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模型被广泛研究，用以表示网络中处理器间通信带宽严重受限的情形。每轮中，处理器之间仅能发送各含$O(\log(n))$比特信息的消息。这些模型的量子版本允许处理器在相同带宽限制下以量子比特进行通信与计算。由此引出如下自然研究问题：相较于经典模型，哪些问题能在量子模型中得到更高效的求解？基于现有工作，我们从两方面对此问题做出贡献。首先，我们在量子CONGEST-CLIQUE分布式计算模型中提出两种高成功概率的算法：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，每种算法均使用$\tilde{O}(n^{1/4})$轮通信和$\tilde{O}(n^{9/4})$条消息，其中$n$为网络节点数。因此，这些算法在渐近的轮次与消息复杂度上均优于经典CONGEST-CLIQUE模型中已知的任何算法。从高层次看，我们通过将经典算法框架与量子子程序相结合来实现上述结果。其中，利用分布式版本的Grover搜索算法加速三角查找的现有框架，是实现渐近加速的核心。其次，我们仔细刻画了算法及相关算法中涉及的常数与对数因子（这些因子常被$\tilde{O}$记号所掩盖）。分析表明，为使我们的算法及现有的相关量子与经典算法变得实用，仍需进行某些改进，因为其渐近加速仅在$n$极大时方能体现。