A Machine-Verified Proof of a Quantum-Optimization Conjecture

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.

Fokker-Planck to Callan-Symanzik: evolution of weight matrices under training

The dynamical evolution of a neural network during training has been an incredibly fascinating subject of study. First principal derivation of generic evolution of variables in statistical physics systems has proved useful when used to describe training dynamics conceptually, which in practice means numerically solving equations such as Fokker-Planck equation. Simulating entire networks inevitably runs into the curse of dimensionality. In this paper, we utilize Fokker-Planck to simulate the probability density evolution of individual weight matrices in the bottleneck layers of a simple 2-bottleneck-layered auto-encoder and compare the theoretical evolutions against the empirical ones by examining the output data distributions. We also derive physically relevant partial differential equations such as Callan-Symanzik and Kardar-Parisi-Zhang equations from the dynamical equation we have.