ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-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.

Two-dimensional diagonal HHL inversion

Example

Let $H = diag (1, 1 / κ)$ for $κ \geq 1$ and $∣ b ⟩ = (∣ 0 ⟩ + ∣ 1 ⟩) / \sqrt{2}$ . The support promise holds with gap $1 / κ$ .

Verification

[F1] Ideal inverse amplitude and HHL postselection probability: the ideal well vector is $(2 κ)^{- 1} H^{+} ∣ b ⟩$ and its squared norm is the ideal success probability.

Given: The displayed diagonal matrix and normalized input.

1.1givenalgebra

Direct inversion gives $H^{+} ∣ b ⟩ = (∣ 0 ⟩ + κ ∣ 1 ⟩) / \sqrt{2}$ . Its norm is $\sqrt{1 + κ^{2}} / \sqrt{2}$ , so the target is $(∣ 0 ⟩ + κ ∣ 1 ⟩) / \sqrt{1 + κ^{2}}$ .

2.1F1step 1.1algebra∎

By [F1], the well vector is $(∣ 0 ⟩ + κ ∣ 1 ⟩) / (2 \sqrt{2} κ)$ and its probability is $(1 + κ^{2}) / (8 κ^{2})$ , which satisfies $1 / 8 < p \leq 1 / 4$ . For OP5, the eigenvalue grid spacing is $2 π / t_{0} = π ϵ / (1024 κ)$ . The sine window has nonzero tails, so this spacing is a grid resolution and not a hard error guarantee; the kernel moment and filter proof supply the error bound.

Depends on

Used by

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Sources

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, Quantum algorithm for linear systems of equations (standard reference, not scraped)
Andrew M. Childs, Robin Kothari, and Rolando D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)