ExampleConstruction: Literature-sourcedVerification: 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.

Degree-one QSP phase identity

Example

Take $d = 1$ and phases $ϕ_{0} = ϕ_{1} = 0$ . Then $Z (0) = I$ and the sequence is $U (x) = W_{R} (x)$ .

Facts & Assumptions

Given: $x \in [- 1, 1]$ , $s = \sqrt{1 - x^{2}} \geq 0$ , and the definitions of $W_{R}$ , $Z$ , and the degree-one QSP matrix in QSP signal matrix and ordered phase sequence.

[F1]

QSP signal matrix and ordered phase sequence: $W_{R} (x) = (\begin{array}{cc} x & s \\ s & - x \end{array})$ , $Z (0) = I$ , and the length-one sequence is $Z (0) W_{R} (x) Z (0)$ .

[F2]

Admissible polynomial pair for Hermitian-reflection QSP: an admissible pair has the declared degree/parity bounds and satisfies $P P^{*} + (1 - x^{2}) Q Q^{*} = 1$ .

Verification

1.1F1algebra

Multiplying $Z (0) W_{R} (x) Z (0)$ gives $U (x) = (\begin{array}{cc} x & s \\ s & - x \end{array})$ . Thus the upper-left polynomial is $P (x) = x$ and the upper-right entry is $s (x) Q (x)$ with $Q (x) = 1$ .

2.1F2step 1.1algebra

The polynomial $P$ has degree one and odd parity; $Q$ is constant with degree zero and even parity. Also $P P^{*} + (1 - x^{2}) Q Q^{*} = x^{2} + (1 - x^{2}) = 1$ , so the pair is admissible by [F2].

3.1F1step 1.1algebra∎

At $x = 1$ and $x = - 1$ , $s (x) = 0$ , so $U (1) = diag (1, - 1)$ and $U (- 1) = diag (- 1, 1)$ . These are exactly the endpoint values from the displayed matrix, with no division by $s (x)$ at either endpoint.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, Quantum singular value transformation and beyond (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 29 (standard reference, not scraped)