ExampleConstruction: AI-generatedVerification: 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.

Exact estimation of a binary phase

Example

Take a valid eigenphase-estimation input with $m = 3$ and eigenphase $θ = 5 / 8$ . Compute the complete output law and the returned estimate.

Facts & Assumptions

Given: A valid QPE eigenpair $U ψ = \exp (2 π i (5 / 8)) ψ$ , with $m = 3$ .

[F1]

The algorithm returns the measured label $y$ and estimate $y / Q$ , where $Q = 2^{m}$ (Quantum phase estimation).

[F2]

If $Q θ = t$ is an integer, then the exact-grid theorem gives probability one to $y = t$ and zero to every other label (Exact phase recovery on the QPE grid).

Verification

1.1givenF2algebra

Here $Q = 2^{3} = 8$ and $Q θ = 8 (5 / 8) = 5$ , an integer in ${0, \dots, 7}$ . By [F2], the output probabilities in label order $0, 1, \dots, 7$ are $(0, 0, 0, 0, 0, 1, 0, 0)$ .

2.1F1step 1.1algebra∎

The unique measured label is $y = 5$ ; [F1] therefore returns the exact phase estimate $y / Q = 5 / 8$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

41 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)