DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge 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.

Global Phase Equivalence

Definition

For square matrices $U, V$ of the same size, write $U \sim_{p h a s e} V$ when $U = λ V$ for some $λ \in C$ with $∣ λ ∣ = 1$ . This relation is kept distinct from exact matrix equality.

Facts & Assumptions

Given: Square complex matrices $U, V, W$ of the same size.

Proof

1.1givenalgebra

Taking $λ = 1$ gives $U = 1 \cdot U$ , so $U \sim_{p h a s e} U$ .

1.2givenalgebra

If $U = λ V$ and $∣ λ ∣ = 1$ , then $V = \overline{λ} U$ because $λ \overline{λ} = 1$ ; also $∣ \overline{λ} ∣ = 1$ . Hence $U \sim_{p h a s e} V$ implies $V \sim_{p h a s e} U$ .

2.1step 1.1step 1.2algebra

If $U = λ V$ and $V = μ W$ , then $U = (λ μ) W$ and $∣ λ μ ∣ = 1$ . Thus $U \sim_{p h a s e} V$ and $V \sim_{p h a s e} W$ imply $U \sim_{p h a s e} W$ . This proves transitivity; together the three properties establish an equivalence relation.

3.1givenstep 2.1algebra∎

The $1 \times 1$ matrices $[1]$ and $[- 1]$ are phase-equivalent by the scalar $- 1$ , but they are not equal because their entries differ. Thus phase equivalence is distinct from exact equality.

Used by

Ancilla-free Rz approximation problem and T-count Definition
Circuit Optimization Task Definition
Clifford+T gate library and exact matrices Definition
Determinant-one Clifford+T words are dense in SU(2) Lemma
Global Phase Under Control Lemma
Two-phase reduction for Clifford+T Rz synthesis up to phase Theorem
Two-qubit determinant obstruction to controlled T Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)
Adriano Barenco et al., Elementary gates for quantum computation, arXiv:quant-ph/9503016v1 (standard reference, not scraped)