Unconditional Quantum PAC Learning Advantage in the Large Error Regime

Abstract

We establish an unconditional super-polynomial separation between quantum and classical PAC learning of Boolean functions under the uniform distribution in the large error regime. For every advantage $λ(n) = n^{-ω(1)}$, we construct a concept class that a polynomial-time quantum learner learns to agreement $1/2+λ(n)$ with success probability at least $1-1/n^k$. The learner uses only quantum random examples and outputs a polynomial-size deterministic classical circuit. In contrast, no polynomial-time classical learner achieves agreement $1/2 + λ(n)/n^k$ with success probability greater than $1/n^k$, even with membership queries. To achieve this, we develop a nearly-optimal quantum learner that uses quantum random examples to produce a hypothesis achieving a prescribed correlation with an arbitrary Boolean function. The learner generalizes Fourier sampling from parity hypotheses to quadratic hypotheses. While standard Fourier sampling is insensitive to the sign of the correlation, we use interference to bias sampling toward positively correlated hypotheses. This is combined with a careful analysis of the resulting distribution to boost both the learner's success probability and its agreement with the target function. We believe that the resulting quantum learning algorithm might be of independent interest. Finally, although a robust quantum PAC learning advantage holds unconditionally in the large error regime, we note that an analogous separation between quantum and classical learning in the small error regime would imply that $\mathsf{NP} \nsubseteq \mathsf{BPP}$.