What Can be Observed Locally? Round-based Models for Quantum Distributed Computing

TL;DR

This paper critically examines claims that quantum extensions to the LOCAL model improve distributed computing problems, clarifies misconceptions, and identifies fundamental limitations of quantum approaches in this context.

Contribution

It clarifies misconceptions about quantum advantages in distributed computing and introduces the concept of physical locality (PLOCAL) to establish limitations of quantum models.

Findings

Quantum effects can reduce round complexity in some problems.

Quantum variants of LOCAL still face non-trivial limitations.

Lower bounds for problems like Maximal Independent Set remain valid with quantum models.

Abstract

Recently, several claims have been made that certain fundamental problems of distributed computing, including Leader Election and Distributed Consensus, begin to admit feasible and efficient solutions when the model of distributed computation is extended so as to apply quantum processing. This has been achieved in one of two distinct ways: (1) by initializing the system in a quantum entangled state, and/or (2) by applying quantum communication channels. In this paper, we explain why some of these prior claims are misleading, in the sense that they rely on changes to the model unrelated to quantum processing. On the positive side, we consider the aforementioned quantum extensions when applied to Linial's well-established LOCAL model of distributed computing. For both types of extensions, we put forward valid proof-of-concept examples of distributed problems whose round complexity is in fact reduced through genuinely quantum effects, in contexts which do not depend on the anonymity of nodes. Finally, we show that even the quantum variants of the LOCAL model have non-trivial limitations, captured by a very simple (purely probabilistic) notion which we call "physical locality" (PLOCAL). While this is strictly weaker than the "computational locality" of the classical LOCAL model, it nevertheless implies that for many distributed combinatorial optimization problems, such as Maximal Independent Set, the best currently known lower time bounds cannot be broken by applying quantum processing, in any conceivable way.