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.

Known-amplitude amplification query cost

Statement

For $0 < a \leq 1$ , the known-amplitude procedure uses $O (a^{- 1 / 2})$ calls to $A$ , $A^{†}$ , and the good-subspace phase operation, with success probability at least $1 / 2$ . State preparation, zero-input reflections, final verification, exact arithmetic, gate synthesis, and any controlled access retain their separately declared costs. The $a = 0$ branch returns fail and is outside this bound.

Facts & Assumptions

Given: The known-amplitude algorithm and the query/gate cost coordinates.

[F1]

Known-amplitude amplification with integer rounding: the selected integer satisfies $k \leq ⌈ 1 / \sqrt{a} ⌉$ , with one optional final verifier.

[F2]

Fixed-round amplitude amplification query count: a fixed $k$ uses $k + 1$ calls to $A$ , $k$ calls to $A^{†}$ , and $k$ calls to each of the good and zero-input phase operations.

[F3]

Known-amplitude success bound: the success probability is at least $\max {a, 1 - a}$ for $0 < a < 1$ , and equals $1$ at $a = 1$ .

[F4]

Quantum query complexity: query cost counts designated oracle calls, while nonoracle gates use separate coordinates.

[F5]

Circuit-family and computation cost model: calls, gates, state preparation, and classical computation are distinct charged resources.

Proof

1.1givenF1F2algebra

For $0 < a < 1$ , [F1] gives $k \leq ⌈ a^{- 1 / 2} ⌉ \leq a^{- 1 / 2} + 1 \leq 2 a^{- 1 / 2}$ . By [F2], at most $k + 1$ forward calls to $A$ , $k$ inverse calls to $A^{†}$ , and $k$ good-phase calls occur. The optional verifier adds at most its one declared call.

2.1F3F4F5step 1.1algebra∎

The count in step 1.1 is $O (a^{- 1 / 2})$ in each of the three named access coordinates. The success lower bound follows from [F3]; at $a = 1$ the direct branch uses one preparation and succeeds, which is also $O (a^{- 1 / 2})$ . By [F4]–[F5], exact selection of $k$ , gate synthesis, and other cost coordinates are not hidden in the query bound.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

45 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)