TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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QAOA objective is an expected feasible score only

Statement

For the finite graph and circuit of QAOA circuit and expected combinatorial objective, the QAOA objective lies between the minimum and maximum feasible cut sizes, and each computational-basis measurement returns a feasible cut. This expected-value statement alone implies no universal approximation ratio, global optimizer, or advantage guarantee.

Facts & Assumptions

Given: A finite computational-basis measurement distribution and the cut-size observable.

[F1]

QAOA circuit and expected combinatorial objective: the objective is the expectation of the cut-size operator and measured strings are cuts.

[F2]

Projective measurement outcome weights are probabilities: each finite projective outcome has its Born probability, and these probabilities sum to one.

Proof

technique · direct

1.1givenF1F2

Write $q_{x}$ for the probability of bitstring $x$ . By [F2], $q_{x} \geq 0$ and $\sum_{x} q_{x} = 1$ . By [F1], every value $C_{G} (x)$ is a feasible cut size and lies in $[C_{\min}, C_{\max}]$ .

2.1givenF1step 1.1algebra∎

Multiply the pointwise inequalities by $q_{x}$ and sum over the finite bitstring set. This gives $C_{\min} \leq \sum_{x} q_{x} C_{G} (x) \leq C_{\max}$ ; [F1] identifies the middle term with the QAOA objective. The argument supplies no positive ratio when $C_{\max} = 0$ and no optimization conclusion.

Depends on

Used by

VQE and MaxCut-QAOA objective calculation Example

Dependency tree · two levels

28 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)
Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, A Quantum Approximate Optimization Algorithm (arXiv:1411.4028v1) (standard reference, not scraped)