An algorithmic Polynomial Freiman-Ruzsa theorem

Abstract: 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.

• The paper introduces randomized polynomial-time algorithms that construct covering subspaces for sets with small doubling.
• It develops a Quadratic Goldreich–Levin algorithm to efficiently uncover quadratic structure using symplectic geometry and Fourier analysis.
• Quantum-inspired techniques further reduce complexity, bridging additive combinatorics and quantum learning for practical applications.

Algorithmic Polynomial Freiman-Ruzsa Theorem: Summary and Implications

Introduction and Context

The Freiman–Ruzsa theorem is central to additive combinatorics, characterizing the algebraic structure of subsets $A\subseteq \mathbb{F}_2^n$ with small doubling ($|A+A| \leq K|A|$), predicting that such sets are efficiently covered by cosets of relatively small subspaces. The Polynomial Freiman–Ruzsa (PFR) conjecture, widely studied for decades, posited that the number of covering translates could be bounded polynomially in $K$—this was recently resolved by Gowers, Green, Manners, and Tao. However, the original PFR theorem is existential and non-constructive, which hinders algorithmic and theoretical applications in areas where explicit structure recovery is needed.

This paper addresses this critical gap by providing algorithmic versions of the PFR theorem and its equivalent formulations. The contributions are twofold: (1) new polynomial-time algorithms that, given sets with bounded doubling, efficiently extract structural decompositions sufficient for the PFR conclusion, and (2) algorithmic counterparts for variants such as the Gowers $U^3$-inverse theorem, classification of approximate Freiman homomorphisms, and optimal quadratic structure-vs-randomness decompositions.

A crucial novelty is the development of an optimal Quadratic Goldreich–Levin (Quadratic GL) algorithm. This forms the core analytical and algorithmic engine, enabling the efficient search for quadratic structure in functions and sets, leveraging connections with quadratic Fourier analysis, symplectic geometry, and modern quantum learning techniques.

Main Results

1. Algorithmic PFR Theorem

The main result gives a randomized classical algorithm which, given a set $A \subseteq \mathbb{F}_2^n$ with $|A+A|\leq K|A|$, returns in polynomial time a subspace $V\leq \mathbb{F}_2^n$ of size $|V| \leq |A|$ such that $A$ is covered by $2K^C$ translates of $|A+A| \leq K|A|$0 (for an absolute constant $|A+A| \leq K|A|$1). The algorithm operates in $|A+A| \leq K|A|$2 time and uses only logarithmic samples and queries to $|A+A| \leq K|A|$3, making the PFR theorem fully effective in computational settings.

Numerical bounds in the analysis further strengthen earlier existential statements by giving explicit exponents and complexity tradeoffs.

2. Algorithmic Polynomial Gowers $|A+A| \leq K|A|$4 Inverse Theorem

The authors convert the combinatorial proof that PFR and the polynomial $|A+A| \leq K|A|$5-inverse theorem (PGI) are equivalent into an efficient reduction. They then provide the first polynomial-time algorithm for the PGI: given $|A+A| \leq K|A|$6 with $|A+A| \leq K|A|$7 (for $|A+A| \leq K|A|$8-bounded $|A+A| \leq K|A|$9), the algorithm finds a quadratic polynomial $K$0 with correlation $K$1 in randomized $K$2 time and $K$3 oracle queries to $K$4.

This result, via explicit list-decoding and structure learning procedures, implies the agnostic self-correction of quadratic Reed-Muller codes and efficient quadratic decomposition for structure-vs-randomness.

3. Core Techniques: Quadratic Structure Discovery via Symplectic Geometry

A unifying technical innovation is the systematic use of quadratic Fourier analysis over symplectic phase spaces. The analysis utilizes the Heisenberg–Weyl group, the correspondence with stabilizer states, and the symplectic and Lagrangian subspace structure of the associated phase space. This perspective is operationalized algorithmically by instances of the Quadratic Goldreich–Levin algorithm, inspired and “dequantized” from quantum learning theory, specifically efficient agnostic learning of stabilizer states.

Figure 1: Forbidden regions for~$K$5.

This figure illustrates the complex-analytic constraints (forbidden regions for parameters $K$6) that arise in the analysis of the uncertainty principle for quadratic Fourier coefficients over finite fields.

4. Quantum Algorithm for Algorithmic PFR

Building on the connection to quantum learning, the paper presents a quantum algorithm for the algorithmic PFR theorem, which achieves a further $K$7 improvement (i.e., $K$8 time and $K$9 quantum queries to $U^3$0). The proof modularly invokes the agnostic quantum stabilizer learning protocol, showing that quantum techniques can both inspire and improve upon purely classical structure-recovery methods.

Technical Insights

• Lagrangian and stabilizer structure extraction: Exploiting the full symplectic geometry of the underlying vector space $U^3$1, the resulting algorithms efficiently sample and reconstruct high-bias Lagrangian subspaces, which correspond to maximal collections of commuting quadratic phases (stabilizer states). This point of view is instrumental in breaking combinatorial entropy bottlenecks in Fourier analysis of sets with small doubling.
• Quadratic Goldreich–Levin as a central primitive: The new Quadratic GL algorithm generalizes the classic homogeneous list-decoding from linear to quadratic phases, controlling the correlation loss and bounding the number of necessary local optima (approximate local maximizers), thus obtaining polynomial query and time complexity.
• Energy boosting and robust generation: To ensure the coverage of structure even in the presence of error or approximate local maxima, an energy-increment (“boosting”) procedure is introduced, iteratively projecting the input function/set onto subspaces where robust quadratic structure manifests.
• Phase-space uncertainty principles: The efficient sampling and integration of quadratic structure leverages sharp uncertainty principles in the Heisenberg–Weyl representation, ensuring that samples from the characteristic distribution concentrate around isotropic subspaces.

Implications and Theoretical Significance

These results bridge the gap between deep structural theorems in additive combinatorics and their algorithmic instantiations. The algorithmic PFR theorem and its equivalents enable:

• Efficient property testing and learning: Applications in testing linearity/quadraticity, local decoding/correction, and randomness extraction (including affine and two-source extractors) where existential results are unhelpful unless made algorithmic.
• Hardness amplification and average-case reductions: The explicit algorithms support worst-case to average-case reductions for linear and quadratic problems in classical and quantum computation, and serve as black-box subroutines in additive communication complexity and code construction.
• Quantum learning theory: The correspondence between quadratic Fourier structure and quantum stabilizer formalism underscores how quantum learning algorithms can inform and guide the development of classical property testers and reconstruction schemes.

Contrasts and Strong Claims

• First polynomial-time construction of covering subspaces for PFR: This closes the gap between existential structure theorems (previously non-algorithmic, exponential-time) and practical computation.
• List size and optimality in correlation loss: The bounds obtained for list-decoding quadratic correlators are essentially optimal (up to polylogarithmic factors).
• Dequantization of quantum protocols: The framework is the first to convert quantum stabilizer learning results into concrete, efficient classical algorithms for central combinatorial problems.

Future Directions

Several avenues for further development are indicated:

• Higher-order analogues: Extending the framework to cubic (and higher) versions of the polynomial Freiman–Ruzsa and inverse Gowers-norm theorems, where classical combinatorial and quantum learning approaches may again inform each other.
• Practical quantum advantage: Empirical separations between query and gate complexities in classical versus quantum algorithms for structure discovery in high-noise or “hard” instances.
• Extensions beyond finite fields: Adapting the algorithmic framework to other group settings (e.g., non-abelian groups), or more general fields and modules, potentially relevant to more general coding and cryptographic constructions.
• Bridging additional theoretical and algorithmic gaps: Providing effective, efficient versions of other structural results in additive combinatorics, such as robust versions of the Balog–Szemerédi–Gowers theorem or the full non-abelian Freiman theorem.

Conclusion

This paper achieves a major advancement in algorithmic additive combinatorics, translating the polynomial Freiman–Ruzsa theorem and its central corollaries into explicit, polynomial-time algorithms with strong complexity bounds. The combination of quadratic Fourier analysis, symplectic geometry, and quantum-inspired techniques yields both practical and conceptual improvements, opening new research vistas at the interface of analysis, combinatorics, and quantum information theory.

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References:

• "An algorithmic Polynomial Freiman-Ruzsa theorem" (2604.04547).
• Gowers et al., "On a Conjecture of Marton" [GowersGMT2025].
• Sitan Chen et al., "Stabilizer Bootstrapping: A Recipe for Efficient Agnostic Tomography and Magic Estimation" [chen2024stabilizer].