Computing the distance parameters of a network, including the diameter, radius, eccentricities and the all-pairs shortest paths (APSP) is a central problem in distributed computing. This paper investigates the distance parameters in the quantum CONGEST models and establishes almost linear lower bounds on eccentricities and APSP, which match the classical upper bounds. Our results imply that there is not quantum speedup for these two problems. In contrast with the diameter and radius, exchanging quantum messages is able to save the communication when the networks have low diameters [F. L. Gall and F. Magniez, Sublinear-time quantum computation of the diameter in CONGEST networks, in Proc. 2018 ACM Symp. Principles of Distributed Computing (PODC), (2018), pp. 337–346; F. Magniez and A. Nayak, arXiv:2002.11795]. We obtain the lower bounds via a reduction from the two-way quantum communication complexity of the set intersection [A. A. Razborov, Izv. Math. 159 (2003)].

中文翻译：

---

量子CONGEST模型中偏心率和全对最短路径的复杂性

计算网络的距离参数，包括直径、半径、偏心率和全对最短路径 (APSP) 是分布式计算的核心问题。本文研究了量子 CONGEST 模型中的距离参数，并建立了与经典上界相匹配的偏心率和 APSP 几乎线性的下界。我们的结果表明这两个问题没有量子加速。与直径和半径相比，交换量子消息能够在网络直径较小时节省通信 [​​F. L. Gall 和 F. Magniez，CONGEST 网络中直径的亚线性时间量子计算，在过程。2018 ACM 研讨会。分布式计算原理 (PODC), (2018), 第 337–346 页；F. Magniez 和 A. Nayak，arXiv:2002.11795]。我们通过减少集合交叉点的双向量子通信复杂度来获得下界 [A. A.拉兹博罗夫，伊兹瓦。数学。159 (2003)]。