Separations in communication complexity using cheat sheets and information complexity

TL;DR

This paper establishes new super-quadratic and other significant separations between various models of communication complexity for total functions, using a novel framework inspired by query complexity cheat sheets.

Contribution

It introduces the first super-quadratic separation between quantum and randomized communication complexity for total functions, and extends the cheat sheet framework to communication complexity via lookup functions.

Findings

First super-quadratic separation (power 2.5) between quantum and randomized communication complexity.

Improved 1.5 power separation between exact quantum and randomized communication complexity.

Nearly optimal quadratic separation between randomized communication complexity and log of partition number.

Abstract

While exponential separations are known between quantum and randomized communication complexity for partial functions (Raz, STOC 1999), the best known separation between these measures for a total function is quadratic, witnessed by the disjointness function. We give the first super-quadratic separation between quantum and randomized communication complexity for a total function, giving an example exhibiting a power 2.5 gap. We further present a 1.5 power separation between exact quantum and randomized communication complexity, improving on the previous ~1.15 separation by Ambainis (STOC 2013). Finally, we present a nearly optimal quadratic separation between randomized communication complexity and the logarithm of the partition number, improving upon the previous best power 1.5 separation due to G\"o\"os, Jayram, Pitassi, and Watson. Our results are the communication analogues of separations in query complexity proved using the recent cheat sheet framework of Aaronson, Ben-David, and Kothari (STOC 2016). Our main technical results are randomized communication and information complexity lower bounds for a family of functions, called lookup functions, that generalize and port the cheat sheet framework to communication complexity.