Quantum algorithms for testing properties of distributions

TL;DR

This paper demonstrates that quantum algorithms can significantly reduce the number of samples needed to test properties of distributions, such as closeness, uniformity, and orthogonality, compared to classical methods.

Contribution

It introduces quantum algorithms that improve query complexity for distribution property testing, notably achieving N^{1/2} for L_1-distance estimation and N^{1/3} for uniformity and orthogonality tests.

Findings

Quantum algorithms estimate L_1-distance with ~N^{1/2} queries.

Quantum algorithms test Uniformity and Orthogonality with O(N^{1/3}) queries.

Classical methods require at least a(N) queries for these problems.

Abstract

Suppose one has access to oracles generating samples from two unknown probability distributions P and Q on some N-element set. How many samples does one need to test whether the two distributions are close or far from each other in the L_1-norm ? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the L_1-distance between P and Q can be estimated with a constant precision using approximately N^{1/2} queries in the quantum settings, whereas classical computers need \Omega(N) queries. We also describe quantum algorithms for testing Uniformity and Orthogonality with query complexity O(N^{1/3}). The classical query complexity of these problems is known to be \Omega(N^{1/2}).