分布式算法在量子路由模型中的紧密通信界限
Tight Communication Bounds for Distributed Algorithms in the Quantum Routing Model
AI总结：
本文提出新的分布式量子算法，在量子路由模型中实现了领导选举、广播、MST和BFS等基本问题的通信优化。
摘要：
我们为基本的分布式计算问题提出了新的分布式量子算法，包括领导者选举、广播、最小生成树（MST）和广度优先搜索（BFS）树，在任意网络中。这些算法在量子路由模型中（在PODC 2025中引入）本质上是最优的，其通信（消息）复杂度。我们的算法消息复杂度为领导选举、广播和MST为~O(n)，BFS为~O(√(mn))（n和m分别是网络中的节点数和边数）。这些消息界限在量子路由模型中几乎紧密，因为我们展示了几乎匹配的相应量子消息下界。我们的结果显著改进了之前在相同模型下的工作，该工作提出了分布式量子算法，其消息复杂度为~O(√(mn))对于领导者选举。我们的算法展示了量子路由在分布式计算中的显著通信优势，因为Ω(m)是领导者选举、广播、MST和BFS的已知经典消息下界，即使适用于随机蒙特卡洛算法[JACM 2015]。因此，我们的量子算法可以一般在这些基本问题中提供通信成本的平方优势。我们设计分布式算法的主要技术工具是基于电路网络的量子行走。我们提出一个框架，用于在分布式设置中使用量子行走来设计通信高效的分布式量子算法。我们的框架可以作为黑盒使用，显著减少通信成本，并可能对其他问题有独立兴趣。此外，我们为建立分布式量子消息下界的技术方法也可以应用于其他问题。
英文摘要：
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.
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