TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generated

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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-count Grover search uses O of the square root of N over t bit queries

Statement

For $0 < t \leq N$ , the exact known-count procedure uses $O (\sqrt{N / t})$ standard bit queries: one query for each Grover iterate and at most one separate verification query. If $t = 0$ and the exact count is supplied, it returns none with no query. This is only a query bound; exact classical maximization can cost exponentially many bit operations in $n$ .

Facts & Assumptions

Given: The exact known-count algorithm and the worst-case quantum query-cost convention.

[F1]

Known-count Grover search by exact integer maximization: if $t = 0$ , the algorithm returns none without a query. If $t > 0$ , it computes the recurrence and every candidate value for $0 \leq k \leq K = ⌈ \sqrt{N / t} ⌉$ , chooses the least maximizer, applies $k$ iterates, then makes one verification query.

[F2]

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

[F3]

Quantum query complexity: query cost counts the designated oracle invocations, separately from gates and classical work.

[F4]

Success guarantee and boundary behavior of known-count Grover search: the finite maximization has the stated success guarantee and the $t = 0$ branch returns none.

[F5]

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 ${x^{2} : x \neq 0}$ : $K = ⌈ \sqrt{N / t} ⌉ \leq \sqrt{N / t} + 1$ and $\sqrt{N / t} \geq 1$ .

[F6]

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

Proof

1.1F1F2

For $t > 0$ , [F1] uses at most $K$ iterates; by [F2] they contribute at most $K$ bit queries, and the separate verification contributes at most one.

2.1step 1.1F1F3F5F6algebra

By [F5]–[F6], $K + 1 \leq \sqrt{N / t} + 2 \leq 3 \sqrt{N / t}$ . This is the claimed worst-case query bound; no nonoracle gate or classical preprocessing cost is included by [F3]. For $t = 1$ , [F1] explicitly computes $K + 1 \geq \sqrt{N} = 2^{n / 2}$ candidate values, so its classical work can be exponential in $n$ .

3.1givenF1F3F4∎

For $t = 0$ , [F4] gives the correct output none, and the explicit early return in [F1] executes it without any oracle invocation. Thus its query cost is zero by [F3].

Depends on

Used by

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Sources

Ronald de Wolf, Quantum Computing: Lecture Notes, §7.2 (standard reference, not scraped)
Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp, Quantum Amplitude Amplification and Estimation, §2 (standard reference, not scraped)