We present new distributed quantum algorithms for fundamental distributed computing problems, namely, leader election, broadcast, Minimum Spanning Tree (MST), and Breadth-First Search (BFS) tree, in arbitrary networks. These algorithms are (essentially) optimal with respect to their communication (message) complexity in the {\em quantum routing model} introduced in [PODC 2025]. The message complexity of our algorithms is $\tilde{O}(n)$ for leader election, broadcast, and MST, and $\tilde{O}(\sqrt{mn})$ for BFS ($n$ and $m$ are the number of nodes and edges of the network, respectively). These message bounds are nearly tight in the quantum routing model since we show almost matching corresponding quantum message lower bounds. Our results significantly improve on the prior work of [PODC 2025], who presented distributed quantum algorithms under the same model that had a message complexity of $\tilde{O}(\sqrt{mn})$ for leader election. Our algorithms demonstrate the significant communication advantage that quantum routing has over classical in distributed computing, since $Ω(m)$ is a well-established classical message lower bound for leader election, broadcast, MST, and BFS that applies even to randomized Monte-Carlo algorithms [JACM 2015]. Thus, our quantum algorithms can, in general, give a quadratic advantage in the communication cost for these fundamental problems. A main technical tool we use to design our distributed algorithms is quantum walks based on electric networks. We posit a framework for using quantum walks in the distributed setting to design communication-efficient distributed quantum algorithms. Our framework can be used as a black box to significantly reduce communication costs and may be of independent interest. Additionally, our lower-bound technique for establishing distributed quantum message lower bounds can also be applied to other problems.

翻译：我们针对任意网络中的基本分布式计算问题——领导者选举、广播、最小生成树和广度优先搜索树——提出了新的分布式量子算法。这些算法在[PODC 2025]提出的量子路由模型中，其通信（消息）复杂度（本质上）是最优的。我们的算法在领导者选举、广播和最小生成树问题上的消息复杂度为$\tilde{O}(n)$，在广度优先搜索问题上为$\tilde{O}(\sqrt{mn})$（其中$n$和$m$分别为网络的节点数和边数）。这些消息界在量子路由模型中近乎紧致，因为我们证明了几乎匹配的相应量子消息下界。我们的结果显著改进了[PODC 2025]的先前工作，后者在同一模型下提出的分布式量子算法在领导者选举问题上的消息复杂度为$\tilde{O}(\sqrt{mn})$。我们的算法展示了量子路由在分布式计算中相对于经典方法的显著通信优势，因为即使对于随机化的蒙特卡洛算法，$\Omega(m)$也是领导者选举、广播、最小生成树和广度优先搜索问题中公认的经典消息下界[JACM 2015]。因此，对于这些基本问题，我们的量子算法通常可以在通信成本上提供二次方的优势。我们设计分布式算法使用的一个主要技术工具是基于电网路的量子行走。我们提出了一个在分布式环境中利用量子行走来设计通信高效的分布式量子算法的框架。该框架可作为黑盒显著降低通信成本，并可能具有独立的研究价值。此外，我们用于建立分布式量子消息下界的技术也可应用于其他问题。