CounterexampleConstruction: 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.

A variational circuit family may miss the optimum

Statement refuted

A variational circuit family is guaranteed to contain a ground state of every finite Hermitian Hamiltonian.

Facts & Assumptions

Given: $H = diag (0, 1)$ on $C^{2}$ and the ansatz consisting of the fixed state $∣ 1 ⟩$ .

[F1]

Variational eigenvalue objective: the VQE objective is the Dirac expectation and $E_{0}$ is the least eigenvalue.

[F2]

VQE Rayleigh–Ritz upper bound and equality case: every ansatz energy is at least $E_{0}$ , with equality only for a ground-space state.

Counterexample

Given: $H = diag (0, 1)$ and the ansatz consisting of $∣ 1 ⟩$ .

Proof technique: direct.

1.1givenF1algebra

The eigenvalues of $H$ are $0, 1$ , so $E_{0} = 0$ and its ground eigenspace is $span {∣ 0 ⟩}$ . The only ansatz state $∣ 1 ⟩$ is orthogonal to that eigenspace.

2.1givenF2step 1.1algebra∎

Direct multiplication gives $H ∣ 1 ⟩ = ∣ 1 ⟩$ , hence the ansatz energy is $1 > 0 = E_{0}$ . It still satisfies [F2], so the universal ansatz-coverage claim is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

https://arxiv.org/pdf/2012.09265v2 (standard reference, not scraped)