  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 $\epsilon$-far apart in the $l_1$-distance. In
the present paper we show that the quantum communication complexity of this
problem is $\tilde{O}(n/(t\epsilon^2))$ qubits when the distributions have low
$l_2$-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.
