Phase aware n qubit pauli group

The displayed tensor powers are tensor maps on the ordered factors, using Tensor products of finitely many linear maps. Fix an integer $n\ge 1$. Let ${\mathcal{H}}_n=Q_2^{\otimes n}$ be the ordered tensor register of $n$ qubits from Ordered tensor registers, their inner product, and wire permutations, and let ${\mathbb{F}}_2=\mathbb{Z}\mathrm{/}2\mathbb{Z}$. We name the wires $1,\dots ,n$; wire $j$ is position $j-1$ in the source's $0$-based factor list. Bit tuples use the big-endian encoding from Big-endian encoding of ordered bit strings, so the first wire is the most significant bit. Write $X_j$ and $Z_j$ for the tensor maps with the displayed one-qubit matrix on wire $j$ and the identity on every other wire. For $x,z\in {\mathbb{F}}_2^n$, set $X^x={⨂}_{j=1}^n{X}^{x_j}$ and $Z^z={⨂}_{j=1}^n{Z}^{z_j}$, where an exponent $0$ means the identity on that factor and an exponent $1$ means the displayed Pauli matrix. For $k\in \mathbb{Z}\mathrm{/}4\mathbb{Z}$, define

$P(k,x,z):=i^k{X}^x{Z}^z\mathrm{.}$

The phase-aware $n$-qubit Pauli set ${\mathcal{P}}_n$ is the set of these operators with matrix multiplication. Its scalar subset is $Z_n:=\{i^k{I}_{{\mathcal{H}}_n}:k\in \mathbb{Z}\mathrm{/}4\mathbb{Z}\}$. The product-and-adjoint theorem proves that ${\mathcal{P}}_n$ is a group and that $Z_n$ is its center.

The Pauli algebra theorem Pauli matrix algebra and trace identities verifies Y=iXZ, so the one-qubit Y is P(1,1,1). The phase coordinate is modulo 4; the binary label (x,z) alone is used only after quotienting by Z_n.