TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Arbitrary-phase amplification query count

Statement

With $m$ arbitrary-phase iterates, the procedure uses one initial forward $A$ , then $m$ forward $A$ calls, $m$ inverse $A^{†}$ calls, $m$ $S_{0} (ϕ)$ calls, and $m$ $S_{g o o d} (φ)$ calls. It uses one final projective instrument and at most one separately supplied classical verification. Controlled forms are charged only under explicit access, and the outer minus sign is retained under any control.

Facts & Assumptions

Given: The finite integer $m$ , the arbitrary-phase algorithm, and the declared access/cost model.

[F1]

Amplitude amplification with independently chosen phase rotations: $Q = - A S_{0} (ϕ) A^{†} S_{g o o d} (φ)$ is applied exactly $m$ times after one initial preparation.

[F2]

Inverse Unitary Access: inverse calls are distinct declared access operations.

[F3]

Circuit-family and computation cost model and Quantum query complexity: query calls, phase gates, measurements, and classical costs are separate coordinates.

[F4]

Controlled Unitary Access and A Controlled Global Phase Becomes a Relative Control Phase: controlled access is explicit, and the scalar phase of a controlled block is observable relative to the other control branch.

Proof

1.1givenF1F2algebra

The initial preparation in [F1] uses one forward call to $A$ . Each of the $m$ factors of $Q$ contributes exactly one $A$ , one $A^{†}$ , one $S_{0} (ϕ)$ , and one $S_{g o o d} (φ)$ , giving the stated counts. The projective instrument and optional verifier occur once after the loop.

2.1F3F4step 1.1∎

The access and cost conventions in [F3]–[F4] keep any controlled access, phase-gate synthesis, state preparation, and verification in their own coordinates. In particular, replacing $Q$ by $- Q$ inside a controlled call changes the relative control phase and is not a free rewrite.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

49 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

Peter Høyer, On Arbitrary Phases in Quantum Amplitude Amplification (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)