Quantum Communication Advantage for Leader Election and Agreement
Fabien Dufoulon, Frédéric Magniez, Gopal Pandurangan
TL;DR
This work shows that quantum communication can substantially reduce the total number of messages in fundamental distributed tasks, namely leader election and implicit agreement, within synchronous CONGEST networks. By developing a framework that quantizes distributed algorithms and leverages Grover search, quantum counting, and quantum walks, the authors construct sublinear quantum-message protocols across complete, diameter-2, and general graphs (including those with small mixing time). The key results include quantum leader election with complexities up to $\tilde{O}(n^{1/3})$ on complete graphs, $\tilde{O}(n^{2/3})$ on diameter-2 graphs, and $\tilde{O}(\tau^{5/3} n^{1/3})$ on graphs with mixing time $\tau$, as well as a general-$m,n$ bound of $\tilde{O}(\sqrt{m n})$ messages for general graphs; a quantum implicit-agreement protocol on complete networks with shared randomness achieves $\tilde{O}(n^{1/5})$ messages. These results illustrate a pathway to sublinear quantum communication in distributed systems and raise important open questions about lower bounds and optimal trade-offs with rounds. The work significantly advances understanding of how quantum subroutines can be embedded in distributed routing to beat classical message-lower-bound barriers, with potential practical impact for large-scale quantum-enabled networks.
Abstract
This work focuses on understanding the quantum message complexity of two central problems in distributed computing, namely, leader election and agreement in synchronous message-passing communication networks. We show that quantum communication gives an advantage for both problems by presenting quantum distributed algorithms that significantly outperform their respective classical counterparts under various network topologies. While prior works have studied and analyzed quantum distributed algorithms in the context of (improving) round complexity, a key conceptual contribution of our work is positing a framework to design and analyze the message complexity of quantum distributed algorithms. We present and show how quantum algorithmic techniques such as Grover search, quantum counting, and quantum walks can make distributed algorithms significantly message-efficient. In particular, our leader election protocol for diameter-2 networks uses quantum walks to achieve the improved message complexity. To the best of our knowledge, this is the first such application of quantum walks in distributed computing.