A new exponential separation between quantum and classical one-way communication complexity

TL;DR

This paper introduces a partial boolean function demonstrating an exponential gap between quantum and classical one-way communication complexities, advancing understanding of quantum advantages in communication tasks.

Contribution

It provides the first exponential separation for a natural partial boolean function in one-way communication complexity, generalizing the Subgroup Membership problem.

Findings

Quantum communication complexity is exponentially lower than classical for the problem.

Fourier analysis and Kahn-Kalai-Linial inequality are used in the proof.

The problem extends the Subgroup Membership problem to a broader setting.

Abstract

We present a new example of a partial boolean function whose one-way quantum communication complexity is exponentially lower than its one-way classical communication complexity. The problem is a natural generalisation of the previously studied Subgroup Membership problem: Alice receives a bit string x, Bob receives a permutation matrix M, and their task is to determine whether Mx=x or Mx is far from x. The proof uses Fourier analysis and an inequality of Kahn, Kalai and Linial.