The computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange $O(\log n)$ bits per round (here $n$ denotes the number of nodes of the network), it is known that exact computation of the diameter requires $\tilde \Omega(n)$ rounds, even in networks with constant diameter. In this paper we investigate quantum distributed algorithms for this problem in the quantum CONGEST model, where two adjacent nodes can exchange $O(\log n)$ quantum bits per round. Our main result is a $\tilde O(\sqrt{nD})$-round quantum distributed algorithm for exact diameter computation, where $D$ denotes the diameter. This shows a separation between the computational power of quantum and classical algorithms in the CONGEST model. We also show an unconditional lower bound $\tilde \Omega(\sqrt{n})$ on the round complexity of any quantum algorithm computing the diameter, and furthermore show a tight lower bound $\tilde \Omega(\sqrt{nD})$ for any distributed quantum algorithm in which each node can use only $\textrm{poly}(\log n)$ quantum bits of memory.

翻译：直径计算是分布式计算中最核心的问题之一。在标准CONGEST模型中，相邻节点每轮可交换$O(\log n)$比特信息（其中$n$表示网络节点数），已知即便在常数直径的网络中，精确计算直径也需要$\tilde \Omega(n)$轮。本文研究量子CONGEST模型中该问题的量子分布式算法，其中相邻节点每轮可交换$O(\log n)$量子比特。我们的主要成果是一个$\tilde O(\sqrt{nD})$轮的量子分布式算法用于精确直径计算（其中$D$表示直径）。这揭示了CONGEST模型中量子算法与经典算法在计算能力上的分离。我们还证明了任何计算直径的量子算法轮复杂度存在无条件下界$\tilde \Omega(\sqrt{n})$，并进一步证明了对于每个节点仅能使用$\textrm{poly}(\log n)$量子比特内存的分布式量子算法，存在紧的下界$\tilde \Omega(\sqrt{nD})$。