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.

Query cost of randomized known-count Grover search

Statement

For $0 < t \leq N / 2$ , the nine-round randomized known-count procedure uses at most $9 K = O (\sqrt{N / t})$ standard bit queries, where $K = ⌈ \sqrt{N / t} ⌉$ ; rejected rounds use no oracle query and each accepted round uses $j \leq K - 1$ iterate queries plus one verification query. For $t > N / 2$ it uses at most nine queries, and for $t = 0$ it uses none. This is a query count only, not a gate or classical preprocessing bound.

Facts & Assumptions

Given: The randomized known-count algorithm, exact $K$ , and worst-case quantum query convention.

[F1]

Known-count search by randomized Grover iteration: there are nine rounds; rejected labels use no query, accepted labels use $j < K$ iterates and one verification, the direct branch makes one query per round, and $t = 0$ returns none.

[F2]

Quantum query complexity: query cost counts oracle calls and excludes gates and classical work.

[F3]

One Grover iterate uses one standard bit query: one Grover iterate uses one bit query.

[F4]

Ceilings give the least positive integer meeting a real threshold and Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ : $K = ⌈ \sqrt{N / t} ⌉ \leq \sqrt{N / t} + 1$ .

[F5]

Squaring is monotone on the nonnegatives: since $N / t \geq 1$ , its nonnegative square root is at least $1$ .

Proof

1.1givenF1F3

In the branch $0 < t \leq N / 2$ , an accepted round uses at most $K - 1$ iterate queries by [F3] and one verification query by [F1], hence at most $K$ queries; rejected rounds use zero. Nine rounds use at most $9 K$ .

2.1step 1.1F1F2F4F5algebra∎

Since $\sqrt{N / t} \geq 1$ by [F5], [F4] gives $K \leq 2 \sqrt{N / t}$ , so $9 K = O (\sqrt{N / t})$ . In the direct branch there are at most nine queries, and for $t = 0$ none; [F2] keeps these counts separate from gates and classical work.

Depends on

Used by

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Sources

Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp, Quantum Amplitude Amplification and Estimation, §2 (standard reference, not scraped)