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.

arXiv:2502.07416ACM Symposium on Principles of Distributed Computing (PODC)Jun 13, 20251 citation

arXiv:2510.01657

Exponential Quantum Advantage for Message Complexity in Distributed Algorithms

François Le Gall, Maël Luce, Marchand, Joseph, Mathieu Roget

Published Oct 2, 2025• 0 citations

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Abstract

welded trees" introduced in the seminal work by Childs, Cleve, Deotto, Farhi, Gutmann and Spielman (STOC 2003), there exists a quantum distributed algorithm that transfers messages from the entrance of the graph to the exit with message complexity exponentially smaller than any classical algorithm. Our quantum algorithm is based on the recent "succinct" implementation of quantum walks over the welded trees by Li, Li and Luo (SODA 2024). Our classical lower bound is obtained by

lifting'' the lower bound from Childs, Cleve, Deotto, Farhi, Gutmann and Spielman (STOC 2003) from query complexity to message complexity.

DOI: 10.48550/arxiv.2510.01657

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\tilde{O}(n)

for leader election, broadcast, and MST, and

\tilde{O}(\sqrt{mn})

for BFS (

n

and

m

are the number of nodes and edges of the network, respectively). These message bounds are nearly tight in the quantum routing model since we show almost matching corresponding quantum message lower bounds. Our results significantly improve on the prior work of [PODC 2025], who presented distributed quantum algorithms under the same model that had a message complexity of

\tilde{O}(\sqrt{mn})

for leader election. Our algorithms demonstrate the significant communication advantage that quantum routing has over classical in distributed computing, since

Ω(m)

is a well-established classical message lower bound for leader election, broadcast, MST, and BFS that applies even to randomized Monte-Carlo algorithms [JACM 2015]. Thus, our quantum algorithms can, in general, give a quadratic advantage in the communication cost for these fundamental problems. A main technical tool we use to design our distributed algorithms is quantum walks based on electric networks. We posit a framework for using quantum walks in the distributed setting to design communication-efficient distributed quantum algorithms. Our framework can be used as a black box to significantly reduce communication costs and may be of independent interest. Additionally, our lower-bound technique for establishing distributed quantum message lower bounds can also be applied to other problems.

n

, any quantum query algorithm requires at least

Ω(2^{n/24})

queries in order to succeed with constant probability. Our proof uses the compressed permutation oracle technique in order to construct databases which track the graph information an algorithm has learned and forgotten, and show that no efficient quantum algorithm can build an entrance-to-exit path in these records.

α

bits of advice per node, wakes up all nodes with a message complexity of

O( \sqrt{\frac{n^3}{2^{\max\{\lfloor (α-1)/2 \rfloor},0\}}}\cdot\log n )

with high probability. Our result breaks the

Ω( \frac{n^2}{2^α} )

barrier known for the classical port numbering model in sufficiently dense graphs. To complement our algorithm, we give a lower bound on the message complexity for distributed quantum algorithms: By leveraging a lower bound result for the single-bit descriptor problem in the query complexity model, we show that wake-up has a quantum message complexity of

Ω( n^{3/2} )

without advice, which holds independently of how much time we allow. In the setting where an adversary decides which nodes start the algorithm, most graph problems of interest implicitly require solving wake-up, and thus the same lower bound also holds for other fundamental problems such as single-source broadcast and spanning tree construction.

Ω(2^{n/12})

queries are required to find a path in a height-

n

tree with constant success probability.