Separating quantum communication and approximate rank

TL;DR

This paper demonstrates the first known separation between quantum communication complexity and the logarithm of approximate rank, showing that quantum complexity can be nearly quadratically larger for certain total functions.

Contribution

It introduces a new total function exhibiting a quadratic separation, using a novel framework combining cheat sheet and communication lookup methods, and establishes bounds via quantum information theory.

Findings

Quantum communication complexity can be nearly quadratically larger than approximate rank.

Constructed a total function with a significant separation in complexity measures.

Bound quantum communication complexity using discrepancy and relate approximate rank to Boolean circuit size.

Abstract

One of the best lower bound methods for the quantum communication complexity of a function H (with or without shared entanglement) is the logarithm of the approximate rank of the communication matrix of H. This measure is essentially equivalent to the approximate gamma_2 norm and generalized discrepancy, and subsumes several other lower bounds. All known lower bounds on quantum communication complexity in the general unbounded-round model can be shown via the logarithm of approximate rank, and it was an open problem to give any separation at all between quantum communication complexity and the logarithm of the approximate rank. In this work we provide the first such separation: We exhibit a total function H with quantum communication complexity almost quadratically larger than the logarithm of its approximate rank. We construct H using the communication lookup function framework of Anshu et al. (FOCS 2016) based on the cheat sheet framework of Aaronson et al. (STOC 2016). From a starting function F, this framework defines a new function H=F_G. Our main technical result is a lower bound on the quantum communication complexity of F_G in terms of the discrepancy of F, which we do via quantum information theoretic arguments. We show the upper bound on the approximate rank of F_G by relating it to the Boolean circuit size of the starting function F.