The Quantum Message Complexity of Distributed Wake-Up with Advice
Peter Robinson, Ming Ming Tan
TL;DR
This work extends the wake-up problem to a distributed quantum routing setting with per-node advice, introducing a concrete advising scheme and proving both upper and lower bounds on quantum message complexity. The upper bound employs an epoch-based protocol where the oracle assigns compact, type-specific advice (including a level-β port range mechanism) and nodes perform iterated quantum searches to wake sleeping neighbors, achieving responsive scaling in terms of $n$ and the advice length $α$, specifically $O\left(\sqrt{\frac{n^3}{2^{\max\{\lfloor(\alpha-1)/2\rfloor,0\}}}} \log n\right)$ messages. The lower bound leverages a reduction to the Descriptor problem in the quantum query model, using a family of lower-bound graphs with hidden matchings to show that wake-up requires $\Omega\left(n^{3/2}\right)$ messages even without advice, with implications for related problems like spanning tree construction and single-source broadcast under adversarial wake-up. Overall, the paper demonstrates a tangible quantum advantage in message complexity for wake-up with limited advice and clarifies fundamental limits through a rigorous lower-bound framework that connects distributed algorithms to quantum query complexities.
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 $α$ 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.