A Classical Quadratic Speedup for Planted $k$XOR

2025-08-12

AI Summary

This work addresses the noisy $k$XOR problem—where $k$ is a large constant—and presents the first classical algorithm achieving quadratic speedup, thereby substantially narrowing the quantum advantage from prior quartic to quadratic. Methodologically, it introduces a novel integration of sublinear-time algorithmic frameworks with polynomial anti-concentration phenomena, while strategically leveraging the birthday paradox to design an efficient solver tailored to semi-random noise settings. The resulting algorithm improves upon the best prior classical time complexity by a quadratic factor. Although quantum algorithms retain asymptotic advantages in space complexity, their temporal superiority is materially diminished. This work advances the understanding of the classical–quantum boundary for the $k$XOR problem, offering new insights into the interplay between noise structure, algorithmic primitives, and computational hardness.

Technology Category

Machine Learning: Quantum Machine LearningSearch and Optimization: Mixed Discrete/Continuous SearchConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Applications of cryptographySearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Improves classical algorithm speed for planted kXOR
• Reduces quantum speedup gap to quadratic for large k
• Works in semirandom cases using sublinear-time techniques

Innovation

• Classical algorithm with quadratic speedup
• Uses sublinear-time techniques
• Employs polynomial anticoncentration