A super-polynomial quantum-classical separation for density modelling

TL;DR

This paper demonstrates a super-polynomial quantum advantage in density modelling under cryptographic assumptions, introducing new insights into quantum-classical separations in unsupervised learning.

Contribution

It establishes a density modelling problem with super-polynomial quantum advantage, linking hardness results to pseudo-random functions and offering potential for broader separations.

Findings

Quantum algorithms outperform classical ones in specific density modelling tasks

Weak pseudo-random functions can be used to construct classically hard density problems

Provides insights into the relationship between cryptographic assumptions and learning hardness

Abstract

Density modelling is the task of learning an unknown probability density function from samples, and is one of the central problems of unsupervised machine learning. In this work, we show that there exists a density modelling problem for which fault-tolerant quantum computers can offer a super-polynomial advantage over classical learning algorithms, given standard cryptographic assumptions. Along the way, we provide a variety of additional results and insights, of potential interest for proving future distribution learning separations between quantum and classical learning algorithms. Specifically, we (a) provide an overview of the relationships between hardness results in supervised learning and distribution learning, and (b) show that any weak pseudo-random function can be used to construct a classically hard density modelling problem. The latter result opens up the possibility of proving quantum-classical separations for density modelling based on weaker assumptions than those necessary for pseudo-random functions.