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.

VQE and MaxCut-QAOA objective calculation

Example

For VQE, take $H = Z_{1} + 2 Z_{2}$ and the two-qubit ansatz $∣ ψ (θ, ϕ) ⟩ = R_{y} (θ) ∣ 0 ⟩ \otimes R_{y} (ϕ) ∣ 0 ⟩$ , where $R_{y} (t) ∣ 0 ⟩ = \cos (t / 2) ∣ 0 ⟩ + \sin (t / 2) ∣ 1 ⟩$ . Its energy is $E (θ, ϕ) = \cos θ + 2 \cos ϕ$ . At $(θ, ϕ) = (π / 2, 0)$ the energy is $2$ , while at $(π, π)$ the ansatz is $∣ 11 ⟩$ and reaches the ground energy $- 3$ .

For QAOA take the one-edge graph $V = {1, 2}$ , $E = {(1, 2)}$ , depth $p = 1$ , $γ = π / 2$ , $β = π / 8$ . Starting from $∣ + + ⟩$ , the cost phase is $diag (1, - i, - i, 1)$ in the order $00, 01, 10, 11$ ; the mixer $R_{x} (2 β) \otimes R_{x} (2 β)$ produces $- i (∣ 01 ⟩ + ∣ 10 ⟩) / \sqrt{2}$ . Thus the output distribution gives probability $1 / 2$ to each cut string $01, 10$ , zero to $00, 11$ , and expected cut value $1$ .

Facts & Assumptions

Given: The Pauli matrices, the declared QAOA product order, and the finite projective measurement law.

[F1]

VQE Rayleigh–Ritz upper bound and equality case: a normalized ansatz has energy at least the least eigenvalue.

[F2]

QAOA circuit and expected combinatorial objective: the one-edge cost operator is $(I - Z_{1} Z_{2}) / 2$ and the depth-one circuit order is $e^{- i β (X_{1} + X_{2})} e^{- i γ (I - Z_{1} Z_{2}) / 2}$ .

[F3]

QAOA objective is an expected feasible score only: measured bitstrings are feasible cuts and expected cut lies between feasible extrema.

[F4]

Projective measurement outcome weights are probabilities: finite projective measurement probabilities are squared basis amplitudes.

Verification

technique · direct

1.1givenF1algebra

The eigenvalues of $Z_{1} + 2 Z_{2}$ on $∣ 00 ⟩, ∣ 01 ⟩, ∣ 10 ⟩, ∣ 11 ⟩$ are $3, - 1, 1, - 3$ , so the ground energy is $- 3$ . The product ansatz has energy $\cos θ + 2 \cos ϕ$ : at $(π / 2, 0)$ this is $2$ , and at $(π, π)$ the state is $∣ 11 ⟩$ with energy $- 3$ . Thus both the displayed trial value and attainment claim are verified.

2.1givenF2F3F4algebra∎

For QAOA, the initial state is $(∣ 00 ⟩ + ∣ 01 ⟩ + ∣ 10 ⟩ + ∣ 11 ⟩) / 2$ . The cost unitary multiplies only $∣ 01 ⟩, ∣ 10 ⟩$ by $e^{- i π / 2} = - i$ . With $R_{x} (π / 4) = (\begin{array}{cc} \cos (π / 8) & - i \sin (π / 8) \\ - i \sin (π / 8) & \cos (π / 8) \end{array})$ on each wire, direct multiplication gives zero amplitudes for $00, 11$ and amplitude $- i / \sqrt{2}$ for each of $01, 10$ . By [F4], the probabilities are $0, 1 / 2, 1 / 2, 0$ in the stated order, so the expected cut value is $1$ , in agreement with [F2]–[F3].

Depends on

Used by

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Dependency tree · two levels

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Sources

https://arxiv.org/pdf/2012.09265v2 (standard reference, not scraped)
Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, A Quantum Approximate Optimization Algorithm (arXiv:1411.4028v1) (standard reference, not scraped)