Algorithmic Polynomial Freiman-Ruzsa Theorems

We prove algorithmic versions of the polynomial Freiman-Ruzsa theorem of Gowers, Green, Manners, and Tao (Annals of Mathematics, 2025) in additive combinatorics. In particular, we give classical and quantum polynomial-time algorithms that, for $A\subseteq {\mathbb{F}}_2^n$ with doubling constant $K$, learn an explicit description of a subspace $V\subseteq {\mathbb{F}}_2^n$ of size $\mathrm{\mid }V\mathrm{\mid }\le \mathrm{\mid }A\mathrm{\mid }$ such that $A$ can be covered by $K^C$ translates of $V$, for a universal constant $C>1$.