the variational principle lower bound — the guarantee beneath VQE
Theorem.the variational principle lower bound — the guarantee beneath VQE — ⟨ψ|H|ψ⟩ ≥ E₀ for every normalised |ψ⟩ — the ground energy is a floor, every ansatz an upper bound.
Proof.the variational wave (the thinnest ray, named by the research): for EVERY state ⟨ψ|H|ψ⟩ ≥ E₀, the ground energy, with equality iff |ψ⟩ is the ground state. Verified over a dense Bloch-sphere grid of qubit states for H = aZ+bX — no expectation falls below E₀ = −√(a²+b²) (exact diagonalisation), and the minimum reaches it. This is the theorem VQE rests on: minimising the energy expectation over any ansatz can never undershoot the ground energy and reaches it exactly when the ansatz spans the ground state — so variationalQuantumEigensolverAndQaoa is correct by this bound, not by luck. Single-qubit demonstration where the grid is the whole state space; the general ⟨ψ|H|ψ⟩ ≥ λ_min (Rayleigh–Ritz) cited; the guarantee, not a speedup — finding the minimising ansatz is the hard part (barren plateaus).
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/2/8/index.ts#theVariationalPrincipleLowerBound
1 · Classification
finite-complete — computed witness within a cited frame (the unbounded form leans on the cited literature)
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"the variational principle lower bound — the guarantee beneath VQE" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation theVariationalPrincipleLowerBound.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (theVariationalPrincipleLowerBound @ src/2/8) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes theVariationalPrincipleLowerBound (src/2/8/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.