Quantum versus Classical Separation in Simultaneous Number-on-Forehead Communication

TL;DR

This paper demonstrates the first exponential separation between quantum and classical communication complexities in a three-player Number-on-Forehead model, introducing a new problem and technique for lower bounds.

Contribution

It presents the Gadgeted Hidden Matching Problem and shows an exponential gap in communication complexity, along with a novel method for proving randomized lower bounds in NOF models.

Findings

Quantum protocol uses O(log n) communication.

Classical protocol requires Omega(n^{1/16}) communication.

New technique for lower bounds may be broadly applicable.

Abstract

Quantum versus classical separation plays a central role in understanding the advantages of quantum computation. In this paper, we present the first exponential separation between quantum and bounded-error randomized communication complexity in a variant of the Number-on-Forehead (NOF) model. Namely, the three-player Simultaneous Number-on-Forehead model. Specifically, we introduce the Gadgeted Hidden Matching Problem and show that it can be solved using only $O(\log n)$ simultaneous quantum communication. In contrast, any simultaneous randomized protocol requires $\Omega(n^{1/16})$ communication. On the technical side, a key obstacle in separating quantum and classical communication in NOF models is that all known randomized NOF lower bound tools, such as the discrepancy method, typically apply to both randomized and quantum protocols. In this regard, our technique provides a new method for proving randomized lower bounds in the NOF setting and may be of independent interest beyond the separation result.