On the Quantum versus Classical Learnability of Discrete Distributions
This addresses the fundamental question of quantum versus classical learnability in machine learning, offering a provable separation for generative modeling, which is not incremental but foundational.
The paper tackles the problem of comparing classical and quantum learners for generative modeling of discrete distributions, showing that under the decisional Diffie-Hellman assumption, there exists a class of distributions that is not efficiently learnable by classical algorithms but is learnable by an efficient quantum learner, providing a concrete quantum advantage.
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.