Marton's conjecture in polynomial time

Abstract:Gowers, Green, Manners, and Tao (Annals '25) recently resolved Marton's polynomial Freiman-Ruzsa conjecture. We give an algorithmic counterpart to their result: given uniform sampling and membership-oracle access to a set $A \subseteq \mathbb{F}_2^n$ with doubling constant at most $K$, our algorithm outputs a subspace of size at most $|A|$ whose $K^{O(1)}$ translates cover $A$. The algorithm runs in $\textsf{poly}(n,K)$ time. As applications, we obtain polynomial-time algorithms for a variety of learning problems, including quadratic Goldreich-Levin, improper agnostic tomography of stabilizer states, and tomography of quantum states with bounded stabilizer extent.

Computational Complexity (cs.CC); Combinatorics (math.CO); Quantum Physics (quant-ph)

arXiv:2609.20771 [cs.CC]

(or arXiv:2609.20771v1 [cs.CC] for this version)

https://doi.org/10.48550/arXiv.2609.20771 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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