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Exponential Quantum Advantage for Message Complexity in Distributed Algorithms

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

TL;DR

This work establishes an exponential quantum advantage in the message complexity of a fundamental distributed task: routing a message from a designated source to a target in welded-tree networks. The authors implement a succinct discrete-time quantum walk in a distributed setting, enabling a quantum algorithm that uses only polynomial-in-$n$ many qubits/messages to route data, in contrast to a proven exponential classical lower bound. The quantum upper bound leverages the Li–Li–Luo succinct walk to ensure poly$(n)$ communication, while the classical lower bound adapts Childs–Cleve–Deotto–Farhi–Gutmann–Spielman’s embedding arguments to the distributed context, ruling out efficient classical routing. The results highlight a potential broad impact of quantum communication in distributed computing, albeit demonstrated on a specific nontrivial topology, and motivate further exploration of quantum advantages for other message-based distributed tasks.

Abstract

We investigate how much quantum distributed algorithms can outperform classical distributed algorithms with respect to the message complexity (the overall amount of communication used by the algorithm). Recently, Dufoulon, Magniez and Pandurangan (PODC 2025) have shown a polynomial quantum advantage for several tasks such as leader election and agreement. In this paper, we show an exponential quantum advantage for a fundamental task: routing information between two specified nodes of a network. We prove that for the family of ``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.

Exponential Quantum Advantage for Message Complexity in Distributed Algorithms

TL;DR

This work establishes an exponential quantum advantage in the message complexity of a fundamental distributed task: routing a message from a designated source to a target in welded-tree networks. The authors implement a succinct discrete-time quantum walk in a distributed setting, enabling a quantum algorithm that uses only polynomial-in- many qubits/messages to route data, in contrast to a proven exponential classical lower bound. The quantum upper bound leverages the Li–Li–Luo succinct walk to ensure poly communication, while the classical lower bound adapts Childs–Cleve–Deotto–Farhi–Gutmann–Spielman’s embedding arguments to the distributed context, ruling out efficient classical routing. The results highlight a potential broad impact of quantum communication in distributed computing, albeit demonstrated on a specific nontrivial topology, and motivate further exploration of quantum advantages for other message-based distributed tasks.

Abstract

We investigate how much quantum distributed algorithms can outperform classical distributed algorithms with respect to the message complexity (the overall amount of communication used by the algorithm). Recently, Dufoulon, Magniez and Pandurangan (PODC 2025) have shown a polynomial quantum advantage for several tasks such as leader election and agreement. In this paper, we show an exponential quantum advantage for a fundamental task: routing information between two specified nodes of a network. We prove that for the family of ``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.

Paper Structure

This paper contains 18 sections, 14 theorems, 49 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Quantum distributed computing.
  3. Our result.
  4. Technical overview of the quantum upper bound (\ref{['main_thm1']}).
  5. Technical overview of the classical lower bound (\ref{['main_thm2']}).
  6. Preliminaries
  7. Distributed computing
  8. Classical deterministic message complexity.
  9. Deterministic configuration of a classical system.
  10. Classical randomized message complexity.
  11. Quantum message complexity.
  12. Quantum walks
  13. Quantum Upper Bound
  14. Definitions.
  15. Basic algorithm.
  16. ...and 3 more sections

Key Result

Theorem 1.1

There exists a quantum distributed algorithm that solves with probability at least $1-1/\exp(n)$ the point-to-point routing problem in $\mathcal{G}_n$ with message complexity $O(b\cdot\mathrm{poly}(n))$. This upper bound holds even when $x$ is a quantum data of $b$ qubits if $\mathtt{source}$ has se

Figures (1)

  • Figure 1: An instance of welded trees in $\mathcal{G}_{n}$ for $n=4$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.1: Li+24
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • ...and 18 more