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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Phase gate pauli action

Statement

For a phase gate $S_{j} = diag (1, i)$ on factor $j$ , the binary update is $(x_{j}, z_{j}) \mapsto (x_{j}, z_{j} + x_{j})$ and the phase update is $k \mapsto k + x_{j}$ modulo $4$ . In particular $S X S^{*} = Y$ and $S Z S^{*} = Z$ .

Facts & Assumptions

Given: n is a positive integer, j is a factor index, the phase-aware Pauli convention is fixed, and the register factors are ordered.

[F1]

The Pauli matrices: the explicit X,Y,Z matrices are defined in the ordered computational basis.

[F2]

Pauli matrix algebra and trace identities: Y=iXZ and YZ=iX.

[F3]

Exact one-qubit gates and rotations: S=diag(1,i) and S*=diag(1,-i).

[F4]

Tensor products of finitely many linear maps and Tensor maps preserve composition, identities, and adjoints: tensor maps act factorwise and preserve composition and adjoints.

[F5]

Ordered tensor registers, their inner product, and wire permutations: the register factors have the declared order.

Proof

1.1F1F2F3algebra

Multiplying the matrices gives $S X S^{*} = Y = i X Z$ and $S Z S^{*} = Z$ . Therefore $S Y S^{*} = i (S X S^{*}) (S Z S^{*}) = i Y Z = - X$ , using $Y Z = i X$ .

2.1F1F3F4F5step 1.1algebra∎

On factor $j$ , $S X^{x_{j}} Z^{z_{j}} S^{*} = (i X Z)^{x_{j}} Z^{z_{j}} = i^{x_{j}} X^{x_{j}} Z^{z_{j} + x_{j}}$ ; other tensor factors are unchanged. Thus the total phase increases by $x_{j}$ modulo $4$ and the binary $z_{j}$ coordinate increases by $x_{j}$ modulo $2$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)
John Watrous, Understanding Quantum Information and Computation, Lesson 14 (standard reference, not scraped)
Scott Aaronson and Daniel Gottesman, Improved Simulation of Stabilizer Circuits (standard reference, not scraped)