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A near-optimal Quadratic Goldreich-Levin algorithm

Jop Briët, Davi Castro-Silva

TL;DR

The paper develops a near-optimal quadratic Goldreich–Levin algorithm by adapting a quantum-inspired stabilizer-learning framework to the classical query model. It achieves a quadratic correlator within ε of the best possible by a randomized procedure with query complexity $n^2\log n\log(1/\delta)(1/\varepsilon)^{O(\log(1/\varepsilon))}$ and running time $O(n^3)$, matching the information-theoretic lower bound up to a polylog factor. Key contributions include a near-optimal self-corrector for quadratic Reed–Muller codes, an algorithmic polynomial inverse theorem for the order-3 Gowers norm, and an efficient quadratic decomposition theorem, all derived without relying on the polynomial Freiman–Ruzsa conjecture. The approach hinges on dequantizing stabilizer-learning methods, exploiting Lagrangian subspaces and isotropy to extract stabilizer states that approximate maximal quadratic correlations, and then extending to full quadratic phases. This yields significant implications for coding theory and additive combinatorics, providing polynomial-time, query-efficient tools for recovering near-maximum quadratic structure in bounded functions.

Abstract

In this paper, we give a quadratic Goldreich-Levin algorithm that is close to optimal in the following ways. Given a bounded function $f$ on the Boolean hypercube $\mathbb{F}_2^n$ and any $\varepsilon>0$, the algorithm returns a quadratic polynomial $q: \mathbb{F}_2^n \to \mathbb{F}_2$ so that the correlation of $f$ with the function $(-1)^q$ is within an additive $\varepsilon$ of the maximum possible correlation with a quadratic phase function. The algorithm runs in $O_\varepsilon(n^3)$ time and makes $O_\varepsilon(n^2\log n)$ queries to $f$, which matches the information-theoretic lower bound of $Ω(n^2)$ queries up to a logarithmic factor. As a result, we obtain a number of corollaries: - A near-optimal self-corrector of quadratic Reed-Muller codes, which makes $O_\varepsilon(n^2\log n)$ queries to a Boolean function $f$ and returns a quadratic polynomial $q$ whose relative Hamming distance to $f$ is within $\varepsilon$ of the minimum distance. - An algorithmic polynomial inverse theorem for the order-3 Gowers uniformity norm. - An algorithm that makes a polynomial number of queries to a bounded function $f$ and decomposes $f$ as a sum of poly$(1/\varepsilon)$ quadratic phase functions and error terms of order $\varepsilon$. Our algorithm is obtained using ideas from recent work on quantum learning theory. Its construction deviates from previous approaches based on algorithmic proofs of the inverse theorem for the order-3 uniformity norm (and in particular does not rely on the recent resolution of the polynomial Freĭman-Ruzsa conjecture).

A near-optimal Quadratic Goldreich-Levin algorithm

TL;DR

The paper develops a near-optimal quadratic Goldreich–Levin algorithm by adapting a quantum-inspired stabilizer-learning framework to the classical query model. It achieves a quadratic correlator within ε of the best possible by a randomized procedure with query complexity and running time , matching the information-theoretic lower bound up to a polylog factor. Key contributions include a near-optimal self-corrector for quadratic Reed–Muller codes, an algorithmic polynomial inverse theorem for the order-3 Gowers norm, and an efficient quadratic decomposition theorem, all derived without relying on the polynomial Freiman–Ruzsa conjecture. The approach hinges on dequantizing stabilizer-learning methods, exploiting Lagrangian subspaces and isotropy to extract stabilizer states that approximate maximal quadratic correlations, and then extending to full quadratic phases. This yields significant implications for coding theory and additive combinatorics, providing polynomial-time, query-efficient tools for recovering near-maximum quadratic structure in bounded functions.

Abstract

In this paper, we give a quadratic Goldreich-Levin algorithm that is close to optimal in the following ways. Given a bounded function on the Boolean hypercube and any , the algorithm returns a quadratic polynomial so that the correlation of with the function is within an additive of the maximum possible correlation with a quadratic phase function. The algorithm runs in time and makes queries to , which matches the information-theoretic lower bound of queries up to a logarithmic factor. As a result, we obtain a number of corollaries: - A near-optimal self-corrector of quadratic Reed-Muller codes, which makes queries to a Boolean function and returns a quadratic polynomial whose relative Hamming distance to is within of the minimum distance. - An algorithmic polynomial inverse theorem for the order-3 Gowers uniformity norm. - An algorithm that makes a polynomial number of queries to a bounded function and decomposes as a sum of poly quadratic phase functions and error terms of order . Our algorithm is obtained using ideas from recent work on quantum learning theory. Its construction deviates from previous approaches based on algorithmic proofs of the inverse theorem for the order-3 uniformity norm (and in particular does not rely on the recent resolution of the polynomial Freĭman-Ruzsa conjecture).
Paper Structure (21 sections, 19 theorems, 43 equations, 1 algorithm)

This paper contains 21 sections, 19 theorems, 43 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our results
  3. QGL and stabilizer learning
  4. Outline of the algorithm
  5. Preliminaries
  6. Algorithms for linear Fourier analysis
  7. Gowers uniformity in $L_2$ and stabilizer states
  8. Symplectic geometry
  9. Probability distributions and Weyl operators
  10. The main algorithm
  11. Sampling a good Lagrangian subspace
  12. Robust Lagrangian generation
  13. Non-robust Lagrangian generation implies energy increment.
  14. Sampling the desired Lagrangian
  15. From a good Lagrangian to a good stabilizer state
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $f: \mathbb{F}_2^n \to [-1, 1]$ be a $1$-bounded function and let $\varepsilon,\, \delta>0$. There is a randomized algorithm $\mathcal{A}$ that makes $n^2 \log n\, \log(1/\delta) (1/\varepsilon)^{O(\log(1/\varepsilon))}$ queries to $f$ and, with probability at least $1-\delta$, outputs a quadra In addition to the $\tilde{O}(n^2)$ queries, the algorithm $\mathcal{A}$ runs in time $O(n^3)$.

Theorems & Definitions (31)

  • Theorem 1.1: Quadratic Goldreich-Levin
  • Corollary 1.2: Optimal self-correction of quadratic Reed-Muller codes
  • Corollary 1.3: Algorithmic polynomial Gowers inverse theorem
  • Corollary 1.4: Efficient quadratic decomposition
  • Theorem 2.1: Goldreich-Levin algorithm
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4: Gowers $U^3$-norm
  • Definition 2.5: Non-classical quadratic phase functions
  • Definition 2.6: Stabilizer states
  • ...and 21 more