Unbounded-error classical and quantum communication complexity

Since the seminal work of Paturi and Simon [26, FOCS'84 & JCSS'861, the unbounded-error classical communication complexity of a Boolean function has been studied based on the arrangement of points and hyperplanes. Recently; [14, ICALP'07] found that the unbounded-error quantum communication complexity in the 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 two-way and 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 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 weakly unbounded-error quantum and classical communication complexities is at most a factor of three.