Unbounded-Error Classical and Quantum Communication Complexity

TL;DR

This paper extends the arrangement-based approach to analyze unbounded-error classical and quantum communication complexities across various models, establishing tight bounds and demonstrating their near equivalence up to constant factors.

Contribution

It generalizes the arrangement argument to two-way and SMP models, providing tight bounds and showing the close relationship between classical and quantum complexities.

Findings

Quantum and classical unbounded-error communication complexities are within a factor of four.

The gap between weakly unbounded-error quantum and classical complexities is at most three.

All models' complexities are essentially equivalent up to a constant factor.

Abstract

Since the seminal work of Paturi and Simon \cite[FOCS'84 & JCSS'86]{PS86}, the unbounded-error classical communication complexity of a Boolean function has been studied based on the arrangement of points and hyperplanes. Recently, \cite[ICALP'07]{INRY07} found that the unbounded-error {\em quantum} communication complexity in the {\em one-way communication} model can also be investigated using the arrangement, and showed that it is exactly (without a difference of even one qubit) half of the classical one-way communication complexity. In this paper, we extend the arrangement argument to the {\em two-way} and {\em simultaneous message passing} (SMP) models. As a result, we show similarly tight bounds of the unbounded-error two-way/one-way/SMP quantum/classical communication complexities for {\em any} partial/total Boolean function, implying that all of them are equivalent up to a multiplicative constant of four. Moreover, the arrangement argument is also used to show that the gap between {\em weakly} unbounded-error quantum and classical communication complexities is at most a factor of three.