Quantum and Circuits

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1 \section{Proof details and auxiliary derivations}
2 \label{s:app:proofs}
3 
4 \subsection{Residue projector formula}
5 Let $Q\ket{\psi_k}=k\ket{\psi_k}$. Then
6 \[
7 \frac1M\sum_{\ell=0}^{M-1}
8 \omega_M^{-r\ell}e^{2\pi i\ell Q/M}\ket{\psi_k}
9 =
10 \left(
11 \frac1M\sum_{\ell=0}^{M-1}
12 \omega_M^{(k-r)\ell}
13 \right)\ket{\psi_k}.
14 \]
15 The bracket equals one when $k\equiv r\pmod M$ and zero otherwise. This proves \cref{s:eq:residue-projector} and the orthogonality of the residue packets.
16 
17 \subsection{Circuit leakage telescoping}
18 Let $V_j=PG_jP$. Insert and subtract one factor at a time:
19 \[
20 G_m\cdots G_1P-V_m\cdots V_1P
21 =
22 \sum_{k=1}^m
23 G_m\cdots G_{k+1}(G_k-V_k)V_{k-1}\cdots V_1P.
24 \]
25 Every $G_j$ has norm one and every $V_j$ has norm at most one. On $P\Hh$,
26 \[
27 (G_k-V_k)P=(I-P)G_kP.
28 \]
29 Taking norms gives \cref{s:prop:sum-leakage}.
30 
31 \subsection{Hypercube quotient coupling}
32 Let
33 \[
34 \ket k
35 =
36 \binom nk^{-1/2}
37 \sum_{|x|=k}\ket x.
38 \]
39 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
40 \[
41 \bra{k+1}A\ket k
42 =
43 \frac{\binom nk(n-k)}
44 {\sqrt{\binom nk\binom n{k+1}}}
45 =
46 \sqrt{(k+1)(n-k)},
47 \]
48 proving \cref{s:eq:hypercube-coupling}.
49 
50 \subsection{Symmetric-subspace dimension}
51 A symmetric basis state of $n$ qudits is indexed by occupation numbers
52 \[
53 (n_1,\ldots,n_d),
54 \qquad
55 n_1+\cdots+n_d=n.
56 \]
57 The stars-and-bars count is
58 \[
59 \binom{n+d-1}{d-1},
60 \]
61 proving \cref{s:eq:symmetric-qudit-dim}.
62 
63 \subsection{Bosonic and fermionic dimensions}
64 Bosonic occupations are weak compositions of $N$ into $m$ parts, giving $\binom{N+m-1}{N}$. Fermionic occupations are $N$-element subsets of $m$ modes, giving $\binom mN$.
65 
66 \subsection{Acceptance-probability stability}
67 If normalized states $\ket\psi,\ket\phi$ satisfy
68 \[
69 \|\ket\psi-\ket\phi\|\le\varepsilon
70 \]
71 and $0\le M\le I$, then
72 \[
73 |\bra\psi M\ket\psi-\bra\phi M\ket\phi|
74 \le 2\varepsilon.
75 \]
76 Indeed, add and subtract $\bra\phi M\ket\psi$ and use Cauchy--Schwarz. This converts state-vector leakage bounds into measurement-error bounds.
77 
78 \subsection{Orbit quotient invariance}
79 For an automorphism group $G$, the adjacency matrix commutes with the permutation representation. The fixed subspace of the action is spanned by normalized orbit states. Therefore adjacency preserves that span. Polynomial-time simulation additionally requires direct computation of the quotient entries.
80 
81 \section{Python API and reproduction commands}
82 \label{s:app:python}
83 
84 The release code is intentionally compact. Its purpose is to verify the algebra and provide a reusable seed rather than to compete with optimized quantum simulators.
85 
86 \subsection{Core API}
87 \begin{lstlisting}[style=pythonstyle,caption={Core packet API.}]
88 from symmetry_packets import (
89     weight_basis,
90     residue_basis,
91     symmetric_subspace_dimension,
92     boson_fixed_number_dimension,
93     fermion_fixed_number_dimension,
94     PacketPlan,
95     rz,
96     xy_exchange,
97     hypercube_quotient,
98     circuit_leakage_bound,
99     hamiltonian_leakage_bound,
100 )
101 \end{lstlisting}
102 
103 \subsection{Exact packet plan}
104 \begin{lstlisting}[style=pythonstyle,caption={Fixed-weight packet compilation.}]
105 plan = PacketPlan.fixed_weight(n=10, k=2)
106 packet_state = random_packet_state(plan, seed=12)
107 full_state = plan.embed(packet_state)
108 
109 restricted_gate = plan.restrict_gate(
110     xy_exchange(theta=0.2),
111     qubits=(3, 4),
112 )
113 packet_state = restricted_gate @ packet_state
114 \end{lstlisting}
115 
116 \subsection{Validation}
117 From the project root, run
118 \begin{lstlisting}[style=pythonstyle]
119 PYTHONPATH=code python3 code/validate.py
120 PYTHONPATH=code python3 code/validate_exact.py
121 PYTHONPATH=code python3 code/validate_boundaries.py
122 \end{lstlisting}
123 The expected output reports successful formula checks and reconstruction errors near machine precision for the representative fixed-weight circuits.
124 
125 \subsection{Figure generation}
126 All figures in the PDF are generated by
127 \begin{lstlisting}[style=pythonstyle]
128 python3 code/make_figures.py
129 \end{lstlisting}
130 The generator writes vector PDF figures into the \texttt{figures} directory.
131 
132 \subsection{Build}
133 The complete rebuild command is
134 \begin{lstlisting}[style=pythonstyle]
135 ./run_all.sh
136 \end{lstlisting}
137 It validates the Python code, regenerates figures, compiles the LaTeX source, and updates the checksums.
138 
139 \section{Benchmark data ledger}
140 \label{s:app:benchmark-ledger}
141 
142 \subsection{Exact weight two}
143 The JSON file \texttt{data/benchmark\_records.json} contains the values in \cref{s:tab:weight-two}, together with the parity, half-filled, and leakage records.
144 
145 \subsection{Half filling}
146 \begin{longtable}{@{}rrrrrrr@{}}
147 \caption{Recorded half-filled benchmark.}\\
148 \toprule
149 $n$ & $k$ & full dim. & sector dim. & apply gain & one-shot gain & memory gain\\
150 \midrule
151 \endfirsthead
152 \toprule
153 $n$ & $k$ & full dim. & sector dim. & apply gain & one-shot gain & memory gain\\
154 \midrule
155 \endhead
156 8 & 4 & 256 & 70 & 1.212 & 1.083 & 3.657\\
157 10 & 5 & 1,024 & 252 & 1.234 & 0.7513 & 4.063\\
158 12 & 6 & 4,096 & 924 & 2.083 & 0.5092 & 4.433\\
159 14 & 7 & 16,384 & 3,432 & 3.24 & 0.2092 & 4.774\\
160 16 & 8 & 65,536 & 12,870 & 7.041 & 1.501 & 5.092\\
161 18 & 9 & 262,144 & 48,620 & 7.757 & 0.7458 & 5.392\\
162 20 & 10 & 1,048,576 & 184,756 & 7.396 & 0.6529 & 5.675\\
163 \bottomrule
164 \end{longtable}
165 
166 All recorded relative errors were zero at the displayed precision. The one-shot result can be below one because compilation dominates a single application.
167 
168 \subsection{Parity}
169 \begin{longtable}{@{}rrrrrr@{}}
170 \caption{Recorded even-parity benchmark.}\\
171 \toprule
172 $n$ & full dim. & packet dim. & apply gain & one-shot gain & memory gain\\
173 \midrule
174 \endfirsthead
175 \toprule
176 $n$ & full dim. & packet dim. & apply gain & one-shot gain & memory gain\\
177 \midrule
178 \endhead
179 8 & 256 & 128 & 1.06 & 1.19 & 2.0\\
180 10 & 1,024 & 512 & 1.15 & 1.33 & 2.0\\
181 12 & 4,096 & 2,048 & 1.49 & 1.37 & 2.0\\
182 14 & 16,384 & 8,192 & 1.78 & 1.75 & 2.0\\
183 16 & 65,536 & 32,768 & 2.0 & 3.7 & 2.0\\
184 \bottomrule
185 \end{longtable}
186 
187 \subsection{Symmetry-breaking controls}
188 \begin{center}
189 \begin{tabular}{@{}lrr@{}}
190 \toprule
191 Test & retained probability & leakage\\
192 \midrule
193 Break exact weight with pair gate & 0.891 & 0.109\\
194 Break parity with $X$ rotation & 0.869 & 0.131\\
195 Break weight modulo four with pair gate & 0.928 & 0.0723\\
196 \bottomrule
197 \end{tabular}
198 \end{center}
199 
200 These controls demonstrate that a packet restriction must be audited after every symmetry-breaking operation.
201 
202 \section{Theorem and evidence-status ledger}
203 \label{s:app:status}
204 
205 \begin{longtable}{@{}L{.31\textwidth}L{.16\textwidth}L{.43\textwidth}@{}}
206 \caption{Claim-status ledger.}\\
207 \toprule
208 Statement & Status & Qualification\\
209 \midrule
210 \endfirsthead
211 \toprule
212 Statement & Status & Qualification\\
213 \midrule
214 \endhead
215 Invariant-basis simulation & Proved & Exact gatewise subspace invariance.\\
216 Polynomial packet simulation & Proved & Requires efficient discovery, updates, measurement, and polynomial representation size.\\
217 Fixed $k$ number-preserving circuits lie in $P$ & Proved corollary & $k$ fixed and local restricted gates efficiently applicable.\\
218 Approximate circuit leakage bounded by sum of gate leakages & Proved & Operator-norm state-error certificate.\\
219 Hamiltonian leakage bounded by $|t|\|[H,P]\|$ & Proved & Bounded self-adjoint Hamiltonian.\\
220 Permutation-symmetric qubits lie in $P$ & Derived theorem & Gate matrices and measurement efficient in Dicke coordinates.\\
221 Fixed-$d$ symmetric qudits lie in $P$ & Derived theorem & Fixed local dimension and efficient occupation updates.\\
222 Polynomial-orbit quantum walks lie in $P$ & Derived theorem & Quotient data computable from succinct input.\\
223 Fixed-particle boson and fermion circuits lie in $P$ & Derived theorem & Particle number fixed and updates efficient.\\
224 Compressed-packet theorem & Proved abstraction & Representation class closed under gates with polynomial size.\\
225 Weight-two runtime and memory gains & Recorded computation & Implementation-specific archival benchmark.\\
226 Half-filling remains exponential & Proved dimension statement & Runtime constants remain implementation dependent.\\
227 Parity gives only a factor-two dimension reduction & Proved dimension statement & Does not preclude further internal compression.\\
228 Quantum Latin order-six product realization absent, entangled realization present & External theorem & Contextual existence defect, not a simulation theorem.\\
229 Adaptive discarded-state error certificate & Proved & Unnormalized trajectories, explicitly accounted discarded norms.\\
230 Universal polynomial packetization & Open conjecture & Would imply $P=BQP$.\\
231 Representation-independent interference lower bound & Open & Required for a full hard-side complexity conclusion.\\
232 \bottomrule
233 \end{longtable}
234 
235 \subsection{Publication discipline}
236 Every extracted theorem paper should declare the circuit model, input encoding, gate precision, measurement, symmetry discovery method, packet representation, error norm, and complexity parameter. Benchmarks should be accompanied by code and frozen data. Open conjectures should not be phrased as consequences of finite experiments.
237 

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