Phase-aware binary Clifford tableau

Fix the generator order $X_1,\dots ,X_n,Z_1,\dots ,Z_n$ and label order $(x,z)$, with wire indices as in Phase aware n qubit pauli group. For an actual Clifford unitary $U\in {\mathcal{C}}_n$, write each exact generator image uniquely as

$U{X}_j{U}^{\ast }=P(k_j,x_j,z_j),U{Z}_j{U}^{\ast }=P(k_{n+j},x_{n+j},z_{n+j})\mathrm{.}$

Its phase-aware binary Clifford tableau is the pair $(S,p)$, where $S\in M_{2n}({\mathbb{F}}_2)$ has as its ordered columns the $2n$ labels $(x_j,z_j)$ just listed and $p=(k_1,\dots ,k_{2n})\in (\mathbb{Z}\mathrm{/}4\mathbb{Z}{)}^{2n}$. The phase vector retains the exact mod-$4$ generator-image phases; it is not discarded with the binary labels. The columns define the unique linear map on $V_n$ taking the ordered standard basis to those columns. Only pairs arising from a Clifford are called Clifford tableaux; arbitrary binary matrices and phase vectors need not be valid. A scalar global phase on $U$ does not change its tableau.