# Distributional property testing in a quantum world

## TL;DR

This paper introduces quantum algorithms that significantly accelerate the testing of distribution properties, such as closeness, independence, and entropy estimation, applicable to both classical and quantum distributions.

## Contribution

It presents the first quantum algorithms with speed-ups for property testing of density operators and improves precision dependence for classical distribution testing.

## Key findings

- Quantum algorithms outperform classical methods in distribution testing tasks.
- Speed-ups achieved for testing closeness, independence, and entropy of distributions.
- Algorithms work with coherent quantum access to data.

## Abstract

A fundamental problem in statistics and learning theory is to test properties of distributions. We show that quantum computers can solve such problems with significant speed-ups. In particular, we give fast quantum algorithms for testing closeness between unknown distributions, testing independence between two distributions, and estimating the Shannon / von Neumann entropy of distributions. The distributions can be either classical or quantum, however our quantum algorithms require coherent quantum access to a process preparing the samples. Our results build on the recent technique of quantum singular value transformation, combined with more standard tricks such as divide-and-conquer. The presented approach is a natural fit for distributional property testing both in the classical and the quantum case, demonstrating the first speed-ups for testing properties of density operators that can be accessed coherently rather than only via sampling; for classical distributions our algorithms significantly improve the precision dependence of some earlier results.