TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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.

Fixed-round amplitude amplification query count

Statement

For a fixed $k$ -iterate procedure, the displayed factorization of $Q$ uses $k + 1$ forward calls to $A$ , $k$ inverse calls to $A^{†}$ , $k$ good-subspace phase operations, and $k$ zero-input phase operations. It then uses one projective good/bad instrument. If classical output verification is requested, count its supplied verifier separately (one call when the task model assigns one query to verification). Controlled $A$ , state-preparation synthesis, gates, and classical costs are separate access coordinates and are not inferred to be free.

Facts & Assumptions

Given: The fixed- $k$ procedure and access contract.

[F1]

Amplitude amplification with a fixed integer iteration count: one initial preparation $A ∣ 0 ⟩$ , followed by exactly $k$ factors $Q = - A S_{0} A^{†} S_{g o o d}$ , then one final measurement.

[F2]

Quantum query complexity: query cost counts designated calls; gates and other cost coordinates are separate.

[F3]

Circuit-family and computation cost model: named oracle calls, gates, preparation, and classical computation are distinct coordinates, and no operation is silently assigned zero cost.

Proof

1.1givenF1algebra

The initial preparation in [F1] contributes one forward $A$ call. No inverse call occurs before the iterates.

1.2givenF1F2algebra

Each of the $k$ displayed $Q$ -factors contains exactly one $A$ , one $A^{†}$ , one $S_{0}$ , and one $S_{g o o d}$ . Adding these to the initial preparation gives $k + 1$ forward $A$ calls, $k$ inverse $A^{†}$ calls, and $k$ of each phase operation. The final instrument is not an oracle call unless the chosen model explicitly assigns it one.

2.1F2F3step 1.2∎

By [F2]–[F3], any verifier, controlled access, gate synthesis, state-preparation construction, and classical postprocessing retains its own declared charge; none changes the factor counts in step 1.2.

Depends on

Used by

Known-amplitude amplification query cost Theorem

Dependency tree · two levels

28 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

Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp, Quantum Amplitude Amplification and Estimation (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)