Quantum Algorithms for Testing Properties of Distributions

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 $|p-q|_1$ can be estimated with a constant precision using only $O(N^{1/2})$ queries in the quantum settings, whereas classical computers need $Omega(N^{1-o(1)})$ 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 $Omega(N^{1/2})$. A quantum algorithm for testing Uniformity has been recently independently discovered by Chakraborty et al. cite{CFMW09}