LemmaStatement: 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.

QC24 Prefix Norm Rotation Identities

Statement

For a prefix $p$ with $0 \leq ∣ p ∣ < n$ and $r_{p} > 0$ , the angle $θ_{p} = 2 atan2 (r_{p 1}, r_{p 0})$ in QC24 State Preparation satisfies

$\cos (θ_{p} / 2) = r_{p 0} / r_{p}, \sin (θ_{p} / 2) = r_{p 1} / r_{p} .$

Facts & Assumptions

Given: A valid normalized amplitude table and a prefix $p$ with $0 \leq ∣ p ∣ < n$ and $r_{p} > 0$ .

[F1]

The procedure defines $r_{p}$ as the square root of the sum of squared amplitudes over strings extending $p$ , and uses $θ_{p} = 2 atan2 (r_{p 1}, r_{p 0})$ (QC24 State Preparation).

Proof

Given: As in Facts & Assumptions.

1.1F1givenalgebra

Every string extending $p$ extends exactly one of $p 0$ or $p 1$ , so splitting the finite sum in the definition of the prefix radii gives $r_{p}^{2} = r_{p 0}^{2} + r_{p 1}^{2}$ . Both child radii are nonnegative and $r_{p} > 0$ , so the pair $(r_{p 0}, r_{p 1})$ is not $(0, 0)$ . By the declared $atan2$ convention, $ϕ = atan2 (r_{p 1}, r_{p 0})$ lies in $[0, π / 2]$ and has cosine $r_{p 0} / r_{p}$ and sine $r_{p 1} / r_{p}$ .

2.1F1step 1.1algebra∎

Because $θ_{p} / 2 = ϕ$ , substituting gives both displayed identities, including the cases where exactly one child radius is zero.

Depends on

QC24 State Preparation

Used by

QC24 State Preparation Rotation Level Upper Bound Theorem

Dependency tree · two levels

11 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

Vivek V. Shende, Stephen S. Bullock, and Igor L. Markov, Synthesis of Quantum Logic Circuits (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 5, Classical and Quantum Circuits (standard reference, not scraped)