TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Hermitian Pauli phase criterion

Statement

For $P (k, x, z) = i^{k} X^{x} Z^{z} \in P_{n}$ , the operator is Hermitian exactly when $k \equiv x \cdot z (m o d 2)$ . In that case $P (k, x, z)^{2} = I$ .

Facts & Assumptions

Given: $k \in Z / 4 Z$ , $x, z \in F_{2}^{n}$ , and the phase-aware Pauli convention.

[F1]

Pauli product and adjoint: $P (k, x, z)^{*} = P (2 (x \cdot z) - k, x, z)$ and $P (k, x, z)^{2} = P (2 k + 2 (x \cdot z), 0, 0)$ .

Proof

1.1givenF1algebra

Put $p = x \cdot z \in F_{2}$ and use its representative $0$ or $1$ . By [F1], equality $P^{*} = P$ is equivalent to $2 p - k \equiv k (m o d 4)$ , or $2 (k - p) \equiv 0 (m o d 4)$ , which holds exactly when $k - p$ is even, that is, $k \equiv p (m o d 2)$ .

2.1givenF1step 1.1algebra∎

The same product formula gives $P^{2} = i^{2 (k + p)} I = (- 1)^{k + p} I$ , which equals $I$ exactly under the parity condition in 1.1. Conversely, if $P$ is Hermitian then 1.1 gives that condition.

Depends on

Used by

Exact generator-tableau simulator for stabilizer circuits Algorithm
Stabilizer encoder by symplectic elimination Algorithm
Pauli measurement branch semantics Definition
Stabilizer code space Definition
Stabilizer subgroup Definition
An anticommuting stabilizer measurement is fair Lemma
Clifford circuits lift binary symplectic transvections Lemma
Deterministic Pauli measurement on a pure stabilizer state Lemma
Random Pauli measurement updates stabilizer generators Lemma
Stabilizer labels form an isotropic space of the same rank Lemma
Tableau row reduction returns the deterministic Pauli sign Lemma

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)
Scott Aaronson and Daniel Gottesman, Improved Simulation of Stabilizer Circuits (standard reference, not scraped)