Binary phase representation of a Pauli string ($P=i^s{⨂}_j{X}^{a_j}{Z}^{b_j}$)

Store a Pauli group element with binary exponents $a_j,b_j$ and a phase exponent $s$ modulo four. This ordered $XZ$ convention uses $2n+2$ bits for $n$ qubits. Since $XZ=-iY$, phase factors are essential when converting to Hermitian Pauli operators and evaluating measurement probabilities.

Modulo scalar phases, the n-qubit Pauli group is the additive space ${\mathbb{F}}_2^{2n}$, with label $(a,b)$ for $X^a{Z}^b$. Multiplication adds labels, and commutation contributes the sign $(-1{)}^{\omega }$. The displayed alternating form is nondegenerate. An isotropic subspace $L$ satisfies $L\subseteq L^{\perp }$, hence $\dim L\le n$; it labels commuting stabilizers.

A subspace $L$ is isotropic when the symplectic pairing vanishes between every pair of its vectors. Its binary Pauli labels therefore commute. Nondegeneracy identifies the ambient space with its dual, so the map to $L^{\ast }$ is surjective and $\dim L^{\perp }=2n-\dim L$. Since $L\subseteq L^{\perp }$, it follows that $\dim L\le n$. This explains the maximum number of independent generators of a stabilizer group.

Conjugation by a Controlled-Z gate leaves the $a$ bits unchanged and updates the displayed $b$ bits and phase, using the old bits. The phase occurs when the extra $Z$ from one line is moved past $X$ on the other line. These constant-size updates support Heisenberg propagation of a Pauli observable through a Clifford circuit.