Quantum Communication Complexity of Distribution Testing

TL;DR

This paper demonstrates that quantum communication protocols can significantly reduce the amount of communication needed to test the closeness of distributions with classical samples, especially for distributions with low $l_2$-norm.

Contribution

It establishes the quantum communication complexity for distribution testing, showing a quadratic improvement over classical methods and providing matching lower bounds.

Findings

Quantum protocol achieves $ ilde{O}(n/(t ext{ } ext{epsilon}^2))$ qubits complexity.

Quantum protocols outperform classical protocols for distribution testing with classical samples.

Matching lower bounds confirm the optimality of the quantum approach.

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 $\epsilon$-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.