On the Quantum versus Classical Learnability of Discrete Distributions

TL;DR

This paper demonstrates a specific class of discrete distributions that quantum algorithms can learn efficiently while classical algorithms cannot, under certain cryptographic assumptions, highlighting a provable quantum advantage in generative modeling.

Contribution

The paper constructs a class of distributions that are not classically PAC learnable but are efficiently learnable by quantum algorithms, under the decisional Diffie-Hellman assumption.

Findings

Quantum learners outperform classical ones for certain distributions.

Classical PAC learnability hardness results are discussed.

Relationship between Boolean function and distribution learnability is analyzed.

Abstract

Here we study the comparative power of classical and quantum learners for generative modelling within the Probably Approximately Correct (PAC) framework. More specifically we consider the following task: Given samples from some unknown discrete probability distribution, output with high probability an efficient algorithm for generating new samples from a good approximation of the original distribution. Our primary result is the explicit construction of a class of discrete probability distributions which, under the decisional Diffie-Hellman assumption, is provably not efficiently PAC learnable by a classical generative modelling algorithm, but for which we construct an efficient quantum learner. This class of distributions therefore provides a concrete example of a generative modelling problem for which quantum learners exhibit a provable advantage over classical learning algorithms. In addition, we discuss techniques for proving classical generative modelling hardness results, as well as the relationship between the PAC learnability of Boolean functions and the PAC learnability of discrete probability distributions.