact mathematical passage
```
c(n)=\zeta^n+\zeta^{5n}+\zeta^{7n}+\zeta^{11n}
=2\cos(\pi n/6)+2\cos(5\pi n/6).
```

\zeta^{7n}Use in source · 1 passage

Search aliases: ζ^7n zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 688
c(n)=ζ^n+ζ^5n+ζ^7n+ζ^11n =2cos(π n/6)+2cos(5π n/6).
Exact mathematical passage
```
c(n)=\zeta^n+\zeta^{5n}+\zeta^{7n}+\zeta^{11n}
=2\cos(\pi n/6)+2\cos(5\pi n/6).
```

\zeta^{9n}Use in source · 1 passage

Search aliases: ζ^9n zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 853
3cos(π n/2)=frac32(ζ^3n+ζ^9n).
Exact mathematical passage
```
3\cos(\pi n/2)=\frac32(\zeta^{3n}+\zeta^{9n}).
```

\zeta^aUse in source · 2 passages

Search aliases: ζ^a zeta

Link to this entry ·

Use in source · Read passage · q_visual.tex, line 21
Coprimality selects primitive roots. addcontentslinetocsectionCoprimality selects primitive roots leadA selected residue is a label for a wave whose phase has exact order twelve. Set ζ=e^2π i/12. The phase ζ^a first returns to 1 after 12/gcd(a,12) powers. Thus the coprime labels 1,5,7,11 select exactly the primitive twelfth roots. fig01_primitive_roots0.91The circle is the complex unit circle. Teal points are the four primitive roots; pale points are the other eight roots. Labels are residues a,…
Exact mathematical passage
```
\zeta^a
```

\zeta^{an}Use in source · 1 passage

Search aliases: ζ^an zeta

Link to this entry ·

Use in source · Read passage · q_visual.tex, line 30
The selected phases add and cancel. addcontentslinetocsectionThe selected phases add and cancel leadThe same four waves can reinforce, cancel, or reinforce with the opposite sign. At each sample n, add the four numbers ζ^an with a∈ U_12. Each has magnitude one. Their sum need not have magnitude one. fig02_phase_addition1.0At n=0 the sum is 4. At n=1 the four phases cancel. At n=6 the sum is -4. Tiny parallel drawing offsets expose coincident arrows at n=0 and n=6; each arrow's displacement is th…
Exact mathematical passage
```
\zeta^{an}
```

\zeta^nUse in source · 1 passage

Search aliases: ζ^n zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 688
c(n)=ζ^n+ζ^5n+ζ^7n+ζ^11n =2cos(π n/6)+2cos(5π n/6).
Exact mathematical passage
```
c(n)=\zeta^n+\zeta^{5n}+\zeta^{7n}+\zeta^{11n}
=2\cos(\pi n/6)+2\cos(5\pi n/6).
```

\zeta^{na}Use in source · 2 passages

Search aliases: ζ^na zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 914
frac12sqrt12∑_a∈ U_12ζ^na =fracc(n)sqrt48.
Exact mathematical passage
```
\frac1{2\sqrt{12}}\sum_{a\in U_{12}}\zeta^{na}
=\frac{c(n)}{\sqrt{48}}.
```

\zeta_d^{b(n-m)}Use in source · 1 passage

Search aliases: ζ_d^b(n-m) zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 381
(F_ND_dF_N^†)_nm · =frac1N∑_a∈ S_dζ_N^a(n-m) · =frac1N∑_b∈ U_dζ_d^b(n-m) =frac1N c_d(n-m).
Exact mathematical passage
```
\begin{aligned}
(F_ND_dF_N^\dagger)_{nm}
&=\frac1N\sum_{a\in S_d}\zeta_N^{a(n-m)}\\
&=\frac1N\sum_{b\in U_d}\zeta_d^{b(n-m)}
=\frac1N c_d(n-m).
\end{aligned}
```

\zeta_M^{(k-\ell)m}Use in source · 1 passage

Search aliases: ζ_M^(k-ℓ)m zeta ell

Link to this entry ·

Use in source · Read passage · q_main.tex, line 635
widehatx_r(ℓ) · =∑_m=0^M-1x(r+Lm)ζ_M^-ℓ m · =frac1N∑_k=0^N-1widehatx(k)ζ_N^kr ∑_m=0^M-1ζ_M^(k-ℓ)m.
Exact mathematical passage
```
\begin{aligned}
\widehat{x_r}(\ell)
&=\sum_{m=0}^{M-1}x(r+Lm)\zeta_M^{-\ell m}\\
&=\frac1N\sum_{k=0}^{N-1}\widehat{x}(k)\zeta_N^{kr}
\sum_{m=0}^{M-1}\zeta_M^{(k-\ell)m}.
\end{aligned}
```

\zeta_M^{-km}Use in source · 1 passage

Search aliases: ζ_M^-km zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 622
widehatx(k) · =∑_r=0^L-1∑_m=0^M-1 x_r(m)ζ_N^-k(r+Lm) · =∑_rζ_N^-kr∑_mx_r(m)ζ_M^-km.
Exact mathematical passage
```
\begin{aligned}
\widehat{x}(k)
&=\sum_{r=0}^{L-1}\sum_{m=0}^{M-1}
x_r(m)\zeta_N^{-k(r+Lm)}\\
&=\sum_r\zeta_N^{-kr}\sum_mx_r(m)\zeta_M^{-km}.
\end{aligned}
```

\zeta_M^{-\ell m}Use in source · 1 passage

Search aliases: ζ_M^-ℓ m zeta ell

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Use in source · Read passage · q_main.tex, line 635
widehatx_r(ℓ) · =∑_m=0^M-1x(r+Lm)ζ_M^-ℓ m · =frac1N∑_k=0^N-1widehatx(k)ζ_N^kr ∑_m=0^M-1ζ_M^(k-ℓ)m.
Exact mathematical passage
```
\begin{aligned}
\widehat{x_r}(\ell)
&=\sum_{m=0}^{M-1}x(r+Lm)\zeta_M^{-\ell m}\\
&=\frac1N\sum_{k=0}^{N-1}\widehat{x}(k)\zeta_N^{kr}
\sum_{m=0}^{M-1}\zeta_M^{(k-\ell)m}.
\end{aligned}
```

\zeta_N^{-a}Use in source · 1 passage

Search aliases: ζ_N^-a zeta

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Use in source · Read passage · q_main.tex, line 411
Let A be the cyclic shift (Ax)(n)=x(n-1). A Fourier column f_a(n)=N^-1/2ζ_N^na satisfies Af_a=ζ_N^-af_a. Therefore
Exact mathematical passage
```
Af_a=\zeta_N^{-a}f_a
```

\zeta_N^{-k(r+Lm)}Use in source · 1 passage

Search aliases: ζ_N^-k(r+Lm) zeta

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Use in source · Read passage · q_main.tex, line 622
widehatx(k) · =∑_r=0^L-1∑_m=0^M-1 x_r(m)ζ_N^-k(r+Lm) · =∑_rζ_N^-kr∑_mx_r(m)ζ_M^-km.
Exact mathematical passage
```
\begin{aligned}
\widehat{x}(k)
&=\sum_{r=0}^{L-1}\sum_{m=0}^{M-1}
x_r(m)\zeta_N^{-k(r+Lm)}\\
&=\sum_r\zeta_N^{-kr}\sum_mx_r(m)\zeta_M^{-km}.
\end{aligned}
```

\zeta_N^{-kr}Use in source · 2 passages

Search aliases: ζ_N^-kr zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 606
widehatx(k) =∑_r=0^L-1ζ_N^-krwidehatx_r(kbmod M).
Exact mathematical passage
```
\boxed{\widehat{x}(k)
=\sum_{r=0}^{L-1}\zeta_N^{-kr}\widehat{x_r}(k\bmod M).}
```

\zeta_N^{a(n-m)}Use in source · 1 passage

Search aliases: ζ_N^a(n-m) zeta

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Use in source · Read passage · q_main.tex, line 381
(F_ND_dF_N^†)_nm · =frac1N∑_a∈ S_dζ_N^a(n-m) · =frac1N∑_b∈ U_dζ_d^b(n-m) =frac1N c_d(n-m).
Exact mathematical passage
```
\begin{aligned}
(F_ND_dF_N^\dagger)_{nm}
&=\frac1N\sum_{a\in S_d}\zeta_N^{a(n-m)}\\
&=\frac1N\sum_{b\in U_d}\zeta_d^{b(n-m)}
=\frac1N c_d(n-m).
\end{aligned}
```

\zeta_N^{a\ell}Use in source · 1 passage

Search aliases: ζ_N^aℓ zeta ell

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Use in source · Read passage · q_main.tex, line 414
E_a=frac1N∑_ℓ=0^N-1ζ_N^aℓA^ℓ
Exact mathematical passage
```
E_a=\frac1N\sum_{\ell=0}^{N-1}\zeta_N^{a\ell}A^\ell
```

\zeta_N^{kr}Use in source · 1 passage

Search aliases: ζ_N^kr zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 635
widehatx_r(ℓ) · =∑_m=0^M-1x(r+Lm)ζ_M^-ℓ m · =frac1N∑_k=0^N-1widehatx(k)ζ_N^kr ∑_m=0^M-1ζ_M^(k-ℓ)m.
Exact mathematical passage
```
\begin{aligned}
\widehat{x_r}(\ell)
&=\sum_{m=0}^{M-1}x(r+Lm)\zeta_M^{-\ell m}\\
&=\frac1N\sum_{k=0}^{N-1}\widehat{x}(k)\zeta_N^{kr}
\sum_{m=0}^{M-1}\zeta_M^{(k-\ell)m}.
\end{aligned}
```

\zeta_N^{na}Use in source · 1 passage

Search aliases: ζ_N^na zeta

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Use in source · Read passage · q_main.tex, line 411
Let A be the cyclic shift (Ax)(n)=x(n-1). A Fourier column f_a(n)=N^-1/2ζ_N^na satisfies Af_a=ζ_N^-af_a. Therefore
Exact mathematical passage
```
f_a(n)=N^{-1/2}\zeta_N^{na}
```

\zeta_N^{r(\ell+jM)}Use in source · 1 passage

Search aliases: ζ_N^r(ℓ+jM) zeta ell

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Use in source · Read passage · q_main.tex, line 613
widehatx_r(ℓ) =frac1L∑_j=0^L-1 ζ_N^r(ℓ+jM)widehatx(ℓ+jM).
Exact mathematical passage
```
\boxed{\widehat{x_r}(\ell)
=\frac1L\sum_{j=0}^{L-1}
\zeta_N^{r(\ell+jM)}\widehat{x}(\ell+jM).}
```

\zeta_N^{r\ell}Use in source · 1 passage

Search aliases: ζ_N^rℓ zeta ell

Link to this entry ·

Use in source · Read passage · q_main.tex, line 653
Every lane transform combines an alias class of L full-spectrum coefficients. One decimated lane generally cannot distinguish those coefficients. All L offsets together retain the original signal. For a fixed alias class ℓ, the equations across r are an invertible Fourier mixing system after the nonzero phases ζ_N^rℓ are removed. The apparent collapse is resolved by preserving the complete collection of offsets.
Exact mathematical passage
```
\zeta_N^{r\ell}
```

\zeta_qUse in source · 1 passage

Search aliases: ζ_q zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 109
ζ_q=e^2π i/q.
Exact mathematical passage
```
\zeta_q=e^{2\pi i/q}.
```

\zeta_q^{(a-k)n}Use in source · 1 passage

Search aliases: ζ_q^(a-k)n zeta

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Use in source · Read passage · q_main.tex, line 311
widehatc_q(k) =∑_a∈ U_q∑_n=0^q-1ζ_q^(a-k)n.
Exact mathematical passage
```
\widehat{c_q}(k)
=\sum_{a\in U_q}\sum_{n=0}^{q-1}\zeta_q^{(a-k)n}.
```

\zeta_q^{-kn}Use in source · 1 passage

Search aliases: ζ_q^-kn zeta

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Use in source · Read passage · q_main.tex, line 123
widehatx(k)=∑_n=0^q-1x(n)ζ_q^-kn.
Exact mathematical passage
```
\widehat{x}(k)=\sum_{n=0}^{q-1}x(n)\zeta_q^{-kn}.
```

\zeta_q^aUse in source · 2 passages

Search aliases: ζ_q^a zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 142
For a∈ I_q, the multiplicative order of ζ_q^a is
Exact mathematical passage
```
\zeta_q^a
```

\zeta_q^{an}Use in source · 1 passage

Search aliases: ζ_q^an zeta

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Use in source · Read passage · q_main.tex, line 229
c_q(n)=∑_a∈ U_qζ_q^an, n∈ Z.
Exact mathematical passage
```
c_q(n)=\sum_{a\in U_q}\zeta_q^{an},
\qquad n\in\mathbb Z.
```

\zeta_q^hUse in source · 1 passage

Search aliases: ζ_q^h zeta

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Use in source · Read passage · q_main.tex, line 167
Proof. If qmid h, every summand is one. Otherwise r=ζ_q^h≠1 and r^q=1. Multiplying the sum by 1-r gives 1-r^q=0, so the sum is zero. This argument also covers q=1, where only the first case occurs. square
Exact mathematical passage
```
r=\zeta_q^h\ne1
```

\zeta_q^{hn}Use in source · 1 passage

Search aliases: ζ_q^hn zeta

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Use in source · Read passage · q_main.tex, line 161
∑_n=0^q-1ζ_q^hn =q, · qmid h, 0, · qnmid h.
Exact mathematical passage
```
\sum_{n=0}^{q-1}\zeta_q^{hn}
=\begin{cases}q,&q\mid h,\\0,&q\nmid h.\end{cases}
```

\zeta_q^{kn}Use in source · 1 passage

Search aliases: ζ_q^kn zeta

Link to this entry ·

Use in source · Read passage · q_main.tex, line 184
x(n)=frac1q∑_k=0^q-1widehatx(k)ζ_q^kn, ∑_n|x(n)|^2=frac1q∑_k|widehatx(k)|^2.
Exact mathematical passage
```
x(n)=\frac1q\sum_{k=0}^{q-1}\widehat{x}(k)\zeta_q^{kn},
\qquad
\sum_n|x(n)|^2=\frac1q\sum_k|\widehat{x}(k)|^2.
```

\zeta_q^{n(b-a)}Use in source · 1 passage

Search aliases: ζ_q^n(b-a) zeta

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Use in source · Read passage · q_main.tex, line 192
frac1q∑_n=0^q-1ζ_q^n(b-a).
Exact mathematical passage
```
\frac1q\sum_{n=0}^{q-1}\zeta_q^{n(b-a)}.
```

\zeta_q^{na}Use in source · 1 passage

Search aliases: ζ_q^na zeta

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Use in source · Read passage · q_main.tex, line 129
(F_q)_na=q^-1/2ζ_q^na.
Exact mathematical passage
```
(F_q)_{na}=q^{-1/2}\zeta_q^{na}.
```

\etaUse in source · 5 passages

Search aliases: η eta

Link to this entry ·

Use in source · Read passage · s_main.tex, line 661
[Adaptive discarded-state certificate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ_m⟩-⟨ϕ_m,Mϕ_m⟩| ≤2∑_t=1^mη_t. No renormalization of ϕ_t is performed. The second quadratic form is an unnormalized a…
Exact mathematical passage
```
\psi_t=G_t\psi_{t-1},\qquad
\phi_t=P_tG_t\phi_{t-1},\qquad
\eta_t=\|(I-P_t)G_t\phi_{t-1}\|.
```

\eta_1Use in source · 2 passages

Search aliases: η_1 eta

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Use in source · Read passage · s_main.tex, line 700
0. Using G_1=G_2=G and P_1=P_2=P gives ϕ_2=9/25 0, ψ_2=-7/25 24/25, η_1=4/5, η_2=12/25. The squared state error is 832/625, below (η_1+η_2)^2=1024/625. The rational validation script checks these values exactly. The deliberately large tail illustrates a valid but loose certificate, not an efficient approximation.
Exact mathematical passage
```
\phi_2=\begin{pmatrix}9/25\\0\end{pmatrix},\qquad
\psi_2=\begin{pmatrix}-7/25\\24/25\end{pmatrix},\qquad
\eta_1=4/5,\quad\eta_2=12/25.
```

\eta_2Use in source · 2 passages

Search aliases: η_2 eta

Link to this entry ·

Use in source · Read passage · s_main.tex, line 700
0. Using G_1=G_2=G and P_1=P_2=P gives ϕ_2=9/25 0, ψ_2=-7/25 24/25, η_1=4/5, η_2=12/25. The squared state error is 832/625, below (η_1+η_2)^2=1024/625. The rational validation script checks these values exactly. The deliberately large tail illustrates a valid but loose certificate, not an efficient approximation.
Exact mathematical passage
```
\phi_2=\begin{pmatrix}9/25\\0\end{pmatrix},\qquad
\psi_2=\begin{pmatrix}-7/25\\24/25\end{pmatrix},\qquad
\eta_1=4/5,\quad\eta_2=12/25.
```

\eta_tUse in source · 3 passages

Search aliases: η_t eta

Link to this entry ·

Use in source · Read passage · s_main.tex, line 661
[Adaptive discarded-state certificate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ_m⟩-⟨ϕ_m,Mϕ_m⟩| ≤2∑_t=1^mη_t. No renormalization of ϕ_t is performed. The second quadratic form is an unnormalized a…
Exact mathematical passage
```
\psi_t=G_t\psi_{t-1},\qquad
\phi_t=P_tG_t\phi_{t-1},\qquad
\eta_t=\|(I-P_t)G_t\phi_{t-1}\|.
```

\ThetaUse in source · 1 passage

Search aliases: Θ Theta

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 463
L3 · Maximum unitriangular defect is Θ(nlog n). · Open · Matching upper bound or larger interacting-catalyst family.
Exact mathematical passage
```
\Theta(n\log n)
```

\thetaUse in source · 30 passages

Search aliases: θ theta

Link to this entry ·

Use in source · Read passage · c_visual.tex, line 699
visualfigureoverlap_gf16A concrete field point z=θ with θ≠0,1. The two columns are unequal in both rows.
Exact mathematical passage
```
z=\theta
```

\theta^2Use in source · 1 passage

Search aliases: θ^2 theta

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Use in source · Read passage · c_visual.tex, line 703
The field is F_16=F_2[θ]/(θ^4+θ+1). The checker stores field elements as four-bit coefficient vectors: θ is integer 2 and θ^2 is integer 4. These integers encode field elements; they are not ordinary integer multiplication. Exact field multiplication gives the reported mismatch.
Exact mathematical passage
```
\theta^2
```

\theta^4Use in source · 1 passage

Search aliases: θ^4 theta

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Use in source · Read passage · c_visual.tex, line 701
The field is F_16=F_2[θ]/(θ^4+θ+1). The checker stores field elements as four-bit coefficient vectors: θ is integer 2 and θ^2 is integer 4. These integers encode field elements; they are not ordinary integer multiplication. Exact field multiplication gives the reported mismatch.
Exact mathematical passage
```
\F_{16}=\F_2[\theta]/(\theta^4+\theta+1)
```

\theta_jUse in source · 1 passage

Search aliases: θ_j theta

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Use in source · Read passage · g_main.tex, line 3932
The simplest symbolic sandwich is C(∏_j=1^m R_P_j(θ_j))C^† = ∏_j=1^m R_CP_jC^†(θ_j). If the P_j commute, their conjugates commute. A catalyst can therefore change the geometry of the support without changing the algebra of the angles.
Exact mathematical passage
```
C\left(\prod_{j=1}^m R_{P_j}(\theta_j)\right)C^\dagger
=
\prod_{j=1}^m R_{CP_jC^\dagger}(\theta_j).
```

\iotaUse in source · 11 passages

Search aliases: ι iota

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Use in source · Read passage · g_appendix.tex, line 164
Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).
Exact mathematical passage
```
\iota(A)=\diag(A,A^{-T})\in\Sp(2n,\R).
```

\LambdaUse in source · 1 passage

Search aliases: Λ Lambda

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Use in source · Read passage · s_main.tex, line 536
Sparse packet-transfer graphs. Let I=∑_α∈ΛP_α be an orthogonal packet decomposition. Write the gate blocks as G_t;βα=P_β G_tP_α. Starting from an active packet set S_0, define recursively S_t = { β:∃α∈ S_t-1 with G_t;βα≠0 }.
Exact mathematical passage
```
I=\sum_{\alpha\in\Lambda}P_\alpha
```

\lambda_0Use in source · 1 passage

Search aliases: λ_0 lambda

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Use in source · Read passage · g_main.tex, line 3951
This makes phase-polynomial synthesis a close relative of the snapshot families in the linear core. A parity row may evolve through λ_0,λ_1,…,λ_m, with selected states consumed by phase gates. The difference is that consumption need not copy the parity to another row; it can deposit phase while leaving the carrier unchanged.
Exact mathematical passage
```
\lambda_0,\lambda_1,\ldots,\lambda_m,
```

\lambda_1Use in source · 1 passage

Search aliases: λ_1 lambda

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Use in source · Read passage · g_main.tex, line 3951
This makes phase-polynomial synthesis a close relative of the snapshot families in the linear core. A parity row may evolve through λ_0,λ_1,…,λ_m, with selected states consumed by phase gates. The difference is that consumption need not copy the parity to another row; it can deposit phase while leaving the carrier unchanged.
Exact mathematical passage
```
\lambda_0,\lambda_1,\ldots,\lambda_m,
```

\lambda_iUse in source · 1 passage

Search aliases: λ_i lambda

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Use in source · Read passage · s_main.tex, line 1157
Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).
Exact mathematical passage
```
u_\lambda
=
\prod_{1\le i<j\le d}
\frac{\lambda_i-\lambda_j+j-i}{j-i}
=O_d\left(n^{d(d-1)/2}\right).
\label{s:eq:weyl-dimension-bound}
```

\lambda_jUse in source · 1 passage

Search aliases: λ_j lambda

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1157
Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).
Exact mathematical passage
```
u_\lambda
=
\prod_{1\le i<j\le d}
\frac{\lambda_i-\lambda_j+j-i}{j-i}
=O_d\left(n^{d(d-1)/2}\right).
\label{s:eq:weyl-dimension-bound}
```

\lambda_mUse in source · 1 passage

Search aliases: λ_m lambda

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Use in source · Read passage · g_main.tex, line 3951
This makes phase-polynomial synthesis a close relative of the snapshot families in the linear core. A parity row may evolve through λ_0,λ_1,…,λ_m, with selected states consumed by phase gates. The difference is that consumption need not copy the parity to another row; it can deposit phase while leaving the carrier unchanged.
Exact mathematical passage
```
\lambda_0,\lambda_1,\ldots,\lambda_m,
```

\muUse in source · 23 passages

Search aliases: μ mu

Link to this entry ·

Use in source · Read passage · g_main.tex, line 512
Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint series after Fourier inversion.
Exact mathematical passage
```
\mu_k(h)
```

\mu_kUse in source · 6 passages

Search aliases: μ_k mu

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Use in source · Read passage · g_main.tex, line 512
Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint series after Fourier inversion.
Exact mathematical passage
```
\mu_k(h)
```

\mu_LUse in source · 2 passages

Search aliases: μ_L mu

Link to this entry ·

Use in source · Read passage · g_main.tex, line 549
Word maps and product loci. Let K be a field and let X⊆ G be an algebraic gate family. For a fixed length L, multiplication defines μ_L:X^L→ G, (x_L,…,x_1)↦ x_L⋯ x_1. The exact length-L locus is μ_L(X^L), and the length-at-most-L locus is a finite union of such images when the gate family has finitely many components.
Exact mathematical passage
```
\mu_L:X^L\to G,
\qquad
(x_L,\ldots,x_1)\mapsto x_L\cdots x_1.
```

\PiUse in source · 8 passages

Search aliases: Π Pi

Link to this entry ·

Use in source · Read passage · g_main.tex, line 489
Why eigenvalues alone are insufficient. If A_S=∑_λλΠ_λ, then G_S,g(z) = ∑_λ frac⟨δ_g,Π_λδ_e⟩1-λ z. The eigenvalues and multiplicities do not determine the endpoint coefficients ⟨δ_g,Π_λδ_e⟩. Two Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.
Exact mathematical passage
```
A_S=\sum_\lambda\lambda\Pi_\lambda,
```

\piUse in source · 45 passages

Search aliases: π pi

Link to this entry ·

Use in source · Read passage · g_main.tex, line 399
When defects are impossible. A useful positive theorem is a criterion guaranteeing geodesic convexity. [One-Lipschitz retraction criterion] Suppose there is a group homomorphism π:G→ H with π|_H=id_H such that every ambient generator s∈ S_G satisfies ℓ_H(π(s))≤ c(s). Then ℓ_H(h)≤ℓ_G(h) for every h∈ H, hence the two metrics agree on H. For an ambient word h=s_L⋯ s_1, multiplicativity gives h=π(h)=π(s_L)⋯π(s_1). Subadditivity and the generator bounds give ℓ_H(h)≤∑_j c(s_j). Taking the infimum over…
Exact mathematical passage
```
\pi:G\to H
```

\pi^{-1}Use in source · 2 passages

Search aliases: π^-1 pi

Link to this entry ·

Use in source · Read passage · g_main.tex, line 4262
Architecture and routing. On a line or grid, Gaussian and matchgate circuits may be dominated by routing. Let G_conn be the hardware graph. A mode permutation π can be treated as a catalyst if it is opened, many interactions are executed in the reordered frame, and π^-1 restores the logical labeling. The benefit is routing saved across consumers - 2 cost(π). This is the same amortization law as the linear snapshot family, now applied to mode geometry.
Exact mathematical passage
```
\pi^{-1}
```

\pi_1Use in source · 1 passage

Search aliases: π_1 pi

Link to this entry ·

Use in source · Read passage · g_main.tex, line 608
Direct sums and monoidal amplification. Suppose targets admit a direct sum U⊕ V and a homomorphism from the large restricted group onto the two factor groups sends U⊕ V to (U,V). Require the aggregate generator bound ellRes(π_1(s))+ellRes(π_2(s))≤ c(s) for every large restricted generator s. Assume that both restricted and ambient factor circuits embed into the corresponding large model without increasing cost. Then ellRes(U⊕ V) = ellRes(U)+ellRes(V). Indeed, projecting any word and summing the …
Exact mathematical passage
```
\ellRes(\pi_1(s))+\ellRes(\pi_2(s))\le c(s)
```

\pi_2Use in source · 1 passage

Search aliases: π_2 pi

Link to this entry ·

Use in source · Read passage · g_main.tex, line 608
Direct sums and monoidal amplification. Suppose targets admit a direct sum U⊕ V and a homomorphism from the large restricted group onto the two factor groups sends U⊕ V to (U,V). Require the aggregate generator bound ellRes(π_1(s))+ellRes(π_2(s))≤ c(s) for every large restricted generator s. Assume that both restricted and ambient factor circuits embed into the corresponding large model without increasing cost. Then ellRes(U⊕ V) = ellRes(U)+ellRes(V). Indeed, projecting any word and summing the …
Exact mathematical passage
```
\ellRes(\pi_1(s))+\ellRes(\pi_2(s))\le c(s)
```

\Pi_AUse in source · 3 passages

Search aliases: Π_A Pi

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Use in source · Read passage · s_main.tex, line 1697
Error accounting for truncated channels. Let Π_A project operator space onto an active set of coherence blocks. A reduced evolution uses L_A = Π_A LΠ_A. The discarded transfer is (I-Π_A) LΠ_A. Duhamel's formula yields an error bound analogous to the Hamiltonian leakage theorem. The resulting estimate depends on the induced norm of the discarded superoperator transfer and the integration time.
Exact mathematical passage
```
\Pi_A
```

\Pi_\lambdaUse in source · 5 passages

Search aliases: Π_λ Pi lambda

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Use in source · Read passage · g_main.tex, line 489
Why eigenvalues alone are insufficient. If A_S=∑_λλΠ_λ, then G_S,g(z) = ∑_λ frac⟨δ_g,Π_λδ_e⟩1-λ z. The eigenvalues and multiplicities do not determine the endpoint coefficients ⟨δ_g,Π_λδ_e⟩. Two Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.
Exact mathematical passage
```
A_S=\sum_\lambda\lambda\Pi_\lambda,
```

\rho_a^{\rm Gauss}Use in source · 1 passage

Search aliases: ρ_a^rm Gauss rho rm

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Use in source · Read passage · s_main.tex, line 1573
Low-rank non-Gaussian corrections. One research direction is to represent a weakly non-Gaussian state as ρ = ∑_a=1^R c_aρ_a^rm Gauss with small Gaussian rank R, or as a Gaussian reference plus sparse packet corrections. A polynomial collapse requires R and the update cost to remain polynomial.
Exact mathematical passage
```
\rho
=
\sum_{a=1}^R c_a\rho_a^{\rm Gauss}
```

\rho_{\mathrm{diag}}Use in source · 3 passages

Search aliases: ρ_diag rho mathrm

Link to this entry ·

Use in source · Read passage · q_main.tex, line 1383
ρ_diag=diag(c(n)^2/48)_n=0^11.
Exact mathematical passage
```
\rho_{\mathrm{diag}}=\operatorname{diag}\big(c(n)^2/48\big)_{n=0}^{11}.
```

\rho_iUse in source · 4 passages

Search aliases: ρ_i rho

Link to this entry ·

Use in source · Read passage · g_main.tex, line 4611
Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.
Exact mathematical passage
```
\rho_i
```

\rho_i^{-1}Use in source · 1 passage

Search aliases: ρ_i^-1 rho

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Use in source · Read passage · g_main.tex, line 4612
Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.
Exact mathematical passage
```
\label{g:eq:transition-fusion}
\sum_{i<j}
\left(
P_{\rho_i\rho_j^{-1}}
+
P_{\rho_j\rho_i^{-1}}
\right)
=n(J-I).
```

\rho_jUse in source · 1 passage

Search aliases: ρ_j rho

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Use in source · Read passage · g_main.tex, line 4612
Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.
Exact mathematical passage
```
\label{g:eq:transition-fusion}
\sum_{i<j}
\left(
P_{\rho_i\rho_j^{-1}}
+
P_{\rho_j\rho_i^{-1}}
\right)
=n(J-I).
```

\rho_j^{-1}Use in source · 2 passages

Search aliases: ρ_j^-1 rho

Link to this entry ·

Use in source · Read passage · g_main.tex, line 4612
Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.
Exact mathematical passage
```
\label{g:eq:transition-fusion}
\sum_{i<j}
\left(
P_{\rho_i\rho_j^{-1}}
+
P_{\rho_j\rho_i^{-1}}
\right)
=n(J-I).
```

\rho_\alphaUse in source · 1 passage

Search aliases: ρ_α rho alpha

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1728
Projective measurement. For projectors {P_α}, the probability of outcome α is p_α = Tr(P_αρ), and the conditioned state is ρ_α = fracP_αρ P_αp_α when p_α>0. Measurement is therefore packet selection followed by normalization.
Exact mathematical passage
```
\rho_\alpha
=
\frac{P_\alpha\rho P_\alpha}{p_\alpha}
```

\rho_\lambdaUse in source · 6 passages

Search aliases: ρ_λ rho lambda

Link to this entry ·

Use in source · Read passage · s_main.tex, line 1104
[Permutation-invariant state normal form] A density operator ρ commutes with every permutation if and only if it can be written as ρ = ⊕_λ p_λ ρ_λ⊗ fracI_ V_λf_λ, where p_λ≥0, ∑_λ p_λ=1, and each ρ_λ is a density operator on U_λ whenever p_λ>0. The weights p_λ and the unnormalized blocks p_λρ_λ are unique. The normalized block ρ_λ is unique only when p_λ>0 and is omitted when p_λ=0.
Exact mathematical passage
```
\rho
=
\bigoplus_\lambda
p_\lambda\,\rho_\lambda\otimes
\frac{I_{\mathcal V_\lambda}}{f_\lambda},
\label{s:eq:commutant-state-record}
```

\sigmaUse in source · 7 passages

Search aliases: σ sigma

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Use in source · Read passage · g_main.tex, line 2405
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\sigma=((t_1,s_1),\ldots,(t_L,s_L)),
\qquad t_i\ne s_i,
```

\sigma_a^{(q)}Use in source · 1 passage

Search aliases: σ_a^(q) sigma

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Use in source · Read passage · s_main.tex, line 936
Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1, J_-ketD_n^k = sqrtk(n-k+1)ketD_n^k-1. Thus co…
Exact mathematical passage
```
J_a=\frac12\sum_{q=1}^n\sigma_a^{(q)},
\qquad a\in\{x,y,z\}.
```

\sigma_zUse in source · 2 passages

Search aliases: σ_z sigma

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Use in source · Read passage · s_main.tex, line 940
Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1, J_-ketD_n^k = sqrtk(n-k+1)ketD_n^k-1. Thus co…
Exact mathematical passage
```
\sigma_z\ket0=\ket0
```

\tau_*Use in source · 3 passages

Search aliases: τ_* tau

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Use in source · Read passage · g_main.tex, line 2893
[Stable tensor length] The limit τ_*(A) = lim_r→∞ fracellAll(A^⊗ r)r m^r-1 = inf_r≥1 fracellAll(A^⊗ r)r m^r-1 exists. If A is tree tight, then the stable catalytic capacity κ_⊗(A) = ellDown(A)-τ_*(A) satisfies κ_⊗(A)≥DeltaDef(A).
Exact mathematical passage
```
\tau_*(A)
=
\lim_{r\to\infty}
\frac{\ellAll(A^{\otimes r})}{r m^{r-1}}
=
\inf_{r\ge1}
\frac{\ellAll(A^{\otimes r})}{r m^{r-1}}
```

\tau_{1,0}Use in source · 2 passages

Search aliases: τ_1,0 tau

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Use in source · Read passage · g_appendix.tex, line 30
1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).
Exact mathematical passage
```
\tau_{1,2}(1),
\tau_{3,1}(1),
\tau_{1,0}(1),
\tau_{4,1}(1),
\tau_{5,1}(1),
\tau_{1,2}(-1).
```

\tau_{1,2}Use in source · 1 passage

Search aliases: τ_1,2 tau

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Use in source · Read passage · g_appendix.tex, line 30
1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).
Exact mathematical passage
```
\tau_{1,2}(1),
\tau_{3,1}(1),
\tau_{1,0}(1),
\tau_{4,1}(1),
\tau_{5,1}(1),
\tau_{1,2}(-1).
```

\tau_{2,1}Use in source · 1 passage

Search aliases: τ_2,1 tau

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Use in source · Read passage · g_appendix.tex, line 39
1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).
Exact mathematical passage
```
\tau_{1,0}(1),
\tau_{2,1}(1),
\tau_{3,0}(-1),
\tau_{3,2}(1),
\tau_{4,2}(1),
\tau_{5,2}(1),
\tau_{2,1}(-1).
```

\tau_{3,0}Use in source · 1 passage

Search aliases: τ_3,0 tau

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Use in source · Read passage · g_appendix.tex, line 39
1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).
Exact mathematical passage
```
\tau_{1,0}(1),
\tau_{2,1}(1),
\tau_{3,0}(-1),
\tau_{3,2}(1),
\tau_{4,2}(1),
\tau_{5,2}(1),
\tau_{2,1}(-1).
```

\tau_{3,1}Use in source · 1 passage

Search aliases: τ_3,1 tau

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Use in source · Read passage · g_appendix.tex, line 30
1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).
Exact mathematical passage
```
\tau_{1,2}(1),
\tau_{3,1}(1),
\tau_{1,0}(1),
\tau_{4,1}(1),
\tau_{5,1}(1),
\tau_{1,2}(-1).
```

\tau_{3,2}Use in source · 1 passage

Search aliases: τ_3,2 tau

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 39
1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).
Exact mathematical passage
```
\tau_{1,0}(1),
\tau_{2,1}(1),
\tau_{3,0}(-1),
\tau_{3,2}(1),
\tau_{4,2}(1),
\tau_{5,2}(1),
\tau_{2,1}(-1).
```

\tau_{4,1}Use in source · 1 passage

Search aliases: τ_4,1 tau

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 30
1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).
Exact mathematical passage
```
\tau_{1,2}(1),
\tau_{3,1}(1),
\tau_{1,0}(1),
\tau_{4,1}(1),
\tau_{5,1}(1),
\tau_{1,2}(-1).
```

\tau_{4,2}Use in source · 1 passage

Search aliases: τ_4,2 tau

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 39
1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).
Exact mathematical passage
```
\tau_{1,0}(1),
\tau_{2,1}(1),
\tau_{3,0}(-1),
\tau_{3,2}(1),
\tau_{4,2}(1),
\tau_{5,2}(1),
\tau_{2,1}(-1).
```

\tau_{5,1}Use in source · 1 passage

Search aliases: τ_5,1 tau

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 30
1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).
Exact mathematical passage
```
\tau_{1,2}(1),
\tau_{3,1}(1),
\tau_{1,0}(1),
\tau_{4,1}(1),
\tau_{5,1}(1),
\tau_{1,2}(-1).
```

\tau_{5,2}Use in source · 1 passage

Search aliases: τ_5,2 tau

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 39
1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).
Exact mathematical passage
```
\tau_{1,0}(1),
\tau_{2,1}(1),
\tau_{3,0}(-1),
\tau_{3,2}(1),
\tau_{4,2}(1),
\tau_{5,2}(1),
\tau_{2,1}(-1).
```

\tau_{\mathrm{exist}}Use in source · 1 passage

Search aliases: τ_exist tau mathrm

Link to this entry ·

Use in source · Read passage · g_main.tex, line 148
Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.
Exact mathematical passage
```
\tau_{\mathrm{metric}}&=\min\{n:\exists U,\ 0<\Delta(U)<\infty\},\\
\tau_{\mathrm{exist}}&=\min\{n:\exists U,\ \ellRes(U)=\infty,\ \ellAmb(U)<\infty\},\\
\tau_{q}&=\min\{n:\exists U,\ \ell_{\le q}(U)<\ell_{\le0}(U)\},
```

\tau_{\mathrm{metric}}Use in source · 1 passage

Search aliases: τ_metric tau mathrm

Link to this entry ·

Use in source · Read passage · g_main.tex, line 148
Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.
Exact mathematical passage
```
\tau_{\mathrm{metric}}&=\min\{n:\exists U,\ 0<\Delta(U)<\infty\},\\
\tau_{\mathrm{exist}}&=\min\{n:\exists U,\ \ellRes(U)=\infty,\ \ellAmb(U)<\infty\},\\
\tau_{q}&=\min\{n:\exists U,\ \ell_{\le q}(U)<\ell_{\le0}(U)\},
```

\tau_{q}Use in source · 1 passage

Search aliases: τ_q tau

Link to this entry ·

Use in source · Read passage · g_main.tex, line 148
Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.
Exact mathematical passage
```
\tau_{\mathrm{metric}}&=\min\{n:\exists U,\ 0<\Delta(U)<\infty\},\\
\tau_{\mathrm{exist}}&=\min\{n:\exists U,\ \ellRes(U)=\infty,\ \ellAmb(U)<\infty\},\\
\tau_{q}&=\min\{n:\exists U,\ \ell_{\le q}(U)<\ell_{\le0}(U)\},
```

\tau_TUse in source · 1 passage

Search aliases: τ_T tau

Link to this entry ·

Use in source · Read passage · g_main.tex, line 4059
Tensor and repeated-module questions. For targets U and V, independent synthesis gives ℓ_T(U⊗ V) ≤ ℓ_T(U)+ℓ_T(V) when the systems are disjoint and local Clifford cost is ignored. Joint synthesis may be better because phase identities can couple terms across modules. This motivates a stable regularization τ_T(U)= liminf_r→∞ fracℓ_T(U^⊗ r)r. A one-copy rigid phase target could still display tensor activation.
Exact mathematical passage
```
\tau_T(U)=
\liminf_{r\to\infty}
\frac{\ell_T(U^{\otimes r})}{r}.
```

\tau_{t,s}Use in source · 8 passages

Search aliases: τ_t,s tau

Link to this entry ·

Use in source · Read passage · g_appendix.tex, line 8
Row-transvection convention. Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection τ_t,s(a)=I+aE_t,s performs R_t← R_t+aR_s. In the binary CNOT interpretation, s is the control and t is the target.
Exact mathematical passage
```
\tau_{t,s}(a)=I+aE_{t,s}
```

\tau_{t_1,s_1}Use in source · 2 passages

Search aliases: τ_t_1,s_1 tau

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1170
Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_1(a_1).
Exact mathematical passage
```
w=(\tau_{t_m,s_m}(a_m),\ldots,\tau_{t_1,s_1}(a_1))
```

\tau_{t_m,s_m}Use in source · 2 passages

Search aliases: τ_t_m,s_m tau

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1170
Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_1(a_1).
Exact mathematical passage
```
w=(\tau_{t_m,s_m}(a_m),\ldots,\tau_{t_1,s_1}(a_1))
```

\tau_zUse in source · 4 passages

Search aliases: τ_z tau

Link to this entry ·

Use in source · Read passage · c_visual.tex, line 691
guidepageOverlap breaks the attempted context-free product rule One shared witness bit already defeats multiplicativity. With τ_z(P)=P(0)+zP(1), τ_z(w)=z, τ_z(1+w)=1, τ_z(w(1+w))=0. The product of the separate summaries is z. Also w^2=w in the Boolean ring, so τ_z(w^2)=z, whereas τ_z(w)^2=z^2.
Exact mathematical passage
```
\tau_z(P)=P(0)+zP(1)
```

\tau_{z_1}Use in source · 1 passage

Search aliases: τ_z_1 tau

Link to this entry ·

Use in source · Read passage · c_visual.tex, line 668
For A=w_1 and B=w_2, τ_z_1(A)=z_1, τ_z_2(B)=z_2, τ_z_1,z_2(AB)=z_1z_2. Tracing both factors over the full two-bit ambient domain introduces unused variable sums: τ_z_1,z_2(A)=z_1(1+z_2), τ_z_1,z_2(B)=z_2(1+z_1). Their product generally differs from the union trace. At z_1=z_2=1, each full ambient factor is zero, while the trace of AB is one. Disjoint dependence alone does not justify multiplying traces with mismatched domains.
Exact mathematical passage
```
 \tau_{z_1}(A)=z_1,\qquad \tau_{z_2}(B)=z_2,\qquad
 \tau_{z_1,z_2}(AB)=z_1z_2.
```

\tau_{z_1,z_2}Use in source · 2 passages

Search aliases: τ_z_1,z_2 tau

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Use in source · Read passage · c_visual.tex, line 668
For A=w_1 and B=w_2, τ_z_1(A)=z_1, τ_z_2(B)=z_2, τ_z_1,z_2(AB)=z_1z_2. Tracing both factors over the full two-bit ambient domain introduces unused variable sums: τ_z_1,z_2(A)=z_1(1+z_2), τ_z_1,z_2(B)=z_2(1+z_1). Their product generally differs from the union trace. At z_1=z_2=1, each full ambient factor is zero, while the trace of AB is one. Disjoint dependence alone does not justify multiplying traces with mismatched domains.
Exact mathematical passage
```
 \tau_{z_1}(A)=z_1,\qquad \tau_{z_2}(B)=z_2,\qquad
 \tau_{z_1,z_2}(AB)=z_1z_2.
```

\tau_{z_2}Use in source · 1 passage

Search aliases: τ_z_2 tau

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Use in source · Read passage · c_visual.tex, line 668
For A=w_1 and B=w_2, τ_z_1(A)=z_1, τ_z_2(B)=z_2, τ_z_1,z_2(AB)=z_1z_2. Tracing both factors over the full two-bit ambient domain introduces unused variable sums: τ_z_1,z_2(A)=z_1(1+z_2), τ_z_1,z_2(B)=z_2(1+z_1). Their product generally differs from the union trace. At z_1=z_2=1, each full ambient factor is zero, while the trace of AB is one. Disjoint dependence alone does not justify multiplying traces with mismatched domains.
Exact mathematical passage
```
 \tau_{z_1}(A)=z_1,\qquad \tau_{z_2}(B)=z_2,\qquad
 \tau_{z_1,z_2}(AB)=z_1z_2.
```

\phiUse in source · 20 passages

Search aliases: ϕ phi

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Use in source · Read passage · g_main.tex, line 4367
Let U act on the data register. A clean-ancilla realization is a unitary W satisfying W(ketψ⊗ket0) =(Uketψ)⊗ket0 for every ketψ. A dirty-ancilla realization requires W(ketψ⊗ketϕ) =(Uketψ)⊗ketϕ for every ketψ,ketϕ. The second condition is strictly stronger.
Exact mathematical passage
```
W(\ket\psi\otimes\ket\phi)
=(U\ket\psi)\otimes\ket\phi
\quad\text{for every }\ket\psi,\ket\phi.
```

\phi_0Use in source · 3 passages

Search aliases: ϕ_0 phi

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Use in source · Read passage · s_main.tex, line 660
[Adaptive discarded-state certificate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ_m⟩-⟨ϕ_m,Mϕ_m⟩| ≤2∑_t=1^mη_t. No renormalization of ϕ_t is performed. The second quadratic form is an unnormalized a…
Exact mathematical passage
```
\psi_0=\phi_0
```

\phi_2Use in source · 1 passage

Search aliases: ϕ_2 phi

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Use in source · Read passage · s_main.tex, line 700
0. Using G_1=G_2=G and P_1=P_2=P gives ϕ_2=9/25 0, ψ_2=-7/25 24/25, η_1=4/5, η_2=12/25. The squared state error is 832/625, below (η_1+η_2)^2=1024/625. The rational validation script checks these values exactly. The deliberately large tail illustrates a valid but loose certificate, not an efficient approximation.
Exact mathematical passage
```
\phi_2=\begin{pmatrix}9/25\\0\end{pmatrix},\qquad
\psi_2=\begin{pmatrix}-7/25\\24/25\end{pmatrix},\qquad
\eta_1=4/5,\quad\eta_2=12/25.
```

\Phi_mUse in source · 1 passage

Search aliases: Φ_m Phi

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Use in source · Read passage · g_main.tex, line 4383
Measurement and feedforward. A measurement-assisted realization is described by an instrument with classical outcomes m, conditional channels Φ_m, and feedforward operations. Exact deterministic implementation of a unitary target requires every branch, after correction, to induce the same channel.
Exact mathematical passage
```
\Phi_m
```

\phi_mUse in source · 4 passages

Search aliases: ϕ_m phi

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Use in source · Read passage · s_main.tex, line 667
ate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ_m⟩-⟨ϕ_m,Mϕ_m⟩| ≤2∑_t=1^mη_t. No renormalization of ϕ_t is performed. The second quadratic form is an unnormalized acceptance estimate.
Exact mathematical passage
```
\|\psi_m-\phi_m\|\le\sum_{t=1}^m\eta_t.
```

\phi_tUse in source · 3 passages

Search aliases: ϕ_t phi

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Use in source · Read passage · s_main.tex, line 661
[Adaptive discarded-state certificate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ_m⟩-⟨ϕ_m,Mϕ_m⟩| ≤2∑_t=1^mη_t. No renormalization of ϕ_t is performed. The second quadratic form is an unnormalized a…
Exact mathematical passage
```
\psi_t=G_t\psi_{t-1},\qquad
\phi_t=P_tG_t\phi_{t-1},\qquad
\eta_t=\|(I-P_t)G_t\phi_{t-1}\|.
```

\phi_{t-1}Use in source · 3 passages

Search aliases: ϕ_t-1 phi

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Use in source · Read passage · s_main.tex, line 661
[Adaptive discarded-state certificate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ_m⟩-⟨ϕ_m,Mϕ_m⟩| ≤2∑_t=1^mη_t. No renormalization of ϕ_t is performed. The second quadratic form is an unnormalized a…
Exact mathematical passage
```
\psi_t=G_t\psi_{t-1},\qquad
\phi_t=P_tG_t\phi_{t-1},\qquad
\eta_t=\|(I-P_t)G_t\phi_{t-1}\|.
```

\Phi_{\gamma\delta,\alpha\beta}Use in source · 1 passage

Search aliases: Φ_γδ,αβ Phi gamma delta alpha beta

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Use in source · Read passage · s_main.tex, line 1666
Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.
Exact mathematical passage
```
\Phi_{\gamma\delta,\alpha\beta}
=
\mathcal P_{\gamma\delta}
\,\Phi\,
\mathcal P_{\alpha\beta}.
```

\Phi_\sigmaUse in source · 3 passages

Search aliases: Φ_σ Phi sigma

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Use in source · Read passage · g_main.tex, line 2410
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\Phi_\sigma:\Aff^L\longrightarrow\SL_n,
\qquad
(a_1,\ldots,a_L)
\longmapsto
(I+a_LE_{t_L,s_L})\cdots(I+a_1E_{t_1,s_1}).
```

\chiUse in source · 59 passages

Search aliases: χ chi

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Use in source · Read passage · g_main.tex, line 3987
gTT-depth and commuting layers Two phase gadgets may share a T layer when a suitable Clifford network maps their parities to distinct wires or when ancilla resources permit parallel extraction. Define the conflict graph Γ_P whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the chosen ancilla and connectivity rules. Then T-depth(P)≥χ(Γ_P) for any conflict graph whose edge relation is sound. Catalysis can alter Γ_P by changing the Pauli fra…
Exact mathematical passage
```
T\text{-depth}(P)\ge\chi(\Gamma_P)
```

\chi_0Use in source · 2 passages

Search aliases: χ_0 chi

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Use in source · Read passage · q_main.tex, line 1321
It is a normalized state with the same original-basis probabilities for every θ. The endpoints are χ_0=χ and χ_π=ψ.
Exact mathematical passage
```
\chi_0=\chi
```

\chi_nUse in source · 2 passages

Search aliases: χ_n chi

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Use in source · Read passage · q_visual.tex, line 165
Identical histograms can hide orthogonal states. addcontentslinetocsectionIdentical histograms can hide orthogonal states leadErasing the signs leaves sample probabilities unchanged and changes the state. Let χ_n=|c_12(n)|/sqrt48. Both states are normalized and |ψ_n|^2=|χ_n|^2 for every n, ⟨ψ,χ⟩=0. fig13_same_histogram_distinct_states1.0The top panel shows the phase information carried by negative amplitudes. The bottom panel is shared by the two states because measurement in this basis depends …
Exact mathematical passage
```
\chi_n=|c_{12}(n)|/\sqrt{48}
```

\chi_{\rm internal}Use in source · 5 passages

Search aliases: χ_rm internal chi rm

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Use in source · Read passage · s_main.tex, line 890
Entanglement inside a packet. A small number of symmetry labels does not imply low entanglement. The half-filled packet can support volume-law entanglement. Tensor-network bond dimension and packet dimension are independent resources. A robust simulator should therefore track both (D_rm packet,χ_rm internal), where χ_rm internal measures the cost of representing the state inside the packet.
Exact mathematical passage
```
(D_{\rm packet},\chi_{\rm internal}),
```

\chi_\thetaUse in source · 11 passages

Search aliases: χ_θ chi theta

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Use in source · Read passage · q_main.tex, line 1316
χ_θ=v_++e^iθv_-.
Exact mathematical passage
```
\chi_\theta=v_++e^{i\theta}v_-.
```

\chi_\piUse in source · 2 passages

Search aliases: χ_π chi pi

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Use in source · Read passage · q_main.tex, line 1322
It is a normalized state with the same original-basis probabilities for every θ. The endpoints are χ_0=χ and χ_π=ψ.
Exact mathematical passage
```
\chi_\pi=\psi
```

\PsiUse in source · 5 passages

Search aliases: Ψ Psi

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Use in source · Read passage · g_main.tex, line 845
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
\Psi\subseteq\Phi
```

\psiUse in source · 86 passages

Search aliases: ψ psi

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Use in source · Read passage · g_main.tex, line 4361
Let U act on the data register. A clean-ancilla realization is a unitary W satisfying W(ketψ⊗ket0) =(Uketψ)⊗ket0 for every ketψ. A dirty-ancilla realization requires W(ketψ⊗ketϕ) =(Uketψ)⊗ketϕ for every ketψ,ketϕ. The second condition is strictly stronger.
Exact mathematical passage
```
W(\ket\psi\otimes\ket0)
=(U\ket\psi)\otimes\ket0
\quad\text{for every }\ket\psi.
```

\psi_0Use in source · 11 passages

Search aliases: ψ_0 psi

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Use in source · Read passage · s_main.tex, line 247
Gatewise invariance. Let P be an orthogonal projector and suppose the initial state obeys Pketψ_0=ketψ_0. If every gate satisfies G_jP=PG_jP, then G_j maps PHh into itself. For a general linear map, this equation asserts only invariance of the selected subspace. For the finite-dimensional unitary gates considered here, invariance also implies invariance of the orthogonal complement, so it is equivalent to commutation with P.
Exact mathematical passage
```
P\ket{\psi_0}=\ket{\psi_0}
```

\psi_2Use in source · 1 passage

Search aliases: ψ_2 psi

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Use in source · Read passage · s_main.tex, line 700
0. Using G_1=G_2=G and P_1=P_2=P gives ϕ_2=9/25 0, ψ_2=-7/25 24/25, η_1=4/5, η_2=12/25. The squared state error is 832/625, below (η_1+η_2)^2=1024/625. The rational validation script checks these values exactly. The deliberately large tail illustrates a valid but loose certificate, not an efficient approximation.
Exact mathematical passage
```
\phi_2=\begin{pmatrix}9/25\\0\end{pmatrix},\qquad
\psi_2=\begin{pmatrix}-7/25\\24/25\end{pmatrix},\qquad
\eta_1=4/5,\quad\eta_2=12/25.
```

\psi_{j-1}Use in source · 2 passages

Search aliases: ψ_j-1 psi

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Use in source · Read passage · s_main.tex, line 266
Inductively, if Pketψ_j-1=ketψ_j-1, then G_jketψ_j-1 = G_jPketψ_j-1 = PG_jPketψ_j-1, so the next state remains in PHh. Iterating proves the identity.
Exact mathematical passage
```
P\ket{\psi_{j-1}}=\ket{\psi_{j-1}}
```

\psi_kUse in source · 2 passages

Search aliases: ψ_k psi

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Use in source · Read passage · s_appendix.tex, line 5
Residue projector formula. Let Qketψ_k=kketψ_k. Then frac1M∑_ℓ=0^M-1 ω_M^-rℓe^2π iℓ Q/Mketψ_k = ( frac1M∑_ℓ=0^M-1 ω_M^(k-r)ℓ )ketψ_k. The bracket equals one when k≡ rpmod M and zero otherwise. This proves [source reference] and the orthogonality of the residue packets.
Exact mathematical passage
```
Q\ket{\psi_k}=k\ket{\psi_k}
```

\psi_mUse in source · 4 passages

Search aliases: ψ_m psi

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Use in source · Read passage · s_main.tex, line 667
ate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ_m⟩-⟨ϕ_m,Mϕ_m⟩| ≤2∑_t=1^mη_t. No renormalization of ϕ_t is performed. The second quadratic form is an unnormalized acceptance estimate.
Exact mathematical passage
```
\|\psi_m-\phi_m\|\le\sum_{t=1}^m\eta_t.
```

\psi_nUse in source · 2 passages

Search aliases: ψ_n psi

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Use in source · Read passage · q_visual.tex, line 151
The arithmetic signal is a coherent amplitude vector. addcontentslinetocsectionThe arithmetic signal is a coherent amplitude vector leadNormalization makes the signal a unit vector without changing its relative phases. Define u_k=1/2 for k∈ U_12 and zero otherwise. Then u =1, and ψ=Fu, ψ_n=fracc_12(n)sqrt48, F^†ψ=u. fig11_same_state_two_bases1.0Separate panels preserve the different basis labels: k for the frequency-basis coordinates and n for the sample-basis coordinates. Heights are amplitudes…
Exact mathematical passage
```
 \psi=Fu,\qquad\psi_n=\frac{c_{12}(n)}{\sqrt{48}},\qquad F^\dagger\psi=u.
```

\psi_tUse in source · 3 passages

Search aliases: ψ_t psi

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Use in source · Read passage · s_main.tex, line 593
The active dimension is ∑_k=0^Kbinom nk =O(n^K). Let P project onto Hh_≤ K. By the hypothesis on the actual circuit prefixes, Pψ_t=ψ_t at every step. Consequently PG_tPψ_t-1=G_tψ_t-1 on the evolving state, even if the gate does not preserve all of PHh. The restricted update therefore reproduces that trajectory exactly. The hypothesis supplies efficient initialization, updates, and measurement in these polynomially many coordinates.
Exact mathematical passage
```
P\psi_t=\psi_t
```

\psi_{t-1}Use in source · 3 passages

Search aliases: ψ_t-1 psi

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Use in source · Read passage · s_main.tex, line 593
The active dimension is ∑_k=0^Kbinom nk =O(n^K). Let P project onto Hh_≤ K. By the hypothesis on the actual circuit prefixes, Pψ_t=ψ_t at every step. Consequently PG_tPψ_t-1=G_tψ_t-1 on the evolving state, even if the gate does not preserve all of PHh. The restricted update therefore reproduces that trajectory exactly. The hypothesis supplies efficient initialization, updates, and measurement in these polynomially many coordinates.
Exact mathematical passage
```
PG_tP\psi_{t-1}=G_t\psi_{t-1}
```

\OmegaUse in source · 15 passages

Search aliases: Ω Omega

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Use in source · Read passage · g_main.tex, line 367
Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.
Exact mathematical passage
```
D_n^{(1)}=\Omega(n)
```

\omegaUse in source · 9 passages

Search aliases: ω omega

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Use in source · Read passage · g_main.tex, line 3901
Let ω=e^iπ/4. A phase-polynomial circuit on n qubits has the form [source reference] Uketx=ω^P(x)ketAx+b, where A∈GL(n,2), b∈F_2^n, and P:F_2^n→Z_8 is represented as a sum of binary linear forms with coefficients in Z_8. The diagonal case has A=I and b=0.
Exact mathematical passage
```
\omega=e^{i\pi/4}
```

\omega^{-rj}Use in source · 1 passage

Search aliases: ω^-rj omega

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Use in source · Read passage · q_main.tex, line 550
R_rf(z) · =frac1L∑_j=0^L-1ω^-rjf(ω^jz), R_rR_s · =δ_rsR_r, ∑_r=0^L-1R_rf · =f.
Exact mathematical passage
```
\begin{aligned}
R_rf(z)&=\frac1L\sum_{j=0}^{L-1}\omega^{-rj}f(\omega^jz),\\
R_rR_s&=\delta_{rs}R_r,\\
\sum_{r=0}^{L-1}R_rf&=f.
\end{aligned}
```

\omega^jUse in source · 1 passage

Search aliases: ω^j omega

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Use in source · Read passage · q_main.tex, line 550
R_rf(z) · =frac1L∑_j=0^L-1ω^-rjf(ω^jz), R_rR_s · =δ_rsR_r, ∑_r=0^L-1R_rf · =f.
Exact mathematical passage
```
\begin{aligned}
R_rf(z)&=\frac1L\sum_{j=0}^{L-1}\omega^{-rj}f(\omega^jz),\\
R_rR_s&=\delta_{rs}R_r,\\
\sum_{r=0}^{L-1}R_rf&=f.
\end{aligned}
```

\omega^{j(n-r)}Use in source · 1 passage

Search aliases: ω^j(n-r) omega

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Use in source · Read passage · q_main.tex, line 561
fraca_nL∑_j=0^L-1ω^j(n-r).
Exact mathematical passage
```
\frac{a_n}{L}\sum_{j=0}^{L-1}\omega^{j(n-r)}.
```

\omega^{k\lambda(x)}Use in source · 1 passage

Search aliases: ω^kλ(x) omega lambda

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Use in source · Read passage · g_main.tex, line 3945
Phase polynomials and parity carriers. A CNOT network computes linear parities λ(x)=a· x∈F_2. Applying T^k to a wire carrying λ(x) contributes ω^kλ(x) to the phase polynomial. A parity network must create each required parity, expose it to a phase gate, and restore the linear map required at the endpoint.
Exact mathematical passage
```
\omega^{k\lambda(x)}
```

\omega^{P(x)}Use in source · 1 passage

Search aliases: ω^P(x) omega

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Use in source · Read passage · g_main.tex, line 3902
Let ω=e^iπ/4. A phase-polynomial circuit on n qubits has the form [source reference] Uketx=ω^P(x)ketAx+b, where A∈GL(n,2), b∈F_2^n, and P:F_2^n→Z_8 is represented as a sum of binary linear forms with coefficients in Z_8. The diagonal case has A=I and b=0.
Exact mathematical passage
```
\label{g:eq:phase-poly-normal}
U\ket{x}=\omega^{P(x)}\ket{Ax+b},
```

\omega_MUse in source · 1 passage

Search aliases: ω_M omega

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Use in source · Read passage · s_main.tex, line 141
Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis and charge sectors in quantum theory.
Exact mathematical passage
```
V_M=\exp\left(\frac{2\pi i}{M}Q\right),
\qquad
\omega_M=e^{2\pi i/M}.
```

\omega_M^{(k-r)\ell}Use in source · 1 passage

Search aliases: ω_M^(k-r)ℓ omega ell

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Use in source · Read passage · s_appendix.tex, line 6
Residue projector formula. Let Qketψ_k=kketψ_k. Then frac1M∑_ℓ=0^M-1 ω_M^-rℓe^2π iℓ Q/Mketψ_k = ( frac1M∑_ℓ=0^M-1 ω_M^(k-r)ℓ )ketψ_k. The bracket equals one when k≡ rpmod M and zero otherwise. This proves [source reference] and the orthogonality of the residue packets.
Exact mathematical passage
```
\frac1M\sum_{\ell=0}^{M-1}
\omega_M^{-r\ell}e^{2\pi i\ell Q/M}\ket{\psi_k}
=
\left(
\frac1M\sum_{\ell=0}^{M-1}
\omega_M^{(k-r)\ell}
\right)\ket{\psi_k}.
```

\omega_M^{-r\ell}Use in source · 2 passages

Search aliases: ω_M^-rℓ omega ell

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Use in source · Read passage · s_appendix.tex, line 6
Residue projector formula. Let Qketψ_k=kketψ_k. Then frac1M∑_ℓ=0^M-1 ω_M^-rℓe^2π iℓ Q/Mketψ_k = ( frac1M∑_ℓ=0^M-1 ω_M^(k-r)ℓ )ketψ_k. The bracket equals one when k≡ rpmod M and zero otherwise. This proves [source reference] and the orthogonality of the residue packets.
Exact mathematical passage
```
\frac1M\sum_{\ell=0}^{M-1}
\omega_M^{-r\ell}e^{2\pi i\ell Q/M}\ket{\psi_k}
=
\left(
\frac1M\sum_{\ell=0}^{M-1}
\omega_M^{(k-r)\ell}
\right)\ket{\psi_k}.
```

\Omega_nUse in source · 9 passages

Search aliases: Ω_n Omega

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Use in source · Read passage · g_main.tex, line 1066
[Generic rigidity and defect codimension] Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline K overline D_n,≥ d(overline K)≥ d.
Exact mathematical passage
```
\Omega_n\subseteq\U_n
```

\ell^2Use in source · 1 passage

Search aliases: ℓ^2 ell

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Use in source · Read passage · g_main.tex, line 1847
Let G be a finite group and S=S^-1 a symmetric generating set. On ℓ^2(G), let λ be the left regular representation, λ(s)δ_h=δ_sh, and define A_S=∑_s∈ Sλ(s). For g∈ G, define the endpoint Green series G_S,g(z) = ⟨ δ_g, (I-zA_S)^-1δ_e ⟩.
Exact mathematical passage
```
\ell^2(G)
```

\ell_{\mathfrak M}^{\lambda}Use in source · 1 passage

Search aliases: ℓ_ M^λ ell mathfrak lambda

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Use in source · Read passage · g_main.tex, line 97
For a weight vector λ∈R_≥0^r, scalarization gives ℓ_ M^λ(U) = inf_x∈ P_ M(U)λ· x. The scalar defect Δ^λ(U) can vary with λ even when the target and gate alphabets are fixed.
Exact mathematical passage
```
\ell_{\mathfrak M}^{\lambda}(U)
=
\inf_{x\in\mathcal P_{\mathfrak M}(U)}\lambda\cdot x.
```

\ell_GUse in source · 5 passages

Search aliases: ℓ_G ell

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Use in source · Read passage · g_main.tex, line 230
Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry is the left Schreier graph G/H, with edges gH⟼ sgH. For the base-fiber assertion below, assume also S_H= S_G∩ H. This last condition excludes addition…
Exact mathematical passage
```
\ell_H(h)=|h|_{\mathcal S_H},
\qquad
\ell_G(h)=|h|_{\mathcal S_G}.
```

\ell_HUse in source · 9 passages

Search aliases: ℓ_H ell

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Use in source · Read passage · g_main.tex, line 230
Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry is the left Schreier graph G/H, with edges gH⟼ sgH. For the base-fiber assertion below, assume also S_H= S_G∩ H. This last condition excludes addition…
Exact mathematical passage
```
\ell_H(h)=|h|_{\mathcal S_H},
\qquad
\ell_G(h)=|h|_{\mathcal S_G}.
```

\ell_{\mathrm{large,res}}Use in source · 1 passage

Search aliases: ℓ_large,res ell mathrm

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Use in source · Read passage · g_main.tex, line 5159
Template VIII: block retraction. Let a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). This transfers a catalytic atom to architectures containing it as a retract.
Exact mathematical passage
```
\ell_{\mathrm{large,res}}(\iota U)
\ge
\ell_{\mathrm{small,res}}(U).
```

\ell_{\mathrm{res},\varepsilon}Use in source · 1 passage

Search aliases: ℓ_res,ε ell mathrm varepsilon

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Use in source · Read passage · g_main.tex, line 4457
Approximate synthesis. For continuous gate sets, define ℓ_res,ε(U) = min{cost(W): W-U ≤ε}. There are two distinct border phenomena: item algebraic border length, where a target lies in the closure of a bounded exact word locus; item analytic approximation length, where an error tolerance is explicitly permitted. They should not be conflated.
Exact mathematical passage
```
\ell_{\mathrm{res},\varepsilon}(U)
=
\min\{\cost(W):\|W-U\|\le\varepsilon\}.
```

\ell_{\mathrm{small,res}}Use in source · 1 passage

Search aliases: ℓ_small,res ell mathrm

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Use in source · Read passage · g_main.tex, line 5159
Template VIII: block retraction. Let a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). This transfers a catalytic atom to architectures containing it as a retract.
Exact mathematical passage
```
\ell_{\mathrm{large,res}}(\iota U)
\ge
\ell_{\mathrm{small,res}}(U).
```

\ell_\GammaUse in source · 1 passage

Search aliases: ℓ_Γ ell Gamma

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Use in source · Read passage · g_main.tex, line 4418
For a target U, define ℓ_Γ(U) = minimum hardware-respecting cost, and compare it with an enlarged graph Γ'. A matched routing excursion has the form π W π^-1, where π is a logical permutation. The excursion is profitable when many gates in W become local in the permuted frame.
Exact mathematical passage
```
\ell_\Gamma(U)
=
\text{minimum hardware-respecting cost},
```

\ell_\PhiUse in source · 1 passage

Search aliases: ℓ_Φ ell Phi

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Use in source · Read passage · g_main.tex, line 851
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
\ell_\Phi
```

\ell_\PsiUse in source · 1 passage

Search aliases: ℓ_Ψ ell Psi

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Use in source · Read passage · g_main.tex, line 846
Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.
Exact mathematical passage
```
\ell_\Psi(u)
=
\min\{L:u=x_{\beta_L}(t_L)\cdots x_{\beta_1}(t_1),\ \beta_i\in\Psi\},
```

\ell_{\le q}Use in source · 2 passages

Search aliases: ℓ_≤ q ell le

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Use in source · Read passage · g_main.tex, line 148
Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.
Exact mathematical passage
```
\tau_{\mathrm{metric}}&=\min\{n:\exists U,\ 0<\Delta(U)<\infty\},\\
\tau_{\mathrm{exist}}&=\min\{n:\exists U,\ \ellRes(U)=\infty,\ \ellAmb(U)<\infty\},\\
\tau_{q}&=\min\{n:\exists U,\ \ell_{\le q}(U)<\ell_{\le0}(U)\},
```

\ell_{\le0}Use in source · 1 passage

Search aliases: ℓ_≤0 ell le

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Use in source · Read passage · g_main.tex, line 148
Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.
Exact mathematical passage
```
\tau_{\mathrm{metric}}&=\min\{n:\exists U,\ 0<\Delta(U)<\infty\},\\
\tau_{\mathrm{exist}}&=\min\{n:\exists U,\ \ellRes(U)=\infty,\ \ellAmb(U)<\infty\},\\
\tau_{q}&=\min\{n:\exists U,\ \ell_{\le q}(U)<\ell_{\le0}(U)\},
```

\leftarrowUse in source · 15 passages

Search aliases: ← leftarrow

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Use in source · Read passage · c_visual.tex, line 11
gical input wires x, zero-initialized ancillary wires, and a target bit y. A restored oracle is a legal classical circuit satisfying C(x,0^a,y)=(x,0^a,y+f(x)) for every x and both y. The input and scratch restoration conditions hold at the boundary. Wires may change internally, and the target may control an intermediate gate. Each primitive gate is an involution: y← y+1, y← y+x_i, y← y+x_i x_j. In a CNOT the control differs from the target. In a Toffoli all three wires are distinct. These are co…
Exact mathematical passage
```
 y\leftarrow y+1,\quad y\leftarrow y+x_i,\quad
 y\leftarrow y+x_i x_j.
```

\leftrightarrowUse in source · 1 passage

Search aliases: ↔ leftrightarrow

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Use in source · Read passage · g_appendix.tex, line 79
Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.
Exact mathematical passage
```
H_i:e_i\leftrightarrow e_{n+i}.
```

\hookrightarrowUse in source · 1 passage

Search aliases: ↪ hookrightarrow

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Use in source · Read passage · g_main.tex, line 253
Thus Δ_H↪ G(h) = intrinsic fiber cost - shortest holonomy loop cost. This is the geometric content of a dirty-workspace shortcut.
Exact mathematical passage
```
\Delta_{H\hookrightarrow G}(h)
=
\text{intrinsic fiber cost}
-
\text{shortest holonomy loop cost}.
```

\RightarrowUse in source · 1 passage

Search aliases: ⇒ Rightarrow

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Use in source · Read passage · c_lean.tex, line 67
Safe detector · A Boolean h with h(x)=1⇒ f(x)=1.
Exact mathematical passage
```
h(x)=1\Rightarrow f(x)=1
```

\forallUse in source · 2 passages

Search aliases: ∀ forall

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Use in source · Read passage · c_lean.tex, line 87
pageheadWhy sharing does not break the degree boundA STRONGER INVARIANT THAN SINGLE-REGISTER DEGREE For a current state s(x) after t Toffolis, the invariant is ∀ L, Deg (t+|L|, x↦∏_i∈ Ls_i(x)). L may be any finite list, including repeated registers. Initially each coordinate is a variable or a constant, so a product of r coordinates has degree at most r.
Exact mathematical passage
```
\forall L,\qquad
\Deg\!\left(t+|L|,\;x\mapsto\prod_{i\in L}s_i(x)\right).
```

\existsUse in source · 5 passages

Search aliases: ∃ exists

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Use in source · Read passage · c_lean.tex, line 125
pageheadFrom the cut to a numerical lower boundEXISTENCE, BALANCE, AND COMPLETE COVERAGE Write c=pmathbin++q. At this cut, Block 01 defines H_i(x)=run(p)(x,0^a,0)_ix run(q^-1)(x,0^a,0)_i and proves H_i≤ f, f(x)=1iff∃ i H_i(x)=1. Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives deg(H_i)≤max{TC(p),TC(q)}+1.
Exact mathematical passage
```
H_i\le f,\qquad f(x)=1\iff\exists i\;H_i(x)=1.
```

\varnothingUse in source · 5 passages

Search aliases: ∅ varnothing

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Use in source · Read passage · c_visual.tex, line 359
guidepageThe finite evaluation-span dual in full Fix a positive v and integer d≥0. Let v_d(x)=(x^S)_S⊆[n], |S|≤ d, U=f^-1(0), N=Span_F_2{v_d(u):u∈ U}. The constant monomial S=∅ is included. Degree-d coefficient covectors form the finite dual of this evaluation space. A covector is safe exactly when it annihilates N.
Exact mathematical passage
```
S=\varnothing
```

\notinUse in source · 2 passages

Search aliases: ∉ notin

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Use in source · Read passage · c_visual.tex, line 363
If v_d(v)∈ N, every safe covector vanishes at v. If v_d(v)∉ N, extend a basis of N by v_d(v) and then to a basis of the full space. Assign zero to the N basis and one to v_d(v). The resulting covector is a safe detector. Hence β_v(f)>d Longleftrightarrow v_d(v)∈ N. Choose a basis of N from negative evaluation rows. A span representation of the positive row gives a subset W⊆ U with v_d(v)=∑_u∈ Wv_d(u), |W| odd, |W|≤∑_j=0^min(d,n)binom nj. It is a replayable finite obstruction to every safe degree…
Exact mathematical passage
```
v_d(v)\notin N
```

\prod_{1\le i<j\le d}Use in source · 1 passage

Search aliases: ∏_1≤ i<j≤ d prod le le

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Use in source · Read passage · s_main.tex, line 1157
Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).
Exact mathematical passage
```
u_\lambda
=
\prod_{1\le i<j\le d}
\frac{\lambda_i-\lambda_j+j-i}{j-i}
=O_d\left(n^{d(d-1)/2}\right).
\label{s:eq:weyl-dimension-bound}
```

\prod_iUse in source · 1 passage

Search aliases: ∏_i prod

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Use in source · Read passage · c_visual.tex, line 591
guidepageDistinct witness monomials prevent formal cancellation Over a characteristic-two field, define P_R(z)=∑_w∈{0,1}^mR(w)∏_i z_i^w_i. The z_i are formal indeterminates, not Boolean variables. Different witnesses produce different monomials, so P_R=0 exactly when there is no accepting witness.
Exact mathematical passage
```
 P_R(z)=\sum_{w\in\{0,1\}^m}R(w)\prod_i z_i^{w_i}.
```

\prod_{i:v_i=0}Use in source · 1 passage

Search aliases: ∏_i:v_i=0 prod

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Use in source · Read passage · c_main.tex, line 162
[Basic size bounds] For nonzero f on n inputs, α_1(f)≤β(f)≤ n. If f has at least two positive inputs, then β(f)≤ n-1. In particular β^*(f)≤ n. Any polynomial detecting a specified positive input is nonzero, giving the first inequality. For v∈B^n, its point indicator is δ_v(x)=∏_i:v_i=1x_i∏_i:v_i=0(1+x_i). If f(v)=1, then δ_v∈ I_f and has degree n, proving the upper bound.
Exact mathematical passage
```
 \delta_v(x)=\prod_{i:v_i=1}x_i\prod_{i:v_i=0}(1+x_i).
```

\prod_{i:v_i=1}Use in source · 1 passage

Search aliases: ∏_i:v_i=1 prod

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Use in source · Read passage · c_main.tex, line 162
[Basic size bounds] For nonzero f on n inputs, α_1(f)≤β(f)≤ n. If f has at least two positive inputs, then β(f)≤ n-1. In particular β^*(f)≤ n. Any polynomial detecting a specified positive input is nonzero, giving the first inequality. For v∈B^n, its point indicator is δ_v(x)=∏_i:v_i=1x_i∏_i:v_i=0(1+x_i). If f(v)=1, then δ_v∈ I_f and has degree n, proving the upper bound.
Exact mathematical passage
```
 \delta_v(x)=\prod_{i:v_i=1}x_i\prod_{i:v_i=0}(1+x_i).
```

\prod_{i=1}^mUse in source · 3 passages

Search aliases: ∏_i=1^m prod

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Use in source · Read passage · c_main.tex, line 554
Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.
Exact mathematical passage
```
 P_R(z_1,\ldots,z_m)=
 \sum_{w\in\B^m}R(w)\prod_{i=1}^m z_i^{w_i}\in K[z_1,\ldots,z_m].
```

\prod_{i=1}^{n}Use in source · 1 passage

Search aliases: ∏_i=1^n prod

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Use in source · Read passage · c_lean.tex, line 135
The least safe degree exists. For a positive v, take the point indicator δ_v(x)=∏_i=1^n x_i, · v_i=1, 1+x_i, · v_i=0. It is safe, detects v, and has a degree-n certificate. Hence complete safe coverage exists at degree n for every function. Strong induction proves the existence of a least admissible natural degree. The definition pathrefbeta chooses this bound and its specification and minimality are proved.
Exact mathematical passage
```
\delta_v(x)=\prod_{i=1}^{n}
\begin{cases}
x_i,&v_i=1,\\
1+x_i,&v_i=0.
\end{cases}
```

\prod_{i\in L}Use in source · 6 passages

Search aliases: ∏_i∈ L prod in

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Use in source · Read passage · c_lean.tex, line 75
Polynomial certificates are not postulated degree oracles. The inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator polyEval(p,x)=⊕_L∈ p∏_i∈ Lx_i by a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a certificate.
Exact mathematical passage
```
\operatorname{polyEval}(p,x)=\bigoplus_{L\in p}\prod_{i\in L}x_i
```

\prod_{i\in R}Use in source · 3 passages

Search aliases: ∏_i∈ R prod in

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Use in source · Read passage · c_lean.tex, line 94
Suppose the next gate toggles target u by the product of a control list of length q. A product that omits u is unchanged. If u occurs, idempotence removes all its repetitions. With R the remaining list, the updated product is (s_u∏_i∈ Rs_i) x (∏_j∈controlss_j∏_i∈ Rs_i). The two old-state products have certified degrees at most t+1+|R| and t+q+|R|. Since |R|+1≤ |L|, the invariant's budget rises by at most max{q-1,0}: zero for NOT and CNOT, one for Toffoli.
Exact mathematical passage
```
\left(s_u\prod_{i\in R}s_i\right)
\x
\left(\prod_{j\in\mathrm{controls}}s_j\prod_{i\in R}s_i\right).
```

\prod_{i\in S}Use in source · 1 passage

Search aliases: ∏_i∈ S prod in

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Use in source · Read passage · c_main.tex, line 12
For S⊆[n], write x^S=∏_i∈ Sx_i and let 1_S be the input with support S. A minterm of a monotone function means an inclusion-minimal positive support, not an arbitrary positive input.
Exact mathematical passage
```
x^S=\prod_{i\in S}x_i
```

\prod_{j=1}^nUse in source · 2 passages

Search aliases: ∏_j=1^n prod

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Use in source · Read passage · g_main.tex, line 3436
Exact group sizes. The finite state spaces are |P_n| = 2^n(n+1)/2|GL(n,2)|, and |Sp(2n,2)| = 2^n^2∏_j=1^n(2^2j-1). For n=1,2,3 this gives: @rcc@ n · |P_n| · |Sp(2n,2)|
Exact mathematical passage
```
|\Sp(2n,2)|
=
2^{n^2}\prod_{j=1}^n(2^{2j}-1).
```

\prod_{j\in S}Use in source · 1 passage

Search aliases: ∏_j∈ S prod in

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Use in source · Read passage · g_main.tex, line 3712
Algebraic normal form. Every Boolean coordinate function has a unique algebraic normal form f_i(x) = ⊕_S⊆[n]a_i,S∏_j∈ Sx_j. Affine maps have degree at most one. A Toffoli gate targeting t with controls a,b performs x_t⟼ x_t+x_ax_b, which can raise the degree of later coordinates. The final affine target requires every higher-degree coefficient to cancel.
Exact mathematical passage
```
f_i(x)
=
\bigoplus_{S\subseteq[n]}a_{i,S}\prod_{j\in S}x_j.
```

\prod_{j\in\mathrm{controls}}Use in source · 3 passages

Search aliases: ∏_j∈controls prod in mathrm

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Use in source · Read passage · c_lean.tex, line 94
Suppose the next gate toggles target u by the product of a control list of length q. A product that omits u is unchanged. If u occurs, idempotence removes all its repetitions. With R the remaining list, the updated product is (s_u∏_i∈ Rs_i) x (∏_j∈controlss_j∏_i∈ Rs_i). The two old-state products have certified degrees at most t+1+|R| and t+q+|R|. Since |R|+1≤ |L|, the invariant's budget rises by at most max{q-1,0}: zero for NOT and CNOT, one for Toffoli.
Exact mathematical passage
```
\left(s_u\prod_{i\in R}s_i\right)
\x
\left(\prod_{j\in\mathrm{controls}}s_j\prod_{i\in R}s_i\right).
```

\prod_{k>0}Use in source · 1 passage

Search aliases: ∏_k>0 prod

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Use in source · Read passage · g_main.tex, line 854
The Chevalley conjugation formula has the schematic form x_α(a)x_β(b)x_α(a)^-1 = x_β(b) ∏_k>0 x_β+kα(c_k a^k b). A forbidden root α can therefore create several allowed root effects before being removed.
Exact mathematical passage
```
x_\alpha(a)x_\beta(b)x_\alpha(a)^{-1}
=
x_\beta(b)
\prod_{k>0}
x_{\beta+k\alpha}(c_k a^k b).
```

\sum_aUse in source · 1 passage

Search aliases: ∑_a sum

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Use in source · Read passage · s_main.tex, line 1680
Lindblad dynamics. A Markovian open system obeys fracdρdt = L(ρ) = -i[H,ρ] + ∑_a ( L_aρ L_a^† - frac12{L_a^† L_a,ρ} ). Packet selection rules of H and L_a determine a sparse directed graph on coherence pairs. If the reachable graph and each block representation remain polynomial, a classical integrator can evolve the reduced system.
Exact mathematical passage
```
\frac{d\rho}{dt}
=
\mathcal L(\rho)
=
-i[H,\rho]
+
\sum_a
\left(
L_a\rho L_a^\dagger
-
\frac12\{L_a^\dagger L_a,\rho\}
\right).
```

\sum_{a=1}^RUse in source · 1 passage

Search aliases: ∑_a=1^R sum

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Use in source · Read passage · s_main.tex, line 1573
Low-rank non-Gaussian corrections. One research direction is to represent a weakly non-Gaussian state as ρ = ∑_a=1^R c_aρ_a^rm Gauss with small Gaussian rank R, or as a Gaussian reference plus sparse packet corrections. A polynomial collapse requires R and the update cost to remain polynomial.
Exact mathematical passage
```
\rho
=
\sum_{a=1}^R c_a\rho_a^{\rm Gauss}
```

\sum_{a\in S_d}Use in source · 1 passage

Search aliases: ∑_a∈ S_d sum in

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Use in source · Read passage · q_main.tex, line 381
(F_ND_dF_N^†)_nm · =frac1N∑_a∈ S_dζ_N^a(n-m) · =frac1N∑_b∈ U_dζ_d^b(n-m) =frac1N c_d(n-m).
Exact mathematical passage
```
\begin{aligned}
(F_ND_dF_N^\dagger)_{nm}
&=\frac1N\sum_{a\in S_d}\zeta_N^{a(n-m)}\\
&=\frac1N\sum_{b\in U_d}\zeta_d^{b(n-m)}
=\frac1N c_d(n-m).
\end{aligned}
```

\sum_{a\in U_{12}}Use in source · 3 passages

Search aliases: ∑_a∈ U_12 sum in

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Use in source · Read passage · q_main.tex, line 914
frac12sqrt12∑_a∈ U_12ζ^na =fracc(n)sqrt48.
Exact mathematical passage
```
\frac1{2\sqrt{12}}\sum_{a\in U_{12}}\zeta^{na}
=\frac{c(n)}{\sqrt{48}}.
```

\sum_{a\in U_q}Use in source · 2 passages

Search aliases: ∑_a∈ U_q sum in

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Use in source · Read passage · q_main.tex, line 229
c_q(n)=∑_a∈ U_qζ_q^an, n∈ Z.
Exact mathematical passage
```
c_q(n)=\sum_{a\in U_q}\zeta_q^{an},
\qquad n\in\mathbb Z.
```

\sum_{b=0}^{q/d-1}Use in source · 2 passages

Search aliases: ∑_b=0^q/d-1 sum

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Use in source · Read passage · q_main.tex, line 272
c_q(n) · =∑_dmid qμ(d) ∑_b=0^q/d-1e^2π ibn/(q/d) · =∑_substackdmid q (q/d)mid nμ(d)frac qd.
Exact mathematical passage
```
\begin{aligned}
c_q(n)
&=\sum_{d\mid q}\mu(d)
  \sum_{b=0}^{q/d-1}e^{2\pi ibn/(q/d)}\\
&=\sum_{\substack{d\mid q\\(q/d)\mid n}}\mu(d)\frac qd.
\end{aligned}
```

\sum_{b\in U_d}Use in source · 1 passage

Search aliases: ∑_b∈ U_d sum in

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Use in source · Read passage · q_main.tex, line 381
(F_ND_dF_N^†)_nm · =frac1N∑_a∈ S_dζ_N^a(n-m) · =frac1N∑_b∈ U_dζ_d^b(n-m) =frac1N c_d(n-m).
Exact mathematical passage
```
\begin{aligned}
(F_ND_dF_N^\dagger)_{nm}
&=\frac1N\sum_{a\in S_d}\zeta_N^{a(n-m)}\\
&=\frac1N\sum_{b\in U_d}\zeta_d^{b(n-m)}
=\frac1N c_d(n-m).
\end{aligned}
```

\sum_{d=1}^{N}Use in source · 1 passage

Search aliases: ∑_d=1^N sum

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Use in source · Read passage · g_main.tex, line 2566
The defect condition implies ambient length at most N-d, so the number of such targets is at most (Mq)^N-d. Divide by |U_n(F_q)|=q^N. The expectation bound follows from Exp[DeltaDef]=∑_d=1^NProb[DeltaDef≥ d].
Exact mathematical passage
```
\Exp[\DeltaDef]=\sum_{d=1}^{N}\Prob[\DeltaDef\ge d].
```

\sum_{d\mid g}Use in source · 1 passage

Search aliases: ∑_dmid g sum mid

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Use in source · Read passage · q_main.tex, line 256
∑_dmid gμ(d) =1, · g=1, 0, · g>1.
Exact mathematical passage
```
\sum_{d\mid g}\mu(d)
=\begin{cases}1,&g=1,\\0,&g>1.\end{cases}
```

\sum_{d\mid N}Use in source · 3 passages

Search aliases: ∑_dmid N sum mid

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Use in source · Read passage · q_main.tex, line 363

(P_d^(N))^† · =P_d^(N), P_d^(N)P_e^(N) · =δ_deP_d^(N), ∑_dmid NP_d^(N) · =Id_N, rankP_d^(N) · =trP_d^(N)=φ(d).

Exact mathematical passage


\begin{aligned}
(P_d^{(N)})^\dagger&=P_d^{(N)},\\
P_d^{(N)}P_e^{(N)}&=\delta_{de}P_d^{(N)},\\
\sum_{d\mid N}P_d^{(N)}&=\operatorname{Id}_N,\\
\operatorname{rank}P_d^{(N)}&=\operatorname{tr}P_d^{(N)}=\varphi(d).
\end{aligned}

\sum_{d\mid q}Use in source · 2 passages

Search aliases: ∑_dmid q sum mid

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Use in source · Read passage · q_main.tex, line 272
c_q(n) · =∑_dmid qμ(d) ∑_b=0^q/d-1e^2π ibn/(q/d) · =∑_substackdmid q (q/d)mid nμ(d)frac qd.
Exact mathematical passage
```
\begin{aligned}
c_q(n)
&=\sum_{d\mid q}\mu(d)
  \sum_{b=0}^{q/d-1}e^{2\pi ibn/(q/d)}\\
&=\sum_{\substack{d\mid q\\(q/d)\mid n}}\mu(d)\frac qd.
\end{aligned}
```

\sum_{d\mid12}Use in source · 2 passages

Search aliases: ∑_dmid12 sum mid

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Use in source · Read passage · q_visual.tex, line 128
six order classes leadThe primitive-twelve projector is one packet in a complete orthogonal decomposition. A frequency label k has root order d=12/gcd(k,12). The possible orders are the divisors 1,2,3,4,6,12. fig08_order_partition1.0The number inside each tile is its root order d. The horizontal label is frequency k. Every label belongs to one class; no labels are dropped. P_d_nm=fracc_d(n-m)12, P_dP_e=δ_deP_d, ∑_dmid12P_d=I. These are character-order projectors. They select Fourier modes by roo…
Exact mathematical passage
```
 P_d{}_{nm}=\frac{c_d(n-m)}{12},\quad
 P_dP_e=\delta_{de}P_d,\quad
 \sum_{d\mid12}P_d=I.
```

\sum_{d\mid\gcd(a,q)}Use in source · 2 passages

Search aliases: ∑_dmidgcd(a,q) sum mid gcd

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Use in source · Read passage · q_main.tex, line 265
1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d).
Exact mathematical passage
```
\mathbf1_{\gcd(a,q)=1}=\sum_{d\mid\gcd(a,q)}\mu(d).
```

\sum_{d\mid\gcd(q,n)}Use in source · 1 passage

Search aliases: ∑_dmidgcd(q,n) sum mid gcd

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Use in source · Read passage · q_main.tex, line 247
c_q(n)=∑_dmidgcd(q,n)d μ(q/d)∈ Z.
Exact mathematical passage
```
\boxed{c_q(n)=\sum_{d\mid\gcd(q,n)}d\,\mu(q/d)\in\mathbb Z.}
```

\sum_{d\in\mathcal D}Use in source · 2 passages

Search aliases: ∑_d∈ D sum in mathcal

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Use in source · Read passage · q_main.tex, line 445
P_ D=∑_d∈ DP_d^(N)
Exact mathematical passage
```
P_{\mathcal D}=\sum_{d\in\mathcal D}P_d^{(N)}
```

\sum_{e\mid\gcd(q,n)}Use in source · 1 passage

Search aliases: ∑_emidgcd(q,n) sum mid gcd

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Use in source · Read passage · q_visual.tex, line 56
ntentslinetocsectionWhy the phase sum is an integer leadA divisor identity certifies integrality without rounding trigonometric values. For q≥1, insert the M obius coprimality identity into the finite phase sum: 1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d). Write a=db after interchanging the finite sums. Character orthogonality evaluates the inner geometric series: c_q(n) · =∑_dmid qμ(d)∑_b=0^q/d-1e^2π ibn/(q/d) · =∑_substackdmid q q/dmid nμ(d)frac qd =∑_emidgcd(q,n)eμ(q/e)∈ Z. lll Sample · Relevant divisor …
Exact mathematical passage
```
 \begin{aligned}
 c_q(n)&=\sum_{d\mid q}\mu(d)\sum_{b=0}^{q/d-1}e^{2\pi ibn/(q/d)}\\
 &=\sum_{\substack{d\mid q\\q/d\mid n}}\mu(d)\frac qd
 =\sum_{e\mid\gcd(q,n)}e\mu(q/e)\in\mathbb Z.
 \end{aligned}
```

\sum_{g\in G}Use in source · 1 passage

Search aliases: ∑_g∈ G sum in

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Use in source · Read passage · s_main.tex, line 158
Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).
Exact mathematical passage
```
P_\chi
=
\frac{d_\chi}{|G|}
\sum_{g\in G}
\overline{\chi(g)}U_g.
\label{s:eq:character-projector}
```

\sum_{g\in W}Use in source · 2 passages

Search aliases: ∑_g∈ W sum in

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Use in source · Read passage · g_main.tex, line 4450
For stochastic error rates p_g, a first-order weighted cost is C_err(W)=∑_g∈ W-log(1-p_g). This additive approximation can be inserted directly into Dijkstra search. It does not capture coherent or correlated errors, but it produces a declared and reproducible metric.
Exact mathematical passage
```
C_{\mathrm{err}}(W)=\sum_{g\in W}-\log(1-p_g).
```

\sum_{h\in H}Use in source · 2 passages

Search aliases: ∑_h∈ H sum in

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Use in source · Read passage · g_main.tex, line 513
Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint series after Fourier inversion.
Exact mathematical passage
```
\widehat\mu_k(\tau)
=
\sum_{h\in H}\mu_k(h)\tau(h).
```

\sum_{h\ge1}Use in source · 1 passage

Search aliases: ∑_h≥1 sum ge

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Use in source · Read passage · g_main.tex, line 2958
inproof[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^r+120 ≤ frac2120n. Also r≥log_21n-1. Therefore S≥ nlog_21n-frac4120n. Multiplying by 8/21 yields D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n.
Exact mathematical passage
```
r n-S
=
\sum_{j<r}(r-j)d_j21^j
\le
20\sum_{h\ge1}h21^{r-h}
\le
\frac{21^{r+1}}{20}
\le
\frac{21}{20}n.
```

\sum_iUse in source · 8 passages

Search aliases: ∑_i sum

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Use in source · Read passage · g_main.tex, line 302
Many consumers and catalytic excess. Suppose the middle word is k=h_m⋯ h_1, h_i∈ H. The ambient word gh_m⋯ h_1g^-1 has cost at most 2c(g)+∑_i c(h_i). If the restricted images gh_ig^-1 require costs r_i and a lower-bound invariant proves that their obligations cannot be shared, then ℓ_H(gkg^-1)≥∑_i r_i. A sufficient inequality is ∑_i(r_i-c(h_i))>2c(g). The left side is the gross catalytic benefit. The right side is the restoration cost.
Exact mathematical passage
```
2c(g)+\sum_i c(h_i).
```

\sum_{i<j}Use in source · 1 passage

Search aliases: ∑_i<j sum

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Use in source · Read passage · g_main.tex, line 4612
Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.
Exact mathematical passage
```
\label{g:eq:transition-fusion}
\sum_{i<j}
\left(
P_{\rho_i\rho_j^{-1}}
+
P_{\rho_j\rho_i^{-1}}
\right)
=n(J-I).
```

\sum_{i=1}^kUse in source · 1 passage

Search aliases: ∑_i=1^k sum

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Use in source · Read passage · g_main.tex, line 2601
[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).
Exact mathematical passage
```
\ellDown\bigl(\operatorname{diag}(A_1,\ldots,A_k)\bigr)
=
\sum_{i=1}^k\ellDown(A_i).
```

\sum_{i=1}^mUse in source · 1 passage

Search aliases: ∑_i=1^m sum

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Use in source · Read passage · g_main.tex, line 5077
Template II: amortized consumers. Suppose a catalyst costs c_open+c_close and each of m consumers saves at least s_i restricted cost. Then Δ(U) ≥ ∑_i=1^m s_i -c_open-c_close-r, where r is repair or interference cost. A threshold occurs when the right side first becomes positive.
Exact mathematical passage
```
\Delta(U)
\ge
\sum_{i=1}^m s_i
-c_{\mathrm{open}}-c_{\mathrm{close}}-r,
```

\sum_{i=1}^{s}Use in source · 1 passage

Search aliases: ∑_i=1^s sum

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Use in source · Read passage · g_main.tex, line 2005
Fix integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.
Exact mathematical passage
```
\label{g:eq:prefix-state}
w_s=e_p+\sum_{i=1}^{s}e_{x_i}.
```

\sum_jUse in source · 5 passages

Search aliases: ∑_j sum

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Use in source · Read passage · g_main.tex, line 410
:G→ H with π|_H=id_H such that every ambient generator s∈ S_G satisfies ℓ_H(π(s))≤ c(s). Then ℓ_H(h)≤ℓ_G(h) for every h∈ H, hence the two metrics agree on H. For an ambient word h=s_L⋯ s_1, multiplicativity gives h=π(h)=π(s_L)⋯π(s_1). Subadditivity and the generator bounds give ℓ_H(h)≤∑_j c(s_j). Taking the infimum over ambient words proves the claim.
Exact mathematical passage
```
\ell_H(h)\le\sum_j c(s_j)
```

\sum_{j<r}Use in source · 1 passage

Search aliases: ∑_j<r sum

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Use in source · Read passage · g_main.tex, line 2958
inproof[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^r+120 ≤ frac2120n. Also r≥log_21n-1. Therefore S≥ nlog_21n-frac4120n. Multiplying by 8/21 yields D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n.
Exact mathematical passage
```
r n-S
=
\sum_{j<r}(r-j)d_j21^j
\le
20\sum_{h\ge1}h21^{r-h}
\le
\frac{21^{r+1}}{20}
\le
\frac{21}{20}n.
```

\sum_{j=0}^LUse in source · 2 passages

Search aliases: ∑_j=0^L sum

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Use in source · Read passage · g_main.tex, line 596
Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estimate can be sharpened by orbit counting, stabilizers, and the geometry of word-map fibers.
Exact mathematical passage
```
\sum_{j=0}^L[M(q-1)]^j.
```

\sum_{j=0}^{L-1}Use in source · 3 passages

Search aliases: ∑_j=0^L-1 sum

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Use in source · Read passage · q_main.tex, line 550
R_rf(z) · =frac1L∑_j=0^L-1ω^-rjf(ω^jz), R_rR_s · =δ_rsR_r, ∑_r=0^L-1R_rf · =f.
Exact mathematical passage
```
\begin{aligned}
R_rf(z)&=\frac1L\sum_{j=0}^{L-1}\omega^{-rj}f(\omega^jz),\\
R_rR_s&=\delta_{rs}R_r,\\
\sum_{r=0}^{L-1}R_rf&=f.
\end{aligned}
```

\sum_{j=0}^{\min(d,n)}Use in source · 2 passages

Search aliases: ∑_j=0^min(d,n) sum min

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Use in source · Read passage · c_main.tex, line 227
[Finite support of a dual witness] When the right side of [source reference] holds, there is W⊆ U with v_d(v)=∑_u∈ Wv_d(u), |W|≤∑_j=0^min(d,n)binom nj. The cardinality |W| is odd. Choose a basis of the span from negative evaluation vectors and express v_d(v) in that basis. Over F, the nonzero coefficients select a subset W of size at most the ambient dimension. The constant-monomial coordinate of every evaluation vector is one; comparing it on both sides shows that |W| is odd. Such a relation is…
Exact mathematical passage
```
 v_d(v)=\sum_{u\in W}v_d(u),\qquad
 |W|\le\sum_{j=0}^{\min(d,n)}\binom nj.
```

\sum_{j=0}^rUse in source · 1 passage

Search aliases: ∑_j=0^r sum

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Use in source · Read passage · c_main.tex, line 602
tional algorithmic consequence [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The leading coefficient has total degree at most d-r, so it vanishes at a random first (m-1)-tuple with probability at most (d-r)/|K|. Otherwise the remaini…
Exact mathematical passage
```
 P=\sum_{j=0}^r P_j(z_1,\ldots,z_{m-1})z_m^j,
 \qquad P_r\ne0.
```

\sum_{j=0}^{r}Use in source · 2 passages

Search aliases: ∑_j=0^r sum

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Use in source · Read passage · g_main.tex, line 2945
[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^r+120 ≤ frac2120n. Also r≥log_21n-1. Therefore S≥ nlog_21n-frac4120n. Multiplying by 8/21 yields D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n.
Exact mathematical passage
```
n=\sum_{j=0}^{r}d_jm^j,
\qquad
0\le d_j<m.
```

\sum_{j=1}^mUse in source · 1 passage

Search aliases: ∑_j=1^m sum

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Use in source · Read passage · s_main.tex, line 375
[Sum-of-leakages bound] For an initial state in PHh, G_m⋯ G_1ketψ_0 - V_m⋯ V_1ketψ_0 ≤ ∑_j=1^mδ_j.
Exact mathematical passage
```
\|G_m\cdots G_1\ket{\psi_0}
-
V_m\cdots V_1\ket{\psi_0}\|
\le
\sum_{j=1}^m\delta_j.
```

\sum_kUse in source · 3 passages

Search aliases: ∑_k sum

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Use in source · Read passage · q_main.tex, line 184
x(n)=frac1q∑_k=0^q-1widehatx(k)ζ_q^kn, ∑_n|x(n)|^2=frac1q∑_k|widehatx(k)|^2.
Exact mathematical passage
```
x(n)=\frac1q\sum_{k=0}^{q-1}\widehat{x}(k)\zeta_q^{kn},
\qquad
\sum_n|x(n)|^2=\frac1q\sum_k|\widehat{x}(k)|^2.
```

\sum_{k=0}^{K}Use in source · 2 passages

Search aliases: ∑_k=0^K sum

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Use in source · Read passage · s_main.tex, line 589
The active dimension is ∑_k=0^Kbinom nk =O(n^K). Let P project onto Hh_≤ K. By the hypothesis on the actual circuit prefixes, Pψ_t=ψ_t at every step. Consequently PG_tPψ_t-1=G_tψ_t-1 on the evolving state, even if the gate does not preserve all of PHh. The restricted update therefore reproduces that trajectory exactly. The hypothesis supplies efficient initialization, updates, and measurement in these polynomially many coordinates.
Exact mathematical passage
```
\sum_{k=0}^{K}\binom nk
=O(n^K).
```

\sum_{k=0}^nUse in source · 1 passage

Search aliases: ∑_k=0^n sum

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Use in source · Read passage · s_main.tex, line 742
Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.
Exact mathematical passage
```
\sum_{k=0}^n\binom nk=2^n.
```

\sum_{k=0}^{N-1}Use in source · 1 passage

Search aliases: ∑_k=0^N-1 sum

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Use in source · Read passage · q_main.tex, line 635
widehatx_r(ℓ) · =∑_m=0^M-1x(r+Lm)ζ_M^-ℓ m · =frac1N∑_k=0^N-1widehatx(k)ζ_N^kr ∑_m=0^M-1ζ_M^(k-ℓ)m.
Exact mathematical passage
```
\begin{aligned}
\widehat{x_r}(\ell)
&=\sum_{m=0}^{M-1}x(r+Lm)\zeta_M^{-\ell m}\\
&=\frac1N\sum_{k=0}^{N-1}\widehat{x}(k)\zeta_N^{kr}
\sum_{m=0}^{M-1}\zeta_M^{(k-\ell)m}.
\end{aligned}
```

\sum_{k=0}^{q-1}Use in source · 1 passage

Search aliases: ∑_k=0^q-1 sum

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Use in source · Read passage · q_main.tex, line 184
x(n)=frac1q∑_k=0^q-1widehatx(k)ζ_q^kn, ∑_n|x(n)|^2=frac1q∑_k|widehatx(k)|^2.
Exact mathematical passage
```
x(n)=\frac1q\sum_{k=0}^{q-1}\widehat{x}(k)\zeta_q^{kn},
\qquad
\sum_n|x(n)|^2=\frac1q\sum_k|\widehat{x}(k)|^2.
```

\sum_{k=1}^mUse in source · 2 passages

Search aliases: ∑_k=1^m sum

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Use in source · Read passage · s_appendix.tex, line 19
Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].
Exact mathematical passage
```
G_m\cdots G_1P-V_m\cdots V_1P
=
\sum_{k=1}^m
G_m\cdots G_{k+1}(G_k-V_k)V_{k-1}\cdots V_1P.
```

\sum_{k\equiv r\pmod M}Use in source · 1 passage

Search aliases: ∑_k≡ rpmod M sum equiv pmod

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Use in source · Read passage · s_main.tex, line 136
Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis and charge sectors in quantum theory.
Exact mathematical passage
```
B_r^{(M)}=
\sum_{k\equiv r\pmod M}P_k.
```

\sum_{k\ge0}Use in source · 5 passages

Search aliases: ∑_k≥0 sum ge

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Use in source · Read passage · g_main.tex, line 479
[Endpoint Green valuation] The word distance is d_S(e,g) = ord_z=0 ⟨ δ_g,(I-zA_S)^-1δ_e ⟩. Expand the resolvent as a formal power series: (I-zA_S)^-1=∑_k≥0z^kA_S^k. The first nonzero endpoint coefficient occurs at the shortest word length.
Exact mathematical passage
```
(I-zA_S)^{-1}=\sum_{k\ge0}z^kA_S^k.
```

\sum_mUse in source · 1 passage

Search aliases: ∑_m sum

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Use in source · Read passage · q_main.tex, line 622
widehatx(k) · =∑_r=0^L-1∑_m=0^M-1 x_r(m)ζ_N^-k(r+Lm) · =∑_rζ_N^-kr∑_mx_r(m)ζ_M^-km.
Exact mathematical passage
```
\begin{aligned}
\widehat{x}(k)
&=\sum_{r=0}^{L-1}\sum_{m=0}^{M-1}
x_r(m)\zeta_N^{-k(r+Lm)}\\
&=\sum_r\zeta_N^{-kr}\sum_mx_r(m)\zeta_M^{-km}.
\end{aligned}
```

\sum_{m=0}^{M-1}Use in source · 3 passages

Search aliases: ∑_m=0^M-1 sum

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Use in source · Read passage · q_main.tex, line 590
∑_n=0^N-1|x(n)|^2 =∑_r=0^L-1∑_m=0^M-1|x_r(m)|^2.
Exact mathematical passage
```
\sum_{n=0}^{N-1}|x(n)|^2
=\sum_{r=0}^{L-1}\sum_{m=0}^{M-1}|x_r(m)|^2.
```

\sum_{m\ge0}Use in source · 1 passage

Search aliases: ∑_m≥0 sum ge

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Use in source · Read passage · q_main.tex, line 534
E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).
Exact mathematical passage
```
E_rf(t)=\sum_{m\ge0}a_{r+Lm}t^m,
\qquad
R_rf(z)=z^rE_rf(z^L).
```

\sum_nUse in source · 5 passages

Search aliases: ∑_n sum

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Use in source · Read passage · q_main.tex, line 184
x(n)=frac1q∑_k=0^q-1widehatx(k)ζ_q^kn, ∑_n|x(n)|^2=frac1q∑_k|widehatx(k)|^2.
Exact mathematical passage
```
x(n)=\frac1q\sum_{k=0}^{q-1}\widehat{x}(k)\zeta_q^{kn},
\qquad
\sum_n|x(n)|^2=\frac1q\sum_k|\widehat{x}(k)|^2.
```

\sum_{n=0}^{11}Use in source · 5 passages

Search aliases: ∑_n=0^11 sum

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Use in source · Read passage · q_main.tex, line 889
|ψ⟩=frac1sqrt48∑_n=0^11c(n)|n⟩, |χ⟩=frac1sqrt48∑_n=0^11|c(n)| |n⟩.
Exact mathematical passage
```
|\psi\rangle=\frac1{\sqrt{48}}\sum_{n=0}^{11}c(n)|n\rangle,
\qquad
|\chi\rangle=\frac1{\sqrt{48}}\sum_{n=0}^{11}|c(n)|\,|n\rangle.
```

\sum_{n=0}^{N-1}Use in source · 1 passage

Search aliases: ∑_n=0^N-1 sum

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Use in source · Read passage · q_main.tex, line 590
∑_n=0^N-1|x(n)|^2 =∑_r=0^L-1∑_m=0^M-1|x_r(m)|^2.
Exact mathematical passage
```
\sum_{n=0}^{N-1}|x(n)|^2
=\sum_{r=0}^{L-1}\sum_{m=0}^{M-1}|x_r(m)|^2.
```

\sum_{n=0}^{q-1}Use in source · 7 passages

Search aliases: ∑_n=0^q-1 sum

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Use in source · Read passage · q_main.tex, line 116
⟨ x,y⟩=∑_n=0^q-1overlinex(n)y(n), x ^2=⟨ x,x⟩.
Exact mathematical passage
```
\langle x,y\rangle=\sum_{n=0}^{q-1}\overline{x(n)}y(n),
\qquad \|x\|^2=\langle x,x\rangle.
```

\sum_{n\ge0}Use in source · 1 passage

Search aliases: ∑_n≥0 sum ge

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Use in source · Read passage · q_main.tex, line 530
Let L≥1 and let f(z)=∑_n≥0a_nz^n be a formal power series over a field containing a primitive Lth root ω and 1/L. For 0≤ r<L, define
Exact mathematical passage
```
f(z)=\sum_{n\ge0}a_nz^n
```

\sum_{q=1}^nUse in source · 2 passages

Search aliases: ∑_q=1^n sum

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Use in source · Read passage · s_main.tex, line 718
Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.
Exact mathematical passage
```
\widehat N
=
\sum_{q=1}^n\frac{I-Z_q}{2}.
```

\sum_{q\ne r}Use in source · 1 passage

Search aliases: ∑_q≠ r sum ne

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Use in source · Read passage · g_main.tex, line 4348
Sector-flow invariants. Let P_q be the projector onto sector q. For an intermediate unitary U_t, define transfer blocks T_q,r(t)=P_q U_t P_r. A restricted path has T_q,r=0 for q≠ r. Ambient paths may populate off-diagonal blocks. Possible excursion measures include ∑_q≠ rrank T_q,r, max_q≠ r T_q,r , bandwidth in |q-r|. These are analogues of strict-upper rank and bandwidth in the unitriangular theory.
Exact mathematical passage
```
\sum_{q\ne r}\rank T_{q,r},
\qquad
\max_{q\ne r}\|T_{q,r}\|,
\qquad
\text{bandwidth in }|q-r|.
```

\sum_rUse in source · 1 passage

Search aliases: ∑_r sum

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Use in source · Read passage · q_main.tex, line 622
widehatx(k) · =∑_r=0^L-1∑_m=0^M-1 x_r(m)ζ_N^-k(r+Lm) · =∑_rζ_N^-kr∑_mx_r(m)ζ_M^-km.
Exact mathematical passage
```
\begin{aligned}
\widehat{x}(k)
&=\sum_{r=0}^{L-1}\sum_{m=0}^{M-1}
x_r(m)\zeta_N^{-k(r+Lm)}\\
&=\sum_r\zeta_N^{-kr}\sum_mx_r(m)\zeta_M^{-km}.
\end{aligned}
```

\sum_{r=0}^{2}Use in source · 1 passage

Search aliases: ∑_r=0^2 sum

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Use in source · Read passage · q_visual.tex, line 211
Phase factors connect the lane spectra. addcontentslinetocsectionPhase factors connect the lane spectra leadEach local Fourier transform contributes to every appropriate global frequency. With four-point negative-sign DFTs in the lanes, widehat x(k)=∑_r=0^2e^-2π ikr/12 widehatE_rx(kbmod4). fig18_polyphase_alias_assembly1.0Left: magnitudes of the three local transforms. Right: the recovered global spectrum. Assembly uses the complex coefficients and phase factors; adding the displayed magnitudes …
Exact mathematical passage
```
 \widehat x(k)=\sum_{r=0}^{2}e^{-2\pi ikr/12}
 \widehat{E_rx}(k\bmod4).
```

\sum_{r=0}^{L-1}Use in source · 4 passages

Search aliases: ∑_r=0^L-1 sum

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Use in source · Read passage · q_main.tex, line 550
R_rf(z) · =frac1L∑_j=0^L-1ω^-rjf(ω^jz), R_rR_s · =δ_rsR_r, ∑_r=0^L-1R_rf · =f.
Exact mathematical passage
```
\begin{aligned}
R_rf(z)&=\frac1L\sum_{j=0}^{L-1}\omega^{-rj}f(\omega^jz),\\
R_rR_s&=\delta_{rs}R_r,\\
\sum_{r=0}^{L-1}R_rf&=f.
\end{aligned}
```

\sum_{r=1}^{c}Use in source · 1 passage

Search aliases: ∑_r=1^c sum

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Use in source · Read passage · g_main.tex, line 785
C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, Q_final=I_t.
Exact mathematical passage
```
C=
\sum_{r=1}^{c}
Q_{>r}e_{i_r}e_{j_r}^{T}P_{<r},
```

\sum_{\substack{d\mid q\\(q/d)\mid n}}Use in source · 1 passage

Search aliases: ∑_substackdmid q (q/d)mid n sum substack mid mid

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Use in source · Read passage · q_main.tex, line 272
c_q(n) · =∑_dmid qμ(d) ∑_b=0^q/d-1e^2π ibn/(q/d) · =∑_substackdmid q (q/d)mid nμ(d)frac qd.
Exact mathematical passage
```
\begin{aligned}
c_q(n)
&=\sum_{d\mid q}\mu(d)
  \sum_{b=0}^{q/d-1}e^{2\pi ibn/(q/d)}\\
&=\sum_{\substack{d\mid q\\(q/d)\mid n}}\mu(d)\frac qd.
\end{aligned}
```

\sum_{\substack{d\mid q\\q/d\mid n}}Use in source · 1 passage

Search aliases: ∑_substackdmid q q/dmid n sum substack mid q mid

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Use in source · Read passage · q_visual.tex, line 56
ntentslinetocsectionWhy the phase sum is an integer leadA divisor identity certifies integrality without rounding trigonometric values. For q≥1, insert the M obius coprimality identity into the finite phase sum: 1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d). Write a=db after interchanging the finite sums. Character orthogonality evaluates the inner geometric series: c_q(n) · =∑_dmid qμ(d)∑_b=0^q/d-1e^2π ibn/(q/d) · =∑_substackdmid q q/dmid nμ(d)frac qd =∑_emidgcd(q,n)eμ(q/e)∈ Z. lll Sample · Relevant divisor …
Exact mathematical passage
```
 \begin{aligned}
 c_q(n)&=\sum_{d\mid q}\mu(d)\sum_{b=0}^{q/d-1}e^{2\pi ibn/(q/d)}\\
 &=\sum_{\substack{d\mid q\\q/d\mid n}}\mu(d)\frac qd
 =\sum_{e\mid\gcd(q,n)}e\mu(q/e)\in\mathbb Z.
 \end{aligned}
```

\sum_{s\in S}Use in source · 3 passages

Search aliases: ∑_s∈ S sum in

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Use in source · Read passage · g_main.tex, line 457
Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.
Exact mathematical passage
```
A_S=\sum_{s\in S}\rho(s).
```

\sum_{t=1}^mUse in source · 3 passages

Search aliases: ∑_t=1^m sum

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Use in source · Read passage · s_main.tex, line 667
ate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ_m⟩-⟨ϕ_m,Mϕ_m⟩| ≤2∑_t=1^mη_t. No renormalization of ϕ_t is performed. The second quadratic form is an unnormalized acceptance estimate.
Exact mathematical passage
```
\|\psi_m-\phi_m\|\le\sum_{t=1}^m\eta_t.
```

\sum_{t=1}^{m}Use in source · 2 passages

Search aliases: ∑_t=1^m sum

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Use in source · Read passage · s_main.tex, line 619
[Dynamic sum-of-leakages bound] For an initial unit vector in P_0Hh, G_m⋯ G_1ketψ_0 - widehat G_m⋯widehat G_1ketψ_0 ≤ ∑_t=1^mδ_t. For any acceptance operator 0≤ M≤ I, the corresponding acceptance probabilities differ by at most 2∑_t=1^mδ_t.
Exact mathematical passage
```
\left\|
G_m\cdots G_1\ket{\psi_0}
-
\widehat G_m\cdots\widehat G_1\ket{\psi_0}
\right\|
\le
\sum_{t=1}^{m}\delta_t.
```

\sum_{T\subseteq S}Use in source · 1 passage

Search aliases: ∑_T⊆ S sum subseteq

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Use in source · Read passage · c_main.tex, line 252
hm:monotone If f is monotone, I_f has a monomial basis indexed by the positive supports of f. For each positive v, β_v(f)=min{|S|: S⊆supp(v), f(1_S)=1}. Consequently, for nonzero f, α_1(f)=minimum minterm size, β(f)=maximum minterm size. For any Boolean polynomial h, its coefficient on x^S is [x^S]h=∑_T⊆ Sh(1_T) over F. If S is negative for monotone f, every subset T is negative. A safe h vanishes at each 1_T, so its x^S coefficient is zero. Conversely, a monomial on a positive support is safe b…
Exact mathematical passage
```
 [x^S]h=\sum_{T\subseteq S}h(1_T)
```

\sum_uUse in source · 1 passage

Search aliases: ∑_u sum

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Use in source · Read passage · c_main.tex, line 634
[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independent and the factorization exact.
Exact mathematical passage
```
 \sum_{u,v}A(u)B(v)z_u^uz_v^v
 =\left(\sum_uA(u)z_u^u\right)
  \left(\sum_vB(v)z_v^v\right).
```

\sum_{u,v}Use in source · 1 passage

Search aliases: ∑_u,v sum

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Use in source · Read passage · c_main.tex, line 634
[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independent and the factorization exact.
Exact mathematical passage
```
 \sum_{u,v}A(u)B(v)z_u^uz_v^v
 =\left(\sum_uA(u)z_u^u\right)
  \left(\sum_vB(v)z_v^v\right).
```

\sum_{u\in W}Use in source · 2 passages

Search aliases: ∑_u∈ W sum in

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Use in source · Read passage · c_main.tex, line 227
[Finite support of a dual witness] When the right side of [source reference] holds, there is W⊆ U with v_d(v)=∑_u∈ Wv_d(u), |W|≤∑_j=0^min(d,n)binom nj. The cardinality |W| is odd. Choose a basis of the span from negative evaluation vectors and express v_d(v) in that basis. Over F, the nonzero coefficients select a subset W of size at most the ambient dimension. The constant-monomial coordinate of every evaluation vector is one; comparing it on both sides shows that |W| is odd. Such a relation is…
Exact mathematical passage
```
 v_d(v)=\sum_{u\in W}v_d(u),\qquad
 |W|\le\sum_{j=0}^{\min(d,n)}\binom nj.
```

\sum_{u\ne v\text{ on the face}}Use in source · 1 passage

Search aliases: ∑_u≠ v on the face sum ne text

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Use in source · Read passage · c_main.tex, line 353
An explicit moment certificate. On the face b=1, every point except v=(a=0,z=(1,…,1)) is negative for G_k. A polynomial of degree at most k sums to zero over the (k+1)-dimensional cube: each monomial omits some coordinate and contributes an even number of ones. Therefore v_k(v)=∑_u≠ v on the facev_k(u). This is the certificate from Theorem~[source reference], proving that no safe degree-k polynomial detects v.
Exact mathematical passage
```
 v_k(v)=\sum_{u\ne v\text{ on the face}}v_k(u).
```

\sum_vUse in source · 4 passages

Search aliases: ∑_v sum

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Use in source · Read passage · c_main.tex, line 533
calar unital summaries are evaluations] Every unital K-linear multiplicative map Φ:K^W→ K is evaluation at exactly one witness. An idempotent scalar in a field is either zero or one. Orthogonality allows at most one Φ(δ_v) to be one, and preservation of the unit requires exactly one. If it occurs at v_0, linearity gives Φ(g)=Φ(∑_v g(v)δ_v)=g(v_0). For m≥1, such a scalar summary loses at least one singleton witness other than v_0. At m=0, the unique empty witness is preserved by the identity map.…
Exact mathematical passage
```
 \Phi(g)=\Phi\left(\sum_v g(v)\delta_v\right)=g(v_0).
```

\sum_{v\in W}Use in source · 2 passages

Search aliases: ∑_v∈ W sum in

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Use in source · Read passage · c_main.tex, line 499
The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).
Exact mathematical passage
```
 \delta_v^2=\delta_v,\qquad
 \delta_v\delta_u=0\quad(u\ne v),\qquad
 \sum_{v\in W}\delta_v=1.
```

\sum_{v\in\B^m}Use in source · 1 passage

Search aliases: ∑_v∈B^m sum in B

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Use in source · Read passage · c_main.tex, line 571
Exact top-coefficient extraction. [Top coefficient as a Boolean-cube sum] Let G(w) be a multilinear polynomial over a characteristic-two field. Then [w_1⋯ w_m]G=∑_v∈B^mG(v). It is enough to check a monomial. A monomial of proper support is one on an even number 2^m-|S| of Boolean inputs, so its cube sum is zero in characteristic two. The full-support monomial is one at exactly one input and has cube sum one. Linearity gives the identity.
Exact mathematical passage
```
 [w_1\cdots w_m]G=\sum_{v\in\B^m}G(v).
```

\sum_{v\in\{0,1\}^m}Use in source · 1 passage

Search aliases: ∑_v∈{0,1}^m sum in

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Use in source · Read passage · c_visual.tex, line 619
guidepageA short expression inside a trace is not a computed trace Let τ_w extract the top multilinear coefficient in the Boolean witness ring. For any G(w) over a characteristic-two field, τ_w(G)=∑_v∈{0,1}^mG(v). A proper-support monomial evaluates to one an even number of times; the full-support monomial does so once. This proves the identity directly.
Exact mathematical passage
```
 \tau_w(G)=\sum_{v\in\{0,1\}^m}G(v).
```

\sum_{w\in\B^m}Use in source · 1 passage

Search aliases: ∑_w∈B^m sum in B

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Use in source · Read passage · c_main.tex, line 554
Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.
Exact mathematical passage
```
 P_R(z_1,\ldots,z_m)=
 \sum_{w\in\B^m}R(w)\prod_{i=1}^m z_i^{w_i}\in K[z_1,\ldots,z_m].
```

\sum_{w\in\{0,1\}^m}Use in source · 1 passage

Search aliases: ∑_w∈{0,1}^m sum in

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Use in source · Read passage · c_visual.tex, line 591
guidepageDistinct witness monomials prevent formal cancellation Over a characteristic-two field, define P_R(z)=∑_w∈{0,1}^mR(w)∏_i z_i^w_i. The z_i are formal indeterminates, not Boolean variables. Different witnesses produce different monomials, so P_R=0 exactly when there is no accepting witness.
Exact mathematical passage
```
 P_R(z)=\sum_{w\in\{0,1\}^m}R(w)\prod_i z_i^{w_i}.
```

\sum_{x,y}Use in source · 1 passage

Search aliases: ∑_x,y sum

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Use in source · Read passage · s_main.tex, line 1881
Forrelation amplitude. Let f,g:{0,1}^n→{-1,1} and N=2^n. The circuit with alternating Hadamards and phase oracles has zero-state amplitude Φ(f,g) = frac1N^3/2 ∑_x,y f(x)(-1)^x· yg(y). The circuit does not evaluate the N^2 terms separately. It preserves phases and lets them interfere.
Exact mathematical passage
```
\Phi(f,g)
=
\frac1{N^{3/2}}
\sum_{x,y}
 f(x)(-1)^{x\cdot y}g(y).
\label{s:eq:forrelation}
```

\sum_{x\in\mathcal O_a}Use in source · 1 passage

Search aliases: ∑_x∈ O_a sum in mathcal

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Use in source · Read passage · s_main.tex, line 1227
Orbit states. Let a group G act by automorphisms on a finite graph with vertex set V. The orbits are O_1,…, O_r. Define normalized orbit states ket O_a = frac1sqrt| O_a| ∑_x∈ O_aket x. If the adjacency or Hamiltonian H commutes with the group action, the span of these states is invariant whenever the partition is equitable.
Exact mathematical passage
```
\ket{\mathcal O_a}
=
\frac1{\sqrt{|\mathcal O_a|}}
\sum_{x\in\mathcal O_a}\ket x.
```

\sum_{x\in X}Use in source · 1 passage

Search aliases: ∑_x∈ X sum in

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Use in source · Read passage · g_main.tex, line 1518
Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_ye_q (y∈ Y). The following lower-directed gate list produces the same rows: R_q← R_q+a^-1R_p; [R_y← R_y+ac_yR_q]_y∈ Y;…
Exact mathematical passage
```
B=\sum_{x\in X}b_xe_x
```

\sum_{\{v,v+a\}}Use in source · 1 passage

Search aliases: ∑_{v,v+a} sum

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Use in source · Read passage · c_visual.tex, line 435
Now extract the quadratic homogeneous ANF component Q_m in the original x_v coordinates. Since P_m(0)=P_m(e_v)=0, for distinct u,v its quadratic coefficient is [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The involution v↦ v+a pairs every variable with one other variable: Q_m=∑_{v,v+a}x_vx_v+a. It is a quadratic inner product on n/2 disjoint pairs. Its alternating polar matrix has full rank n.
Exact mathematical passage
```
 Q_m=\sum_{\{v,v+a\}}x_vx_{v+a}.
```

\sum_{|x|=k}Use in source · 3 passages

Search aliases: ∑_|x|=k sum

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Use in source · Read passage · s_appendix.tex, line 33
Hypercube quotient coupling. Let ket k = binom nk^-1/2 ∑_|x|=kket x. Every weight-k string has n-k neighbors of weight k+1, while every weight-(k+1) string has k+1 neighbors of weight k. Counting the bipartite edges yields brak+1Aket k = fracbinom nk(n-k) sqrtbinom nkbinom nk+1 = sqrt(k+1)(n-k), proving [source reference].
Exact mathematical passage
```
\ket k
=
\binom nk^{-1/2}
\sum_{|x|=k}\ket x.
```

\sum_\alphaUse in source · 1 passage

Search aliases: ∑_α sum alpha

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Use in source · Read passage · s_main.tex, line 88
Orthogonal packet decompositions. Let Hh be a finite-dimensional Hilbert space and let {P_α}_α∈ A be orthogonal projectors satisfying P_α P_β=δ_αβP_α, ∑_α∈ AP_α=I. Every state and operator decompose as ketψ=∑_α P_αketψ, X=∑_α,βP_α X P_β. The diagonal blocks act within packets. The off-diagonal blocks transfer amplitude or coherence between packets.
Exact mathematical passage
```
\ket\psi=\sum_\alpha P_\alpha\ket\psi,
\qquad
X=\sum_{\alpha,\beta}P_\alpha X P_\beta.
```

\sum_{\alpha,\beta}Use in source · 4 passages

Search aliases: ∑_α,β sum alpha beta

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Use in source · Read passage · s_main.tex, line 88
Orthogonal packet decompositions. Let Hh be a finite-dimensional Hilbert space and let {P_α}_α∈ A be orthogonal projectors satisfying P_α P_β=δ_αβP_α, ∑_α∈ AP_α=I. Every state and operator decompose as ketψ=∑_α P_αketψ, X=∑_α,βP_α X P_β. The diagonal blocks act within packets. The off-diagonal blocks transfer amplitude or coherence between packets.
Exact mathematical passage
```
\ket\psi=\sum_\alpha P_\alpha\ket\psi,
\qquad
X=\sum_{\alpha,\beta}P_\alpha X P_\beta.
```

\sum_{\alpha\in A}Use in source · 1 passage

Search aliases: ∑_α∈ A sum alpha in

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Use in source · Read passage · s_main.tex, line 82
Orthogonal packet decompositions. Let Hh be a finite-dimensional Hilbert space and let {P_α}_α∈ A be orthogonal projectors satisfying P_α P_β=δ_αβP_α, ∑_α∈ AP_α=I. Every state and operator decompose as ketψ=∑_α P_αketψ, X=∑_α,βP_α X P_β. The diagonal blocks act within packets. The off-diagonal blocks transfer amplitude or coherence between packets.
Exact mathematical passage
```
P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,
\qquad
\sum_{\alpha\in A}P_\alpha=I.
```

\sum_{\alpha\in A_t}Use in source · 1 passage

Search aliases: ∑_α∈ A_t sum alpha in

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Use in source · Read passage · s_main.tex, line 1857
Hybrid transfer graphs. Suppose packets are nodes of a sparse graph and each node carries an internal representation. A gate may update a node internally or transfer a compact component to neighboring nodes. The simulation cost depends on ∑_α∈ A_tsize(r_α) + #{nonzero transfers at time t}. This model combines symmetry selection rules with compact state classes.
Exact mathematical passage
```
\sum_{\alpha\in A_t}\operatorname{size}(r_\alpha)
+
\#\{\text{nonzero transfers at time }t\}.
```

\sum_{\alpha\in S_t}Use in source · 2 passages

Search aliases: ∑_α∈ S_t sum alpha in

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Use in source · Read passage · s_main.tex, line 557
[Reachable packet-graph collapse] Suppose the packet sets S_t and nonzero transfer blocks are computable in polynomial time, and D_t = ∑_α∈ S_trank P_α ≤poly(n) for every t. If all active block updates and the final measurement are efficiently computable, then the circuit family is classically polynomial-time simulable.
Exact mathematical passage
```
D_t
=
\sum_{\alpha\in S_t}\rank P_\alpha
\le\poly(n)
```

\sum_{\alpha\in\Lambda}Use in source · 1 passage

Search aliases: ∑_α∈Λ sum alpha in Lambda

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Use in source · Read passage · s_main.tex, line 536
Sparse packet-transfer graphs. Let I=∑_α∈ΛP_α be an orthogonal packet decomposition. Write the gate blocks as G_t;βα=P_β G_tP_α. Starting from an active packet set S_0, define recursively S_t = { β:∃α∈ S_t-1 with G_t;βα≠0 }.
Exact mathematical passage
```
I=\sum_{\alpha\in\Lambda}P_\alpha
```

\sum_\lambdaUse in source · 10 passages

Search aliases: ∑_λ sum lambda

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Use in source · Read passage · g_main.tex, line 489
Why eigenvalues alone are insufficient. If A_S=∑_λλΠ_λ, then G_S,g(z) = ∑_λ frac⟨δ_g,Π_λδ_e⟩1-λ z. The eigenvalues and multiplicities do not determine the endpoint coefficients ⟨δ_g,Π_λδ_e⟩. Two Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.
Exact mathematical passage
```
A_S=\sum_\lambda\lambda\Pi_\lambda,
```

\sum_{\lambda}Use in source · 1 passage

Search aliases: ∑_λ sum lambda

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Use in source · Read passage · g_main.tex, line 4546
Spherical packets. Let T_1,…,T_m be commuting normal operators on a finite-dimensional Hilbert space. Their unique coarsest joint spectral resolution is T_j=∑_λt_j(λ)B_λ, ∑_λ B_λ=I. The tuple λ=(t_1(λ),…,t_m(λ)) is the spherical label. The abstract spectral packet records these labels and the ranks of B_λ.
Exact mathematical passage
```
T_j=\sum_{\lambda}t_j(\lambda)B_\lambda,
\qquad
\sum_\lambda B_\lambda=I.
```

\sum_\muUse in source · 1 passage

Search aliases: ∑_μ sum mu

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Use in source · Read passage · s_main.tex, line 1654
Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.
Exact mathematical passage
```
\Phi(\rho)
=
\sum_\mu K_\mu\rho K_\mu^\dagger.
```

\sum_\piUse in source · 1 passage

Search aliases: ∑_π sum pi

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Use in source · Read passage · s_main.tex, line 1077
Using cyclicity of the trace and U_π^*AU_π=A, Tr(A T_n(ρ)) = frac1n!∑_π Tr(AU_πρ U_π^*) = frac1n!∑_π Tr(U_π^*AU_πρ) = Tr(Aρ).
Exact mathematical passage
```
\Tr\bigl(A\mathcal T_n(\rho)\bigr)
=
\frac1{n!}\sum_\pi
\Tr(AU_\pi\rho U_\pi^*)
=
\frac1{n!}\sum_\pi
\Tr(U_\pi^*AU_\pi\rho)
=
\Tr(A\rho).
```

\sum_{\pi\in S_n}Use in source · 1 passage

Search aliases: ∑_π∈ S_n sum pi in

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Use in source · Read passage · s_main.tex, line 1054
The permutation twirl. Define the twirling channel T_n(X) = frac1n! ∑_π∈ S_n U_π XU_π^*. Its image is the commutant of the permutation action.
Exact mathematical passage
```
\mathcal T_n(X)
=
\frac1{n!}
\sum_{\pi\in S_n}
U_\pi XU_\pi^*.
\label{s:eq:permutation-twirl}
```

\sum_{\tau\in\widehat H}Use in source · 1 passage

Search aliases: ∑_τ∈widehat H sum tau in widehat

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Use in source · Read passage · g_main.tex, line 519
Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint series after Fourier inversion.
Exact mathematical passage
```
\mu_k(h)
=
\frac1{|H|}
\sum_{\tau\in\widehat H}
\dim(\tau)
\operatorname{tr}
\bigl(
\tau(h^{-1})\widehat\mu_k(\tau)
\bigr).
```

\sum_\chiUse in source · 1 passage

Search aliases: ∑_χ sum chi

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Use in source · Read passage · s_main.tex, line 167
Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).
Exact mathematical passage
```
P_\chi P_\psi=\delta_{\chi\psi}P_\chi,
\qquad
\sum_\chi P_\chi=I
```

\sum_{\ell=0}^{M-1}Use in source · 2 passages

Search aliases: ∑_ℓ=0^M-1 sum ell

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Use in source · Read passage · s_appendix.tex, line 6
Residue projector formula. Let Qketψ_k=kketψ_k. Then frac1M∑_ℓ=0^M-1 ω_M^-rℓe^2π iℓ Q/Mketψ_k = ( frac1M∑_ℓ=0^M-1 ω_M^(k-r)ℓ )ketψ_k. The bracket equals one when k≡ rpmod M and zero otherwise. This proves [source reference] and the orthogonality of the residue packets.
Exact mathematical passage
```
\frac1M\sum_{\ell=0}^{M-1}
\omega_M^{-r\ell}e^{2\pi i\ell Q/M}\ket{\psi_k}
=
\left(
\frac1M\sum_{\ell=0}^{M-1}
\omega_M^{(k-r)\ell}
\right)\ket{\psi_k}.
```

\sum_{\ell=0}^{N-1}Use in source · 2 passages

Search aliases: ∑_ℓ=0^N-1 sum ell

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Use in source · Read passage · q_main.tex, line 414
E_a=frac1N∑_ℓ=0^N-1ζ_N^aℓA^ℓ
Exact mathematical passage
```
E_a=\frac1N\sum_{\ell=0}^{N-1}\zeta_N^{a\ell}A^\ell
```

\capUse in source · 3 passages

Search aliases: ∩ cap

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Use in source · Read passage · g_main.tex, line 235
rangle. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry is the left Schreier graph G/H, with edges gH⟼ sgH. For the base-fiber assertion below, assume also S_H= S_G∩ H. This last condition excludes additional ambient primitives that already lie in H but have no equally cheap restricted realization.
Exact mathematical passage
```
\mathcal S_H=\mathcal S_G\cap H
```

\intUse in source · 1 passage

Search aliases: ∫ int

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Use in source · Read passage · s_main.tex, line 351
Define F(s)=e^-i(t-s)HPe^-isHP. Then F(0)=e^-itHP and F(t)=Pe^-itHP. Differentiation gives F'(s)=ie^-i(t-s)H[H,P]e^-isHP. Therefore (I-P)e^-itHP =-(I-P)∫_0^tF'(s) ds. The unitary factors and projectors have norm at most one, so the integral is bounded by |t| [H,P] .
Exact mathematical passage
```
(I-P)e^{-itH}P
=-(I-P)\int_0^tF'(s)\,ds.
```

\int_0^tUse in source · 1 passage

Search aliases: ∫_0^t int

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Use in source · Read passage · s_main.tex, line 351
Define F(s)=e^-i(t-s)HPe^-isHP. Then F(0)=e^-itHP and F(t)=Pe^-itHP. Differentiation gives F'(s)=ie^-i(t-s)H[H,P]e^-isHP. Therefore (I-P)e^-itHP =-(I-P)∫_0^tF'(s) ds. The unitary factors and projectors have norm at most one, so the integral is bounded by |t| [H,P] .
Exact mathematical passage
```
(I-P)e^{-itH}P
=-(I-P)\int_0^tF'(s)\,ds.
```

\simUse in source · 1 passage

Search aliases: ∼ sim

Link to this entry ·

Use in source · Read passage · s_main.tex, line 871
Half filling. At half filling, binom nlfloor n/2rfloor ∼ frac2^nsqrtπ n/2. The packet is smaller by a polynomial factor, but it remains exponential. At n=20, the half-filled packet contains 184,756 amplitudes. The archived repeated-application gain is approximately 7.396, while the one-shot gain is 0.6529, meaning the packet path was slower in that single-run test after compilation overhead.
Exact mathematical passage
```
\binom n{\lfloor n/2\rfloor}
\sim
\frac{2^n}{\sqrt{\pi n/2}}.
```

\congUse in source · 4 passages

Search aliases: ≅ cong

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1766
Over F_2, write x=e_0, p=e_1, q=e_2. The canonical ambient word uses the six nonzero vectors x, p, q, p+q, p+q+x, p+x inside Span(x,p,q)≅F_2^3. The only missing nonzero vector is x+q.
Exact mathematical passage
```
\Span(x,p,q)\cong\F_2^3
```

\neqUse in source · 8 passages

Search aliases: ≠ neq

Link to this entry ·

Use in source · Read passage · s_main.tex, line 96
Define the packet-transfer graph of an operator X by placing an edge β→α whenever P_α X P_β≠0. An exactly symmetry-preserving operator has only diagonal edges. A sparse transfer graph may still permit efficient simulation if the active packets and internal states remain compact.
Exact mathematical passage
```
P_\alpha X P_\beta\neq0.
```

\equivUse in source · 4 passages

Search aliases: ≡ equiv

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Use in source · Read passage · q_main.tex, line 565
Character orthogonality makes it a_n when n≡ rpmod L and zero otherwise. This is precisely the coefficient of R_rf. Each coefficient position belongs to a unique lane, so the projector and completeness identities follow coefficientwise. The equality is formal, with no analytic convergence hypothesis. square
Exact mathematical passage
```
n\equiv r\pmod L
```

\leqUse in source · 1 passage

Search aliases: ≤ leq

Link to this entry ·

Use in source · Read passage · q_visual.tex, line 226
ove those formulas universally. For general q, character orthogonality gives widehatc_q(k)=q1_gcd(k,q)=1, ∑_n=0^q-1|c_q(n)|^2=qφ(q). The associated primitive-order projector has rank φ(q). The general mathematical conclusions follow from the finite-sum derivations in the main paper. The source verifier separately checks its declared finite range 2≤ q≤64.
Exact mathematical passage
```
2\leq q\leq64
```

\geqUse in source · 1 passage

Search aliases: ≥ geq

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Use in source · Read passage · q_visual.tex, line 51
Why the phase sum is an integer. addcontentslinetocsectionWhy the phase sum is an integer leadA divisor identity certifies integrality without rounding trigonometric values. For q≥1, insert the M obius coprimality identity into the finite phase sum: 1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d). Write a=db after interchanging the finite sums. Character orthogonality evaluates the inner geometric series: c_q(n) · =∑_dmid qμ(d)∑_b=0^q/d-1e^2π ibn/(q/d)
Exact mathematical passage
```
q\geq1
```

\subsetUse in source · 4 passages

Search aliases: ⊂ subset

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1240
, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0⊂ V_1⊂⋯⊂ V_n-1. An upper-directed gate leaves this fiber of the flag map. A strict defect means that the flag-compatible frame subgraph is not geodesi…
Exact mathematical passage
```
V_0\subset V_1\subset\cdots\subset V_{n-1}.
```

\bigoplusUse in source · 12 passages

Search aliases: ⊕ bigoplus

Link to this entry ·

Use in source · Read passage · c_lean.tex, line 75
Polynomial certificates are not postulated degree oracles. The inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator polyEval(p,x)=⊕_L∈ p∏_i∈ Lx_i by a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a certificate.
Exact mathematical passage
```
\operatorname{polyEval}(p,x)=\bigoplus_{L\in p}\prod_{i\in L}x_i
```

\bigoplus_iUse in source · 1 passage

Search aliases: ⊕_i bigoplus

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Use in source · Read passage · s_main.tex, line 115
[Invariant quantum packets] If a unitary U satisfies [U,Q]=0, then P_iUP_j=0 (i≠ j), and U=⊕_i U_i, U_i=P_iUP_i|_P_iHh. If P_kketψ=ketψ, then P_kUketψ=Uketψ.
Exact mathematical passage
```
U=\bigoplus_i U_i,
\qquad
U_i=P_iUP_i\big|_{P_i\Hh}.
```

\bigoplus_{k=0}^{K}Use in source · 1 passage

Search aliases: ⊕_k=0^K bigoplus

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Use in source · Read passage · s_main.tex, line 579
[Bounded-excitation collapse] Consider n qubits and suppose the initial state and every circuit prefix are certified to lie in Hh_≤ K = ⊕_k=0^KHh_n,k for a fixed constant K. If initialization, the gate actions, and the measurement are computable in polynomial time in the occupation basis, then the circuit family lies in P, even when individual gates change excitation number.
Exact mathematical passage
```
\Hh_{\le K}
=
\bigoplus_{k=0}^{K}\Hh_{n,k}
```

\bigoplus_{k=0}^nUse in source · 1 passage

Search aliases: ⊕_k=0^n bigoplus

Link to this entry ·

Use in source · Read passage · s_main.tex, line 736
Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.
Exact mathematical passage
```
(\C^2)^{\otimes n}
=
\bigoplus_{k=0}^n\Hh_{n,k}
```

\bigoplus_{L\in p}Use in source · 1 passage

Search aliases: ⊕_L∈ p bigoplus in

Link to this entry ·

Use in source · Read passage · c_lean.tex, line 75
Polynomial certificates are not postulated degree oracles. The inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator polyEval(p,x)=⊕_L∈ p∏_i∈ Lx_i by a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a certificate.
Exact mathematical passage
```
\operatorname{polyEval}(p,x)=\bigoplus_{L\in p}\prod_{i\in L}x_i
```

\bigoplus_qUse in source · 1 passage

Search aliases: ⊕_q bigoplus

Link to this entry ·

Use in source · Read passage · g_main.tex, line 4328
The Hilbert space decomposes as H=⊕_q H_q. A restricted circuit is block diagonal. An ambient catalyst may couple sectors, use the enlarged route, and decouple them at the endpoint.
Exact mathematical passage
```
\mathcal H=\bigoplus_q\mathcal H_q.
```

\bigoplus_{S\subseteq[n]}Use in source · 1 passage

Search aliases: ⊕_S⊆[n] bigoplus subseteq

Link to this entry ·

Use in source · Read passage · g_main.tex, line 3712
Algebraic normal form. Every Boolean coordinate function has a unique algebraic normal form f_i(x) = ⊕_S⊆[n]a_i,S∏_j∈ Sx_j. Affine maps have degree at most one. A Toffoli gate targeting t with controls a,b performs x_t⟼ x_t+x_ax_b, which can raise the degree of later coordinates. The final affine target requires every higher-degree coefficient to cancel.
Exact mathematical passage
```
f_i(x)
=
\bigoplus_{S\subseteq[n]}a_{i,S}\prod_{j\in S}x_j.
```

\bigoplus_\lambdaUse in source · 4 passages

Search aliases: ⊕_λ bigoplus lambda

Link to this entry ·

Use in source · Read passage · s_main.tex, line 995
Decomposition For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.
Exact mathematical passage
```
A
=
\bigoplus_\lambda
A_\lambda\otimes I_{\mathcal V_\lambda}.
```

\bigoplus_{\lambda\vdash n,\,\ell(\lambda)\le d}Use in source · 2 passages

Search aliases: ⊕_λvdash n, ℓ(λ)≤ d bigoplus lambda vdash ell lambda le

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Use in source · Read passage · s_main.tex, line 988
Decomposition. For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.
Exact mathematical passage
```
(\C^d)^{\otimes n}
\cong
\bigoplus_{\lambda\vdash n,\,\ell(\lambda)\le d}
\mathcal U_\lambda\otimes\mathcal V_\lambda,
```

\bigcupUse in source · 2 passages

Search aliases: ⋃ bigcup

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2418
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\mathcal R^*_{n,\le L}
=
\U_n\cap
\bigcup_{\sigma\in\{(t,s):t\ne s\}^L}
\operatorname{im}\Phi_\sigma
```

\bigcup_{s<t}Use in source · 1 passage

Search aliases: ⋃_s<t bigcup

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2428
Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.
Exact mathematical passage
```
X_*
=
\bigcup_{t\ne s}\{I+aE_{t,s}:a\in K\},
\qquad
X_\downarrow
=
\bigcup_{s<t}\{I+aE_{t,s}:a\in K\}.
```

\bigcup_{t\ne s}Use in source · 1 passage

Search aliases: ⋃_t≠ s bigcup ne

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Use in source · Read passage · g_main.tex, line 2428
Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.
Exact mathematical passage
```
X_*
=
\bigcup_{t\ne s}\{I+aE_{t,s}:a\in K\},
\qquad
X_\downarrow
=
\bigcup_{s<t}\{I+aE_{t,s}:a\in K\}.
```

\bigcup_{\sigma\in\{(t,s):t\ne s\}^L}Use in source · 1 passage

Search aliases: ⋃_σ∈{(t,s):t≠ s}^L bigcup sigma in ne

Link to this entry ·

Use in source · Read passage · g_main.tex, line 2418
For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.
Exact mathematical passage
```
\mathcal R^*_{n,\le L}
=
\U_n\cap
\bigcup_{\sigma\in\{(t,s):t\ne s\}^L}
\operatorname{im}\Phi_\sigma
```

\langleUse in source · 27 passages

Search aliases: ⟨ langle

Link to this entry ·

Use in source · Read passage · g_main.tex, line 224
Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry is the left Schreier graph G/H, with edges gH⟼ sgH. For the base-fiber assertion below, assume also S_H= S_G∩ H. This last condition excludes addition…
Exact mathematical passage
```
H=\langle\mathcal S_H\rangle
\le
G=\langle\mathcal S_G\rangle.
```

\rangleUse in source · 32 passages

Search aliases: ⟩ rangle

Link to this entry ·

Use in source · Read passage · g_main.tex, line 224
Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry is the left Schreier graph G/H, with edges gH⟼ sgH. For the base-fiber assertion below, assume also S_H= S_G∩ H. This last condition excludes addition…
Exact mathematical passage
```
H=\langle\mathcal S_H\rangle
\le
G=\langle\mathcal S_G\rangle.
```

\longrightarrowUse in source · 4 passages

Search aliases: ⟶ longrightarrow

Link to this entry ·

Use in source · Read passage · g_main.tex, line 1254
Set G=Egrp_n(R), H=U_n(R). Consider the right Schreier graph on the left cosets Hbackslash G: a generator s∈ S_n^*(R) gives an edge Hg⟶ Hgs. A word representing h∈ H traces a based loop at the coset H. Its label product is the holonomy h.
Exact mathematical passage
```
Hg\longrightarrow Hgs.
```

\longmapstoUse in source · 13 passages

Search aliases: ⟼ longmapsto

Link to this entry ·

Use in source · Read passage · c_main.tex, line 34
A restored oracle is a circuit satisfying (x,0^a,y)⟼(x,0^a,y+f(x)) for every logical input x and both initial target values y∈B. The zero-workspace condition is imposed at the boundary, not at intermediate times. The circuit may disturb logical inputs, use output feedback, reuse intermediate values, and make arbitrary reversible excursions before restoring the required registers.
Exact mathematical passage
```
\label{c:eq:oracle}
(x,0^a,y)\longmapsto(x,0^a,y+f(x))
```

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monographs/quantum-and-circuits/README.md · 0 mathematical passages
monographs/quantum-and-circuits/release_tools/README.md · 0 mathematical passages
monographs/quantum-and-circuits/sources/circuits/Komorebi_Circuit_Obstructions_Research_Edition_1.0/historical_companion/Komorebi_Theorem_Companion_Source/source_snapshot/docs/CLAIMS.md · 0 mathematical passages
monographs/quantum-and-circuits/sources/circuits/Komorebi_Circuit_Obstructions_Research_Edition_1.0/source/komorebi-circuit-obstructions-v1.2.0/docs/CLAIMS.md · 0 mathematical passages
monographs/quantum-and-circuits/volume/README.md · 0 mathematical passages
monographs/quantum-and-circuits/volume/generated/c_lean.tex · 107 mathematical passages
monographs/quantum-and-circuits/volume/generated/c_macros.tex · 0 mathematical passages
monographs/quantum-and-circuits/volume/generated/c_main.tex · 447 mathematical passages
monographs/quantum-and-circuits/volume/generated/c_supplement.tex · 5 mathematical passages
monographs/quantum-and-circuits/volume/generated/c_visual.tex · 328 mathematical passages
monographs/quantum-and-circuits/volume/generated/g_appendix.tex · 89 mathematical passages
monographs/quantum-and-circuits/volume/generated/g_macros.tex · 0 mathematical passages
monographs/quantum-and-circuits/volume/generated/g_main.tex · 1907 mathematical passages
monographs/quantum-and-circuits/volume/generated/glossary.tex · 2 mathematical passages
monographs/quantum-and-circuits/volume/generated/q_lean.tex · 65 mathematical passages
monographs/quantum-and-circuits/volume/generated/q_main.tex · 477 mathematical passages
monographs/quantum-and-circuits/volume/generated/q_visual.tex · 161 mathematical passages
monographs/quantum-and-circuits/volume/generated/references.tex · 6 mathematical passages
monographs/quantum-and-circuits/volume/generated/s_appendix.tex · 49 mathematical passages
monographs/quantum-and-circuits/volume/generated/s_macros.tex · 0 mathematical passages
monographs/quantum-and-circuits/volume/generated/s_main.tex · 568 mathematical passages
monographs/quantum-and-circuits/volume/generated/source_notes.tex · 0 mathematical passages
monographs/quantum-and-circuits/volume/monograph_style.tex · 0 mathematical passages
monographs/quantum-and-circuits/volume/preamble.tex · 0 mathematical passages
monographs/quantum-and-circuits/volume/verification_status.tex · 0 mathematical passages
monographs/quantum-and-circuits/volume/volume.tex · 0 mathematical passages

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