A Classical Quadratic Speedup for Planted $k$XOR

TL;DR

This paper presents a new classical algorithm for the noisy planted k-XOR problem that achieves a quadratic speedup over previous classical methods, narrowing the gap with quantum algorithms for large constant k.

Contribution

It introduces a classical algorithm that is quadratically faster than prior classical algorithms for large k, using techniques from sublinear algorithms and anticoncentration.

Findings

Classical algorithm is quadratically faster for large constant k.

The algorithm works in the semirandom case.

Quantum speedup remains, but is reduced to quadratic.

Abstract

A recent work of Schmidhuber et al (QIP, SODA, & Phys. Rev. X 2025) exhibited a quantum algorithm for the noisy planted $k$XOR problem running quartically faster than all known classical algorithms. In this work, we design a new classical algorithm that is quadratically faster than the best previous one, in the case of large constant $k$. Thus for such $k$, the quantum speedup of Schmidhuber et al. becomes only quadratic (though it retains a space advantage). Our algorithm, which also works in the semirandom case, combines tools from sublinear-time algorithms (essentially, the birthday paradox) and polynomial anticoncentration.