The one-way communication complexity of the Boolean Hidden Matching Problem

TL;DR

This paper establishes a tight exponential separation between quantum and classical one-way communication complexities for the Boolean Hidden Matching problem, using Fourier analysis to prove a lower bound.

Contribution

It provides a tight lower bound of Omega(√n) for classical complexity, highlighting an exponential gap with quantum protocols for the problem.

Findings

Classical one-way communication complexity is Omega(√n).

Quantum protocols achieve O(log n) complexity.

Exponential separation between quantum and classical complexities for this problem.

Abstract

We give a tight lower bound of Omega(\sqrt{n}) for the randomized one-way communication complexity of the Boolean Hidden Matching Problem [BJK04]. Since there is a quantum one-way communication complexity protocol of O(\log n) qubits for this problem, we obtain an exponential separation of quantum and classical one-way communication complexity for partial functions. A similar result was independently obtained by Gavinsky, Kempe, de Wolf [GKdW06]. Our lower bound is obtained by Fourier analysis, using the Fourier coefficients inequality of Kahn Kalai and Linial [KKL88].