Quantum Communication Cannot Simulate a Public Coin

TL;DR

This paper demonstrates that quantum communication cannot always replace classical public-coin protocols in the SMP model, showing an exponential separation and exploring the limits of quantum fingerprinting with geometric insights.

Contribution

It provides a counterexample to the universality of quantum simulation of public coins and characterizes quantum fingerprinting using geometric tools from machine learning.

Findings

Classical public-coin protocols can be exponentially more efficient than quantum protocols in some cases.

Quantum fingerprinting's power is characterized through geometric methods.

A nearly tight analysis of the Hamming distance problem is presented.

Abstract

We study the simultaneous message passing model of communication complexity. Building on the quantum fingerprinting protocol of Buhrman et al., Yao recently showed that a large class of efficient classical public-coin protocols can be turned into efficient quantum protocols without public coin. This raises the question whether this can be done always, i.e. whether quantum communication can always replace a public coin in the SMP model. We answer this question in the negative, exhibiting a communication problem where classical communication with public coin is exponentially more efficient than quantum communication. Together with a separation in the other direction due to Bar-Yossef et al., this shows that the quantum SMP model is incomparable with the classical public-coin SMP model. In addition we give a characterization of the power of quantum fingerprinting by means of a connection to geometrical tools from machine learning, a quadratic improvement of Yao's simulation, and a nearly tight analysis of the Hamming distance problem from Yao's paper.