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.

Phase-estimation nearest-grid and tail bounds

Statement

Let $m \geq 1$ , $Q = 2^{m}$ , and $θ \in [0, 1)$ . Every nearest grid label $y$ has $P_{θ} (y) \geq 4 / π^{2}$ . For any $η \in R$ with $K = ⌊ Q η ⌋ \geq 2$ , $\Pr_{y \sim P_{θ}} ⁣ [d_{T} (θ - y / Q) \geq η] \leq \frac{1}{2 (K - 1)} .$ If $Q θ$ is an integer $t$ , then $P_{θ} (t) = 1$ and all other labels have probability $0$ .

Facts & Assumptions

Given: $m \geq 1$ , $Q = 2^{m}$ , $θ \in [0, 1)$ , and, for the tail conclusion, $η \in R$ with $K = ⌊ Q η ⌋ \geq 2$ .

[F1]

A nearest label exists and every nearest label has probability at least $4 / π^{2}$ (Nearest-grid probability in phase estimation).

[F2]

The circular-distance tail for $K \geq 2$ is at most $1 / (2 (K - 1))$ (Circular-distance tail bound in phase estimation, Circular phase distance modulo one).

[F3]

Exact-grid phases are returned with probability one (Exact phase recovery on the QPE grid).

Proof

1.1givenF1

Apply [F1] to the given $θ$ ; it supplies a nearest label $y$ and the lower bound $P_{θ} (y) \geq 4 / π^{2}$ .

1.2givenF2

If $K = ⌊ Q η ⌋ \geq 2$ , all hypotheses of [F2] hold, so the stated circular-distance tail bound follows.

2.1givenF3step 1.1step 1.2∎

If $Q θ = t \in Z$ , [F3] gives $P_{θ} (t) = 1$ and zero probability at every other label. These conclusions prove the theorem.

Depends on

Used by

Dependency tree · two levels

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Sources

Andrew M. Childs, Lecture Notes on Quantum Algorithms, §4.3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, version 6, §4.6 (standard reference, not scraped)
Lin Lin, Lecture Notes on Quantum Algorithms for Scientific Computation, §§16.1–16.3 (standard reference, not scraped)