Marton's conjecture in polynomial time

2026-09-17

AI Summary

该研究通过算法解决了Marton的多项式Freiman-Ruzsa猜想，提出了一种在多项式时间内运行的算法，能够处理特定条件下的集合，并应用于多种学习问题。

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.

Problem

• polynomial time
• doubling constant
• subspace

Innovation

• polynomial time
• subspace
• covering set
• algorithmic counterpart
• learning problems