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.

翻译：我们研究带有建议的分布式唤醒问题，其中节点预先具备关于全局网络的初始知识。在对手唤醒一个节点子集后，预言机为每个节点计算一个比特串（“建议”），目标是高效地唤醒所有休眠节点。我们首次给出了由Dufoulon、Magniez和Pandurangan（PODC 2025）提出的量子路由模型中唤醒问题的消息复杂度上下界。具体而言，我们提出了一种分布式建议方案：给定每个节点$α$比特的建议，该方案以$O( \sqrt{\frac{n^3}{2^{\max\{\lfloor (α-1)/2 \rfloor},0\}}}\cdot\log n )$的消息复杂度（高概率）唤醒所有节点。我们的结果突破了经典端口编号模型在足够稠密图中已知的$Ω( \frac{n^2}{2^α} )$障碍。作为算法的补充，我们证明了分布式量子算法的消息复杂度下界：通过利用查询复杂度模型中单比特描述符问题的下界结果，我们表明无建议的唤醒问题具有$Ω( n^{3/2} )$的量子消息复杂度，该下界与允许的运行时间无关。在由对手决定算法起始节点的场景中，大多数重要图问题隐含地需要解决唤醒问题，因此该下界同样适用于其他基本问题，如单源广播和生成树构建。