Computational basis states have an extended stabilizer tableau

Given: The ordered computational basis of an $n$-qubit register and $x=(x_0,\dots ,x_{n-1})\in {\mathbb{F}}_2^n$.

Finite registers and their ordered computational bases: the ordered tensor register has the lexicographically ordered computational basis; for $n=0$ it is $\mathbb{C}$ with basis $\{1\}$.

Big-endian encoding of ordered bit strings: the first wire is the most significant bit and coordinate $x_j$ occupies factor position $j$, which is Pauli wire $j+1$ by [F9].

Pauli group on the empty register: when $n=0$, the ambient Pauli group is scalar and the empty generator list generates $\{I\}$.

The computational basis is orthonormal: the computational basis of each factor $Q_2$ is orthonormal.

Phase-aware stabilizer tableau: a row $(k,u,v)$ represents $P(k,u,v)=i^k{X}^u{Z}^v$ and row pairings are computed by $\omega$.

Binary symplectic form: $\omega ((u,v),(u^{\mathrm{\prime }},v^{\mathrm{\prime }}))=u\cdot v^{\mathrm{\prime }}+v\cdot u^{\mathrm{\prime }}$.

Binary Pauli quotient isomorphism: a nonzero binary label cannot represent a scalar Pauli.

Pauli matrix algebra and trace identities and The Pauli matrices: on the one-qubit basis, $X\mathrm{\mid }b⟩=\mathrm{\mid }b\oplus 1⟩$ and $Z\mathrm{\mid }b⟩=(-1{)}^b\mathrm{\mid }b⟩$, and both are Hermitian involutions.

Pauli commutator is symplectic pairing: Pauli rows commute exactly when their label pairing is zero.

Phase aware n qubit pauli group: Pauli wire $a\in \{1,\dots ,n\}$ is tensor factor $a-1$; its $X_a$ and $Z_a$ operators act on that factor.

Ordered tensor registers, their inner product, and wire permutations: elementary tensors of factor basis vectors form a basis, and their inner product is the product of factor inner products. For $n=0$ the space is $\mathbb{C}$, with $⟨1,1⟩=1$.