DefinitionDefinition: Literature-sourcedProof: 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.

Variational eigenvalue objective

Definition

Let $H = H^{*}$ be a Hermitian matrix on a nonzero finite-dimensional complex space, let $∣ ψ_{0} ⟩$ be a unit vector, and let $U (θ)$ be a supplied unitary ansatz. The variational eigenvalue objective is $E (θ) = ⟨ ψ (θ) ∣ H ∣ ψ (θ) ⟩_{D}, ∣ ψ (θ) ⟩ = U (θ) ∣ ψ_{0} ⟩ .$ The target ground energy is $E_{0} = \min σ (H)$ . The spectral theorem makes $E_{0}$ well-defined and each normalized state energy real. No ansatz coverage or optimizer convergence follows from this objective definition. Charge input-state preparation, each ansatz and observable call, measurements, shots, optimizer arithmetic, and output precision.

Facts & Assumptions

Given: A finite Hermitian $H$ , normalized $∣ ψ (θ) ⟩$ , and the Dirac/root conversion convention.

[F1]

Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely: a finite-dimensional Hermitian matrix has an orthonormal eigenbasis with real eigenvalues.

[F2]

Dirac Notation Conversion: $⟨ ψ ∣ H ∣ ψ ⟩_{D} = ⟨ H ψ, ψ ⟩_{r o o t}$ .

Verification

technique · direct

1.1givenF1F2algebra

Expand $∣ ψ (θ) ⟩ = \sum_{j} c_{j} ∣ e_{j} ⟩$ in the orthonormal eigenbasis from [F1], with eigenvalues $λ_{j}$ . The root convention in [F2] gives $E (θ) = \sum_{j} λ_{j} ∣ c_{j} ∣^{2}$ , a real number because each $λ_{j}$ is real.

2.1givenstep 1.1algebra∎

The coefficients satisfy $\sum_{j} ∣ c_{j} ∣^{2} = 1$ . Therefore $E (θ)$ is a convex combination of eigenvalues and is at least their minimum $E_{0}$ . This defines a variational upper estimate of the ground energy, without asserting that the ansatz attains it.

Depends on

Used by

A variational circuit family may miss the optimum Counterexample
VQE Rayleigh–Ritz upper bound and equality case Theorem

Dependency tree · two levels

23 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)