TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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.

Amplitude-estimation query and precision complexity

Statement

For $M = 2^{m} \geq 4$ , repeated controlled- $Q$ access uses exactly $M - 1$ controlled- $Q$ applications in one run; each expands to one call each to controlled $A$ , controlled $A^{*}$ , controlled $S_{0}$ , controlled $S_{g o o d}$ , and the control phase. Supplied controlled-power access uses exactly $m$ power queries and reports their separate costs. With fixed success probability, additive amplitude error $ϵ \in (0, 1]$ is obtained with $M = Θ (1 / ϵ)$ , hence $O (1 / ϵ)$ repeated controlled- $Q$ calls; state preparation, ordinary gates, and classical encoding are separate costs.

Facts & Assumptions

Given: The $M$ -label amplitude-estimation circuit and the access mode from Amplitude estimation and counting input/output contract.

[F1]

The phase-estimation circuit uses one controlled power $Q^{2^{j}}$ for each $j = 0, \dots, m - 1$ (QFT-based amplitude estimation).

[F2]

QPE charges exactly $m$ supplied powers or $M - 1$ repeated controlled-unitary calls (Phase-estimation query and gate costs).

[F3]

The quantum query model counts only declared oracle actions (Quantum query complexity).

[F4]

In repeated-unitary mode, each controlled $Q$ expands into separately declared controlled $A$ , controlled $A^{*}$ , controlled $S_{0}$ , controlled $S_{g o o d}$ , and control phase. In power-query mode each supplied controlled power has its own declared cost vector, including any constituent implementations used (Amplitude estimation and counting input/output contract).

[F5]

The error guarantee is the inequality in Amplitude-estimation additive error and success probability.

Proof

1.1givenF1F2algebra

In repeated access, the $j$ th power takes $2^{j}$ calls by [F2]. Summing gives $\sum_{j = 0}^{m - 1} 2^{j} = 2^{m} - 1 = M - 1$ . In supplied-power access, there is one call for each of $m$ indices, hence $m$ queries.

2.1givenF3F4step 1.1

In repeated-unitary mode, [F4] makes each controlled- $Q$ call the charged product listed in the Statement. In power-query mode, the model instead charges each supplied controlled power and its declared cost vector, without inferring any constituent capability from a bare uncontrolled call.

3.1givenF3F5step 1.1algebra∎

For $0 < ϵ \leq 1$ , choose $M$ the least power of two at least $4 (22 / 7) / ϵ$ . By [F5], the $k = 1$ error is at most $π / M + π^{2} / M^{2} < ϵ / 4 + ϵ^{2} / 16 \leq 5 ϵ / 16 < ϵ$ , with success probability at least $8 / π^{2}$ . The least-power choice gives $M < 8 (22 / 7) / ϵ$ , so [F3] yields the $O (1 / ϵ)$ repeated-call bound; other cost coordinates remain separate.

Depends on

Used by

Amplitude-estimation expectation estimator Algorithm
Finite CKS sparse QLSP with variable-time inverse estimation Algorithm
A bare unitary oracle does not supply controlled powers for free Counterexample
Median amplitude-estimation query cost Theorem
Quantum bounded-expectation guarantee and query cost Theorem
Quantum counting query complexity Theorem
Relative probability estimation with a charged finite cap Theorem

Dependency tree · two levels

55 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

Brassard, Høyer, Mosca, and Tapp, Quantum Amplitude Amplification and Estimation, complete article, §§4–5, Theorems 11–18 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, complete lecture notes, §§19.1–19.3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, complete 239-page notes, §7.3 (standard reference, not scraped)