The Quantum Message Complexity of Distributed Wake-Up with Advice

TL;DR

This paper investigates the quantum message complexity of the distributed wake-up problem with advice, providing new upper and lower bounds that outperform classical results in dense graphs and establishing fundamental quantum limits.

Contribution

It introduces the first quantum bounds for wake-up with advice, including an algorithm with improved message complexity and a lower bound based on quantum query complexity.

Findings

Quantum algorithm achieves $O( rac{ ext{poly}(n)}{ ext{advice}})$ message complexity.

Lower bound of $ ilde{ ext{Omega}}(n^{3/2})$ message complexity without advice.

Quantum bounds outperform classical results in dense graph scenarios.

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.