On Quantum Learning Advantage Under Symmetries
Tuyen Nguyen, Mária Kieferová, Amira Abbas
TL;DR
The paper investigates whether symmetry can catalyze a quantum learning advantage within the quantum statistical query framework. It proves an exponential quantum–classical separation for permutation-invariant function classes using quantum Fourier sampling, and develops an orbit-based lower-bound framework showing that, for most common symmetries, quantum and classical query complexities match up to constants, with potential gains arising in highly skewed orbit distributions. It also uncovers a tolerance-based separation where quantum learners succeed at noise levels that defeat classical SQ, highlighting regimes where symmetry facilitates quantum advantages. Collectively, these results delineate when symmetry enables quantum gains in learning and emphasize practical considerations such as state preparation and measurement feasibility, guiding future exploration of symmetry-aware quantum learning.
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.