Lower bounds for quantum communication complexity

TL;DR

This paper establishes new exponential lower bounds for bounded error quantum communication complexity using Fourier transform techniques, advancing understanding of quantum communication limits and separating quantum from nondeterministic models.

Contribution

It generalizes classical lower bound methods to quantum communication and introduces new Fourier-based techniques for deriving lower bounds.

Findings

Exponential separation between quantum and nondeterministic quantum communication complexity.

Fourier transform-based methods provide stronger lower bounds than discrepancy.

First large lower bounds for functions with polynomial quantum speedup.

Abstract

We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other lower bound methods based on the Fourier transform, notably showing that \sqrt{\bar{s}(f)/\log n}, for the average sensitivity \bar{s}(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f(x AND y XOR z), where x is a Boolean word held by Alice and y,z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are O(\log n).