On Quantum Learning Advantage Under Symmetries

TL;DR

This paper investigates whether quantum algorithms can leverage symmetry to outperform classical algorithms in learning tasks, revealing exponential and tolerance-based advantages under certain symmetric structures.

Contribution

It demonstrates exponential separation between quantum and classical statistical query models for symmetric functions and identifies conditions for quantum advantage under symmetry.

Findings

Exponential separation between QSQ and SQ for permutation-invariant functions.

Lower bounds on QSQ complexity match classical SQ bounds for common symmetries.

Quantum learners succeed at noise levels where classical SQ algorithms fail.

Abstract

Symmetry underlies many of the most effective classical and quantum learning algorithms, yet whether quantum learners can gain a fundamental advantage under symmetry-imposed structures remains an open question. Based on evidence that classical statistical query ($\SQ$) frameworks have revealed exponential query complexity in learning symmetric function classes, we ask: can quantum learning algorithms exploit the problem symmetry better? In this work, we investigate the potential benefits of symmetry within the quantum statistical query ($\QSQ$) model, which is a natural quantum analog of classical $\SQ$. Our results uncover three distinct phenomena: (i) we obtain an exponential separation between $\QSQ$ and $\SQ$ on a permutation-invariant function class; (ii) we establish query complexity lower bounds for $\QSQ$ learning that match, up to constant factors, the corresponding classical $\SQ$ lower bounds for most commonly studied symmetries; however, the potential advantages may occur under highly skewed orbit distributions; and (iii) we further identify a tolerance-based separation exists, where quantum learners succeed at noise levels that render classical $\SQ$ algorithms ineffective. Together, these findings provide insight into when symmetry can enable quantum advantage in learning.