We provide algorithmic versions of the Polynomial Freiman-Ruzsa theorem of Gowers, Green, Manners, and Tao (Ann. of Math., 2025). In particular, we give a polynomial-time algorithm that, given a set $A \subseteq \mathbb{F}_2^n$ with doubling constant $K$, returns a subspace $V \subseteq \mathbb{F}_2^n$ of size $|V| \leq |A|$ such that $A$ can be covered by $2K^C$ translates of $V$, for a universal constant $C>1$. We also provide efficient algorithms for several "equivalent" formulations of the Polynomial Freiman-Ruzsa theorem, such as the polynomial Gowers inverse theorem, the classification of approximate Freiman homomorphisms, and quadratic structure-vs-randomness decompositions. Our algorithmic framework is based on a new and optimal version of the Quadratic Goldreich-Levin algorithm, which we obtain using ideas from quantum learning theory. This framework fundamentally relies on a connection between quadratic Fourier analysis and symplectic geometry, first speculated by Green and Tao (Proc. of Edinb. Math. Soc., 2008) and which we make explicit in this paper.

翻译：我们提供了高尔斯、格林、曼纳斯和陶哲轩（《数学年鉴》，2025年）提出的多项式的弗里曼-鲁扎定理的算法化版本。特别地，我们给出一个多项式时间算法：给定一个具有加倍常数$K$的集合$A \subseteq \mathbb{F}_2^n$，该算法返回一个子空间$V \subseteq \mathbb{F}_2^n$满足$|V| \leq |A|$，使得$A$可被$2K^C$个$V$的平移覆盖，其中$C>1$为普适常数。我们还为多项式的弗里曼-鲁扎定理的若干"等价"表述（如多项式的高斯逆定理、近似弗里曼同态的分类以及二次结构-随机性分解）提供了高效算法。我们的算法框架基于二次戈尔德里希-莱文算法的一个新的最优版本，该版本通过量子学习理论的思想获得。这一框架从根本上依赖于二次傅里叶分析与辛几何之间的联系——这一联系最早由格林和陶哲轩（《爱丁堡皇家学会会刊》，2008年）推测，并在本文中予以明确化。