On the communication complexity of XOR functions

TL;DR

This paper investigates the quantum and classical communication complexities of XOR functions, providing complete characterizations for exact one-way protocols and exploring relationships for specific classes of functions, with conjectures and protocols for general cases.

Contribution

It offers a complete characterization of one-way communication complexity for XOR functions and proposes a structural conjecture relating quantum and classical complexities.

Findings

Exact one-way complexity characterized for all f

Quantum and classical complexities are quadratically related for monotone f

Quantum complexity of linear threshold functions is Theta(n)

Abstract

An XOR function is a function of the form g(x,y) = f(x + y), for some boolean function f on n bits. We study the quantum and classical communication complexity of XOR functions. In the case of exact protocols, we completely characterise one-way communication complexity for all f. We also show that, when f is monotone, g's quantum and classical complexities are quadratically related, and that when f is a linear threshold function, g's quantum complexity is Theta(n). More generally, we make a structural conjecture about the Fourier spectra of boolean functions which, if true, would imply that the quantum and classical exact communication complexities of all XOR functions are asymptotically equivalent. We give two randomised classical protocols for general XOR functions which are efficient for certain functions, and a third protocol for linear threshold functions with high margin. These protocols operate in the symmetric message passing model with shared randomness.