Phase-aware stabilizer tableau

For $n\ge 0$, $x,z\in {\mathbb{F}}_2^n$, and $k\in \mathbb{Z}\mathrm{/}4$, write $P(k,x,z)=i^k{X}^x{Z}^z$, with the fixed tensor order and $X^x={⨂}_{j<n}{X}^{x_j}$, $Z^z={⨂}_{j<n}{Z}^{z_j}$. A row $(k,x,z)$ is Hermitian when $k\equiv x\cdot z(\mathrm{m}\mathrm{o}\mathrm{d}2)$; it stores the exact Pauli phase as well as its binary label $(x,z)$.

For $n=0$, use the scalar Pauli convention of Pauli group on the empty register; the generator list is empty.

A generator tableau is an ordered list of $n$ Hermitian rows whose Pauli operators commute, have linearly independent labels, and generate a subgroup not containing $-I$. It represents the unique common $+1$ ray when the rows stabilize a pure state. An extended tableau also stores Hermitian destabilizer rows $d_1,\dots ,d_n$ and stabilizer rows $s_1,\dots ,s_n$ with

$\omega (d_i,s_j)={\delta }_{ij},\omega (d_i,d_j)=0,\omega (s_i,s_j)=0.$

The exact phase and symplectic conventions are those of the cited Pauli suppliers. The generator-only simulator uses the $s_j$ rows; the extended pairing is retained as a second representation invariant.

Aaronson–Gottesman §III, PDF pp. 3–5, defines the phase-bit tableau and its destabilizer/stabilizer pairing. De Wolf §20.3, PDF pp. 187–188, gives the generator-only stabilizer representation used by the simulator: https://arxiv.org/pdf/quant-ph/0406196; https://arxiv.org/pdf/1907.09415.