Provable Advantage in Quantum PAC Learning

TL;DR

This paper demonstrates a generic quantum advantage in PAC learning, showing that quantum algorithms can achieve a square root improvement in sample complexity over classical methods for any concept class with finite VC dimension.

Contribution

The authors extend the definition of quantum PAC learning to establish a universal quantum advantage with near-optimal sample complexity bounds for all concept classes.

Findings

Quantum PAC learners can achieve a square root improvement in sample complexity.

The paper provides tight bounds matching the upper and lower limits up to polylogarithmic factors.

The results apply broadly to any concept class with finite VC dimension.

Abstract

We revisit the problem of characterising the complexity of Quantum PAC learning, as introduced by Bshouty and Jackson [SIAM J. Comput. 1998, 28, 1136-1153]. Several quantum advantages have been demonstrated in this setting, however, none are generic: they apply to particular concept classes and typically only work when the distribution that generates the data is known. In the general case, it was recently shown by Arunachalam and de Wolf [JMLR, 19 (2018) 1-36] that quantum PAC learners can only achieve constant factor advantages over classical PAC learners. We show that with a natural extension of the definition of quantum PAC learning used by Arunachalam and de Wolf, we can achieve a generic advantage in quantum learning. To be precise, for any concept class $\mathcal{C}$ of VC dimension $d$, we show there is an $(\epsilon, \delta)$-quantum PAC learner with sample complexity \[ O\left(\frac{1}{\sqrt{\epsilon}}\left[d+ \log(\frac{1}{\delta})\right]\log^9(1/\epsilon)\right). \] Up to polylogarithmic factors, this is a square root improvement over the classical learning sample complexity. We show the tightness of our result by proving an $\Omega(d/\sqrt{\epsilon})$ lower bound that matches our upper bound up to polylogarithmic factors.