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.

Exact phase recovery on the QPE grid

Statement

Let $m \geq 1$ , $Q = 2^{m}$ , and $θ \in [0, 1)$ . If $Q θ = t \in Z$ , then $t \in {0, \dots, Q - 1}$ , phase estimation returns $y = t$ with probability $1$ , and every other output has probability $0$ .

Facts & Assumptions

Given: The eigenphase input, $m \geq 1$ , $Q = 2^{m}$ , and $Q θ = t \in Z$ .

[F1]

The outcome probability is $P_{θ} (y) = Q^{- 2} ∣ \sum_{j < Q} \exp (2 π i j (θ - y / Q)) ∣^{2}$ (Phase-estimation finite outcome law).

[F2]

For $Q \geq 1$ and integers $k, ℓ$ , $\sum_{j < Q} \exp (2 π i (k - ℓ) j / Q)$ is $Q$ when $k \equiv ℓ (m o d Q)$ and $0$ otherwise (Orthogonality of the characters $x \mapsto e^{2 π i k x / N}$ on $Z / N Z$ ).

Proof

1.1givenF1algebra

Since $0 \leq θ < 1$ , the integer $t = Q θ$ satisfies $0 \leq t < Q$ . At $y = t$ , every summand in [F1] is $1$ , so $P_{θ} (t) = Q^{- 2} Q^{2} = 1$ .

2.1givenF1F2step 1.1algebra∎

If $y \in {0, \dots, Q - 1}$ and $y \neq t$ , then $t - y$ is not divisible by $Q$ , since $∣ t - y ∣ < Q$ and $t - y \neq 0$ . Substituting $θ = t / Q$ in [F1], the sum is the nontrivial cyclic-character sum in [F2], hence zero; therefore $P_{θ} (y) = 0$ .

Depends on

Used by

Szegedy-walk marked-set detection Algorithm
Exact estimation of a binary phase Example
Amplitude-estimation additive error and success probability Theorem
Finite reversible-walk finding cost Theorem
Phase-estimation nearest-grid and tail bounds Theorem
Quantum counting error and boundary cases Theorem
Relative approximate counting query complexity Theorem
Walk detection correctness and cost Theorem

Dependency tree · two levels

40 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

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)