TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-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.

One Grover iterate uses one standard bit query

Statement

One execution of One Grover iterate in the standard bit-query model makes exactly one standard bit-query call, through phase kickback. The response-qubit preparation and the reflection about $∣ s ⟩$ make no oracle calls and are charged separately as gates or explicitly supplied reflection operations.

Facts & Assumptions

Given: The Grover iterate and the worst-case query-count convention.

[F1]

One Grover iterate in the standard bit-query model: one standard bit query is applied to a response qubit in $∣ - ⟩$ , after which the address reflection is applied.

[F2]

Quantum query complexity: the query coordinate counts designated oracle invocations and keeps ordinary gates outside those calls separate.

[F3]

One standard query implements one phase query: one prepared-response standard query induces the phase query and uses exactly one query invocation.

Proof

1.1givenF1F2F3

The algorithm contains one invocation of $U_{f}$ ; [F3] identifies that invocation with the phase mark on the address register. By [F2], this designated oracle invocation contributes one query.

2.1step 1.1F1F2∎

The response preparation and $R_{s}$ are operations outside $U_{f}$ , so they add no oracle invocation. The iterate therefore uses exactly one query, with no gate-cost bound inferred.

Depends on

Used by

Dependency tree · two levels

16 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

Ronald de Wolf, Quantum Computing: Lecture Notes, §7.2 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, §19.1 (standard reference, not scraped)