Exponential Quantum Advantage for Message Complexity in Distributed Algorithms

2025-10-02

AI Summary

This paper investigates the message complexity of routing information between two nodes in distributed networks. Method: Leveraging the compact quantum walk on welded trees introduced by Li, Li, and Luo, we design a quantum routing algorithm achieving message complexity $O(log N)$. To establish a tight classical lower bound, we adapt and extend techniques from query complexity—specifically, the polynomial method and adversary bounds—to the distributed message-passing model. Contribution/Results: We prove that any classical algorithm requires $Omega(N^{1/4})$ messages on the welded tree family, thereby demonstrating an exponential quantum advantage in message complexity. This constitutes the first rigorous separation showing exponential quantum speedup for message complexity in distributed computing, surpassing prior results that only achieved polynomial improvements.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Content-based information diffusion

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.

Problem

• Exponential quantum advantage in distributed message complexity
• Quantum routing outperforms classical algorithms on welded trees
• Lifting query complexity bounds to message complexity analysis

Innovation

• Quantum distributed algorithm for routing information
• Exponential message complexity advantage over classical
• Succinct quantum walks implementation on welded trees