TheoremStatement: 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.

Basic Rewrite Correct And Terminating

Statement

Each registered rewrite preserves the exact matrix; the procedure terminates and returns a local normal form relative only to the declared finite rewrite set. It is not claimed globally minimal.

Facts & Assumptions

Given: a gate word and the fixed finite rewrite list in the algorithm item.

[F1]

Basic Clifford T Rewrite fixes the finite rewrite list and requires identical ordered supports for adjacent gate cancellations.

[F2]

Clifford+T gate library and exact matrices gives $H = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ , $T = diag (1, e^{i π / 4})$ , $T^{†} = T^{- 1}$ , and $CNOT ∣ a, b ⟩ = ∣ a, b \oplus a ⟩$ on ordered wires.

Proof

1.1F1F2givenalgebra

By [F2], $H^{2} = I$ by direct matrix multiplication, $T^{8} = diag (1, e^{2 π i}) = I$ , and $T T^{†} = T^{†} T = I$ . On ordered basis input $(a, b)$ , two consecutive same-support CNOTs send the target bit to $b \oplus a \oplus a = b$ , so ${CNOT}^{2} = I$ . Tensoring each identity with the identity on unaffected wires proves that every rule in [F1] preserves the whole circuit matrix.

2.1F1step 1.1algebra∎

Each rewrite removes at least two gates and inserts none. Thus after at most half the input length many rewrites, the deterministic scan halts. At halt it has found no match, so the output is locally irreducible for the displayed rule list. This proves neither global minimality nor uniqueness of the local normal form.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Barenco et al., Elementary gates for quantum computation (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)