The CONGEST and CONGEST-CLIQUE models have been carefully studied to represent situations where the communication bandwidth between processors in a network is severely limited. Messages of only $O(log(n))$ bits of information each may be sent between processors in each round. The quantum versions of these models allow the processors instead to communicate and compute with quantum bits under the same bandwidth limitations. This leads to the following natural research question: What problems can be solved more efficiently in these quantum models than in the classical ones? Building on existing work, we contribute to this question in two ways. Firstly, we present two algorithms in the Quantum CONGEST-CLIQUE model of distributed computation that succeed with high probability; one for producing an approximately optimal Steiner Tree, and one for producing an exact directed minimum spanning tree, each of which uses $\tilde{O}(n^{1/4})$ rounds of communication and $\tilde{O}(n^{9/4})$ messages, where $n$ is the number of nodes in the network. The algorithms thus achieve a lower asymptotic round and message complexity than any known algorithms in the classical CONGEST-CLIQUE model. At a high level, we achieve these results by combining classical algorithmic frameworks with quantum subroutines. An existing framework for using distributed version of Grover's search algorithm to accelerate triangle finding lies at the core of the asymptotic speedup. Secondly, we carefully characterize the constants and logarithmic factors involved in our algorithms as well as related algorithms, otherwise commonly obscured by $\tilde{O}$ notation. The analysis shows that some improvements are needed to render both our and existing related quantum and classical algorithms practical, as their asymptotic speedups only help for very large values of $n$.

翻译：CONGEST和CONGEST-CLIQUE模型被仔细研究以描述网络中处理器间通信带宽严重受限的情形。在每个回合中，处理器之间只能发送大小为$O(\log n)$比特的消息。这些模型的量子版本允许处理器在相同的带宽限制下用量子比特进行通信和计算。这引出了以下自然的研究问题：哪些问题在量子模型中比在经典模型中能更高效地解决？基于现有工作，我们从两个方面对这一问题作出贡献。首先，我们提出了分布式计算的量子CONGEST-CLIQUE模型中的两种算法，它们以高概率成功：一种用于生成近似最优的斯坦纳树，另一种用于生成精确的有向最小生成树，每种算法使用$\tilde{O}(n^{1/4})$轮通信和$\tilde{O}(n^{9/4})$条消息，其中$n$是网络中的节点数。因此，这些算法在渐近轮数和消息复杂度上均优于经典CONGEST-CLIQUE模型中已知的任何算法。在高层次上，我们通过将经典算法框架与量子子程序结合来实现这些结果。利用分布式版本的Grover搜索算法加速三角形查找的现有框架是渐近加速的核心。其次，我们仔细刻画了本算法及相关算法中涉及的常数和对数因子，这些因子通常被$\tilde{O}$符号掩盖。分析表明，要使我们的算法以及现有的相关量子与经典算法实用化，尚需一些改进，因为它们的渐近加速仅对非常大的$n$值有效。