New Results in the Simultaneous Message Passing Model

TL;DR

This paper establishes optimal direct sum results for the simultaneous message passing model in both quantum and classical settings, demonstrating the communication cost scales linearly with the number of instances and exploring the differences between SMP and one-way models.

Contribution

It provides the first optimal direct sum results for all relations in the SMP model with shared resources, and compares SMP with one-way communication complexity, especially in quantum scenarios.

Findings

Optimal direct sum results for all relations in SMP model.

Linear scaling of communication cost for multiple instances.

Exponential gap between SMP and one-way models in quantum communication.

Abstract

Consider the following Simultaneous Message Passing (SMP) model for computing a relation f subset of X x Y x Z. In this model Alice, on input x in X and Bob, on input y in Y, send one message each to a third party Referee who then outputs a z in Z such that (x,y,z) in f. We first show optimal 'Direct sum' results for all relations f in this model, both in the quantum and classical settings, in the situation where we allow shared resources (shared entanglement in quantum protocols and public coins in classical protocols) between Alice and Referee and Bob and Referee and no shared resource between Alice and Bob. This implies that, in this model, the communication required to compute k simultaneous instances of f, with constant success overall, is at least k-times the communication required to compute one instance with constant success. This in particular implies an earlier Direct sum result, shown by Chakrabarti, Shi, Wirth and Yao, 2001, for the Equality function (and a class of other so-called robust functions), in the classical smp model with no shared resources between any parties. Furthermore we investigate the gap between the smp model and the one-way model in communication complexity and exhibit a partial function that is exponentially more expensive in the former if quantum communication with entanglement is allowed, compared to the latter even in the deterministic case.