The Polynomial-Time Low-Degree Conjecture is False

2026-07-22

AI Summary

This work addresses the low-degree conjecture in the standard polynomial-time setting by investigating whether efficient distinguishers exist for permutation-invariant distributions with bounded low-degree advantage. The authors construct a family of graph distributions that are permutation-invariant and exhibit zero low-degree advantage, yet remain efficiently distinguishable from the uniform null distribution via a deterministic rank test in polynomial time—even when subjected to independent resampling noise. This construction leverages algebraic techniques involving subspaces of Reed–Muller codes, low-bias polynomials, sets of points with no short linear dependencies, and randomly interleaved bilinear forms. To the best of our knowledge, this is the first counterexample to the low-degree conjecture within the standard polynomial-time framework, demonstrating that low-degree indistinguishability, permutation symmetry, and independent noise alone are insufficient to guarantee computational hardness, thereby indicating the need for stronger conditions in future formulations of the conjecture.

Abstract

The low-degree method and its associated lower bounds are widely used to guide algorithm design and to provide evidence of computational hardness in average-case inference, high-dimensional statistics, random optimization, and related problems. This led to the low-degree conjecture, which predicts that when the low-degree advantage between a planted distribution and a uniform null distribution remains bounded, no efficient distinguisher can succeed after independent noise, provided that the planted distribution has permutation symmetry. Several works have produced counterexamples to variants of this conjecture or to versions for algorithms with higher time complexity, but the conjecture remained open in its standard binary, polynomial-time formulation. We disprove the polynomial-time low-degree conjecture by giving a family of examples in this setting. For every fixed integer $r\geq3$, we construct a permutation-invariant distribution $\mathbb{P}_n$ on simple graphs, with $\mathbb{Q}_n=G(n,1/2)$, such that every marginal of $\mathbb{P}_n$ on at most $D_n=Θ((\log n)^{r-1})$ edges is uniform. Therefore, the low-degree advantage is zero through degree $D_n$. Nevertheless, after every edge is independently resampled at a fixed positive rate, a deterministic rank test strongly distinguishes the resulting distribution from $\mathbb{Q}_n$ in polynomial time. The construction chooses a subspace of a Reed--Muller code whose nonzero polynomials have small absolute bias, selects points whose evaluation vectors have no short linear dependencies, and evaluates a random alternating bilinear form on pairs of these vectors. Our result shows that low-degree indistinguishability, a uniform null distribution, permutation invariance, and independent resampling do not by themselves imply polynomial-time hardness, and suggests that a valid general conjecture must impose an additional condition.

Problem

• low-degree conjecture
• polynomial-time hardness
• permutation-invariant distribution
• average-case inference
• computational indistinguishability

Innovation

• low-degree conjecture
• polynomial-time distinguisher
• permutation-invariant distribution
• Reed-Muller code
• rank test