Abstract
The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive t samples from one distribution over [n], and the goal is to decide whether their two distributions are equal, or are ε-far apart in the l₁-distance. In the present paper we show that the quantum communication complexity of this problem is O(n/(tε²)) qubits when the distributions have low l₂-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.