Exponential Separation between Quantum Communication and Logarithm of Approximate Rank

TL;DR

This paper demonstrates that the sink function exhibits a polynomial quantum communication complexity despite having polynomial approximate rank, providing an exponential separation and refuting the quantum log-approximate-rank conjecture.

Contribution

It extends the exponential separation result to the quantum setting and introduces a new proof technique for classical lower bounds using fooling distributions.

Findings

Quantum communication complexity of sink function is polynomial.

Exponential separation between quantum communication complexity and log of approximate rank.

Refutes the quantum log-approximate-rank conjecture.

Abstract

Chattopadhyay, Mande and Sherif (ECCC 2018) recently exhibited a total Boolean function, the sink function, that has polynomial approximate rank and polynomial randomized communication complexity. This gives an exponential separation between randomized communication complexity and logarithm of the approximate rank, refuting the log-approximate-rank conjecture. We show that even the quantum communication complexity of the sink function is polynomial, thus also refuting the quantum log-approximate-rank conjecture. Our lower bound is based on the fooling distribution method introduced by Rao and Sinha (ECCC 2015) for the classical case and extended by Anshu, Touchette, Yao and Yu (STOC 2017) for the quantum case. We also give a new proof of the classical lower bound using the fooling distribution method.