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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Clifford normalizer

Definition

Work with actual unitary matrices, not equivalence classes modulo global phase. The $n$ -qubit Clifford normalizer is

$C_{n} : = {U unitary on H_{n} : U P_{n} U^{*} = P_{n}} .$

It is a group under matrix multiplication. Every scalar unitary $λ I$ with $∣ λ ∣ = 1$ belongs to $C_{n}$ and has the same conjugation action as $I$ ; the chosen convention retains these distinct matrices.

Facts & Assumptions

Given: the finite-dimensional inner-product space $H_{n}$ and the phase-aware Pauli group $P_{n}$ .

[F1]

Pauli product and adjoint: $P_{n}$ is a group under matrix multiplication.

[F2]

Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces: a unitary operator is an invertible linear isometry and its inverse is its adjoint.

Proof

1.1givenF1algebra

The identity conjugates $P_{n}$ to itself. If $U, V \in C_{n}$ , then $(U V) P_{n} (U V)^{*} = U (V P_{n} V^{*}) U^{*} = U P_{n} U^{*} = P_{n}$ , so $U V \in C_{n}$ .

2.1givenF1F2step 1.1algebra∎

If $U P_{n} U^{*} = P_{n}$ , multiplying this equality by $U^{*}$ on the left and $U$ on the right gives $U^{*} P_{n} U = P_{n}$ , so $U^{*} \in C_{n}$ by [F2]. Thus the set is a group. For $∣ λ ∣ = 1$ , $(λ I) P (λ I)^{*} = P$ for every $P$ , and conjugation by $λ U$ equals conjugation by $U$ .

Depends on

Used by

Phase-aware binary Clifford tableau Definition
Three-Ising-anyon braid matrices are Clifford Example
Clifford circuits lift binary symplectic transvections Lemma
Clifford induces symplectic map Theorem
Clifford to binary symplectic action is surjective Theorem
Even-Majorana Ising exchanges normalize logical Paulis Theorem

Dependency tree · two levels

16 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)
John Watrous, Understanding Quantum Information and Computation, Lesson 14 (standard reference, not scraped)