Stabilizer subgroup

For an ordered register of $n$ qubits, a stabilizer subgroup is an abelian subgroup $S$ of the Hermitian $n$-qubit Pauli operators such that $-I\mathrm{\notin }S$. The binary label of a phase-aware Pauli $P(k,x,z)$ is $(x,z)\in {\mathbb{F}}_2^{2n}$. Define the rank of $S$ by

$\mathrm{rank}(S):={\dim }_{{\mathbb{F}}_2}\mathrm{span}\{(x,z):P(k,x,z)\in S\}\mathrm{.}$

The binary label convention and commutation form are supplied by the cited Pauli items. Stabilizer labels form an isotropic space of the same rank ↗ proves that this label span is isotropic, its dimension is independent of any generating set, and $\mathrm{\mid }S\mathrm{\mid }=2^{\mathrm{rank}(S)}$. Matrix multiplication gives the phase-aware Pauli group operation by Pauli product and adjoint. For $n=0$, the Pauli group is the empty-register convention of Pauli group on the empty register.

De Wolf §20.1, PDF pp. 185–186, and Watrous Lesson 14 §§14.1 and 14.3, printed pp. 403–404 and 415–416 (PDF pp. 409–410 and 421–422), describe commuting Pauli stabilizer groups, independence, and exclusion of $-I$: https://arxiv.org/pdf/1907.09415; https://jhwatrous.github.io/UQIC.pdf.