Quantum and Circuits: Gate matrices and symbolic conventions

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C.1 Gate matrices and symbolic conventions

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C.1.1 Row-transvection convention

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Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection

\[ \tau_{t,s}(a)=I+aE_{t,s} \]

performs

\[ R_t\leftarrow R_t+aR_s. \]

In the binary CNOT interpretation, \(s\) is the control and \(t\) is the target.

The canonical six-row target is

\[ C= \begin{pmatrix} 1&0&0&0&0&0\\ 1&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&1&1&1&0&0\\ 1&1&1&0&1&0\\ 1&1&1&0&0&1 \end{pmatrix}. \]

One ambient word is

\[ \tau_{1,2}(1), \tau_{3,1}(1), \tau_{1,0}(1), \tau_{4,1}(1), \tau_{5,1}(1), \tau_{1,2}(-1). \]

A restricted word of length seven is

\[ \tau_{1,0}(1), \tau_{2,1}(1), \tau_{3,0}(-1), \tau_{3,2}(1), \tau_{4,2}(1), \tau_{5,2}(1), \tau_{2,1}(-1). \]

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C.1.2 Qudit SUM gate

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On computational basis labels in \({\mathbb Z}_d\),

\[ {\operatorname{SUM}}^{(a)}_{s\to t} {\lvert x_s,x_t\rangle} = {\lvert x_s,x_t+a x_s\bmod d\rangle}. \]

Its classical linear action is the transvection \(I+aE_{t,s}\) over \({\mathbb Z}_d\).

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C.1.3 Binary symplectic form

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Pauli exponents are ordered as

\[ (x_0,\ldots,x_{n-1},z_0,\ldots,z_{n-1}). \]

The symplectic form is

\[ J= \begin{pmatrix} 0&I_n\\ I_n&0 \end{pmatrix} \]

over \({\mathbb F}_2\). A matrix \(M\) is symplectic when

\[ M^TJM=J. \]

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C.1.4 Clifford generator matrices

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For qubit \(i\), the Hadamard swaps \(x_i\) and \(z_i\):

\[ H_i:e_i\leftrightarrow e_{n+i}. \]

The phase gate has

\[ z_i\leftarrow z_i+x_i. \]

For CNOT with control \(c\) and target \(t\),

\[ x_t\leftarrow x_t+x_c, \qquad z_c\leftarrow z_c+z_t. \]

For CZ on \(i,j\),

\[ z_i\leftarrow z_i+x_j, \qquad z_j\leftarrow z_j+x_i. \]

SWAP exchanges both \(x\) and \(z\) coordinates of the two qubits.

The parabolic subgroup used in the census is generated by \(S_i\), \({\operatorname{CZ}}_{ij}\), and \({\operatorname{CX}}_{i\to j}\). It stabilizes the \(Z\) Lagrangian

\[ L_Z={\operatorname{span}}(e_n,\ldots,e_{2n-1}). \]

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C.1.5 Two-qubit Clifford witness

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The target symplectic matrix is

\[ M_2= \begin{pmatrix} 0&1&0&0\\ 1&0&0&0\\ 1&1&0&1\\ 1&1&1&0 \end{pmatrix}. \]

A restricted path is

\[ {\operatorname{CX}}_{0\to1},\ S_1,\ {\operatorname{CX}}_{1\to0},\ {\operatorname{CX}}_{0\to1}, \]

with three charged entangling gates. An ambient path is

\[ H_1,\ {\operatorname{CX}}_{1\to0},\ {\operatorname{CX}}_{0\to1},\ H_0,\ S_1,\ S_0, \]

with two charged entangling gates when \(H,S\) are free.

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C.1.6 Three-qubit Clifford witness

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The target symplectic matrix is

\[ M_3= \begin{pmatrix} 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 1&0&0&0&0&0\\ 1&0&1&0&1&0\\ 0&1&0&0&0&1\\ 0&1&0&1&0&0 \end{pmatrix}. \]

A restricted total-count path is

\[ {\operatorname{CX}}_{0\to2},\ {\operatorname{CZ}}_{1,2},\ {\operatorname{CX}}_{2\to0},\ {\operatorname{CX}}_{0\to2}, \ {\operatorname{CX}}_{0\to1},\ {\operatorname{CX}}_{1\to0},\ {\operatorname{CX}}_{0\to1}, \]

while an ambient path is

\[ H_1,\ {\operatorname{CX}}_{1\to2},\ {\operatorname{CX}}_{2\to1},\ {\operatorname{CX}}_{2\to0},\ {\operatorname{CX}}_{0\to2},\ H_0. \]

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C.1.7 Classical reversible gates

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For a bit string \(x\in{\mathbb F}_2^n\):

\[ X_i:x_i\leftarrow x_i+1, \]

\[ {\operatorname{CX}}_{c\to t}:x_t\leftarrow x_t+x_c, \]

\[ {\operatorname{TOF}}_{a,b\to t}:x_t\leftarrow x_t+x_ax_b. \]

All other coordinates are unchanged. The first two generate \(\operatorname{AGL}(n,2)\). On three bits, all three generate the full permutation group \(S_8\). For \(n\ge4\) without ancillas they are all even permutations, so the generated group is contained in \(A_{2^n}\) and is not the full symmetric group.

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C.1.8 Bosonic symplectic embedding

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For \(A\in{\operatorname{GL}}(n,{\mathbb R})\),

\[ \iota(A)={\operatorname{diag}}(A,A^{-T})\in{\operatorname{Sp}}(2n,{\mathbb R}). \]

This convention orders quadratures as \((q_1,\ldots,q_n,p_1,\ldots,p_n)\).

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C.1.9 Pauli rotations

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For a Hermitian Pauli \(P\),

\[ R_P(\theta)=e^{-i\theta P/2}. \]

Clifford conjugation gives

\[ C R_P(\theta) C^\dagger=R_{CPC^\dagger}(\theta). \]

The sign of \(CPC^\dagger\) can be absorbed by changing \(\theta\) to \(-\theta\).

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C.1.10 Tensor and direct-sum conventions

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For linear maps, \(A\oplus B\) denotes a block-diagonal direct sum. The Kronecker product \(A\otimes B\) uses lexicographic ordering. In quantum chapters, \(U\otimes V\) denotes the physical tensor product on disjoint systems. These operations have different dimension and cost normalizations and are not interchanged silently.

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