[2606.29687] A Machine-Verified Proof of a Quantum-Optimization Conjecture

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Quantum Physics

arXiv:2606.29687 (quant-ph)

[Submitted on 29 Jun 2026]

Title:A Machine-Verified Proof of a Quantum-Optimization Conjecture

Authors:Uri Kol, Maor Ben-Shahar, Kfir Sulimany, Dirk Englund<br>View a PDF of the paper titled A Machine-Verified Proof of a Quantum-Optimization Conjecture, by Uri Kol and 3 other authors

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Abstract:We report a machine-verified resolution of a problem open for over a decade in quantum optimization: the Farhi, Goldstone and Gutmann (FGG) conjecture that depth-$p$ Quantum Approximate Optimization Algorithm (QAOA) on the ring of disagrees attains approximation ratio $(2p+1)/(2p+2)$ exactly. We found the proof using a large language model, Claude Fable 5, and verified its correctness end-to-end by the Lean 4 proof assistant. Our methodology includes several ingredients: building on a substantial Lean library of quantum information, we formalized the QAOA components and the known parts of the problem, and reduced the conjecture to a single open mathematical statement. The model was then handed the library and our agentic toolkit, and tasked with closing that gap by constructing a proof in Lean. The resulting process is a feedback loop between the model's natural-language reasoning and Lean's mechanical verification, which converged to a machine-verified proof. Human verification is required only for the structural scaffolding - that the formal statement faithfully encodes the intended claim - while the proof itself is supplied by the model and certified mechanically by Lean. The proof is nevertheless striking - the model uncovered a hidden dynamical symmetry of the problem and exploited it, borrowing tools and machinery from an adjacent field to turn a hard existence problem into an explicit construction. This work paves the way for resolving open conjectures in quantum information science and beyond.

Subjects:

Quantum Physics (quant-ph); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Logic in Computer Science (cs.LO); Optimization and Control (math.OC)

Cite as:<br>arXiv:2606.29687 [quant-ph]

(or<br>arXiv:2606.29687v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.29687

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arXiv-issued DOI via DataCite

Submission history<br>From: Uri Kol [view email]<br>[v1]<br>Mon, 29 Jun 2026 01:25:40 UTC (19 KB)

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