The Quantum Message Complexity of Distributed Wake-Up with Advice

2026-02-05

AI Summary

This study addresses the problem of node wake-up in distributed networks with prior information (advice), where an adversary activates a subset of nodes and the goal is to efficiently awaken the remaining dormant nodes. Under the quantum routing model, the work presents the first advice-based quantum distributed wake-up algorithm. By combining query complexity lower-bound techniques with probabilistic analysis, it overcomes the classical message complexity barrier of Ω(n²/2^α) in the port-numbering model. The main contributions include establishing a lower bound of Ω(n^{3/2}) on quantum message complexity without advice and designing an advising scheme that achieves an upper bound of O(√(n³/2^{max{⌊(α−1)/2⌋,0}})·log n), thereby establishing a tight complexity characterization for this problem in the quantum setting.

Technology Category

• Machine Learning: Quantum Machine Learning
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the distributed wake-up problem with advice, where nodes are equipped with initial knowledge about the network at large. After the adversary awakens a subset of nodes, an oracle computes a bit string (``the advice'') for each node, and the goal is to wake up all sleeping nodes efficiently. We present the first upper and lower bounds on the message complexity for wake-up in the quantum routing model, introduced by Dufoulon, Magniez, and Pandurangan (PODC 2025). In more detail, we give a distributed advising scheme that, given $\alpha$ bits of advice per node, wakes up all nodes with a message complexity of $O( \sqrt{\frac{n^3}{2^{\max\{\lfloor (\alpha-1)/2 \rfloor},0\}}}\cdot\log n )$ with high probability. Our result breaks the $\Omega( \frac{n^2}{2^\alpha} )$ 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 $\Omega( n^{3/2} )$ 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.

Problem

• distributed wake-up
• quantum message complexity
• advice
• message complexity
• quantum routing model

Innovation

• quantum message complexity
• distributed wake-up
• advice
• lower bound
• quantum routing model