Entangled simultaneity versus classical interactivity in communication complexity

TL;DR

This paper introduces a partial function called Shape that demonstrates a clear separation between quantum simultaneous-message protocols and classical two-way protocols, showing quantum advantage in communication complexity.

Contribution

It provides the first example of a function with an efficient quantum simultaneous-messages protocol but no efficient classical two-way protocol.

Findings

Shape can be computed with poly-logarithmic quantum simultaneous messages.

Classical two-way complexity for Shape is polynomially lower bounded.

Confirms quantum advantage in communication complexity for this function.

Abstract

In 1999 Raz demonstrated a partial function that had an efficient quantum two-way communication protocol but no efficient classical two-way protocol and asked, whether there existed a function with an efficient quantum one-way protocol, but still no efficient classical two-way protocol. In 2010 Klartag and Regev demonstrated such a function and asked, whether there existed a function with an efficient quantum simultaneous-messages protocol, but still no efficient classical two-way protocol. In this work we answer the latter question affirmatively and present a partial function Shape, which can be computed by a protocol sending entangled simultaneous messages of poly-logarithmic size, and whose classical two-way complexity is lower bounded by a polynomial.