LemmaStatement: AI-adaptedProof: 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.

J-gate basis and exact phase identities

Statement

For every $α, β \in R$ , $J (α)$ is unitary and ${∣ 0 ⟩_{α}, ∣ 1 ⟩_{α}}$ is an orthonormal basis. The exact identities

$J (0) = H, J (0)^{2} = I, J (- β) X = e^{- i β} Z J (β), J (π / 2)^{3} = e^{i π / 4} I$

hold in the fixed phase convention.

Facts & Assumptions

Given: $P_{α} = diag (1, e^{i α})$ , $J (α) = H P_{α}$ , and $∣ b ⟩_{α} = J (α)^{†} ∣ b ⟩$ .

[F1]

The current QC07 draft defines $H = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ and $X = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ in the ordered computational basis (Exact one-qubit gates and rotations). Its complete definition and verification were inspected.

[F2]

The J-gate definition fixes $P_{α} = diag (1, e^{i α})$ , $J (α) = H P_{α}$ , and $∣ b ⟩_{α} = J (α)^{†} ∣ b ⟩$ (J gate and ordered equatorial measurement vectors).

Proof

1.1givenF1F2algebra

Since $∣ e^{i α} ∣ = 1$ , $P_{α}^{†} P_{α} = I$ . Direct multiplication from [F1] gives $H^{†} = H$ and $H^{2} = I$ , so [F2] gives $J (α)^{†} J (α) = P_{α}^{†} H^{†} H P_{α} = I$ . Hence $J (α)^{†}$ maps the orthonormal computational basis to an orthonormal basis.

1.2F1F2algebra

Substituting $α = 0$ in [F2] gives $J (0) = H$ , and therefore $J (0)^{2} = H^{2} = I$ . Matrix multiplication using [F1] and [F2] gives $J (- β) X = \frac{1}{\sqrt{2}} (\begin{array}{cc} e^{- i β} & 1 \\ - e^{- i β} & 1 \end{array})$ and $e^{- i β} Z J (β) = \frac{1}{\sqrt{2}} (\begin{array}{cc} e^{- i β} & 1 \\ - e^{- i β} & 1 \end{array})$ , proving the exact phase identity.

2.1F1F2algebra∎

From [F1] and [F2], $J (π / 2) = \frac{1}{\sqrt{2}} (\begin{array}{cc} 1 & i \\ 1 & - i \end{array})$ . Multiplying this matrix three times gives $J (π / 2)^{3} = \frac{1 + i}{\sqrt{2}} I = e^{i π / 4} I$ . This phase will be retained in the transport normalization.

Depends on

Used by

Dependency tree · two levels

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Sources

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415, §20.6 (standard reference, not scraped)
Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel, Measurement-based quantum computation on cluster states, arXiv:quant-ph/0301052, §§II B and II E (standard reference, not scraped)