Quantum Complexity of Weighted Diameter and Radius in CONGEST Networks

TL;DR

This paper introduces a quantum algorithm for approximating the weighted diameter and radius in CONGEST networks, demonstrating quantum advantages over classical methods and establishing lower bounds for these computations.

Contribution

It presents the first quantum algorithms for weighted diameter and radius approximation in CONGEST networks, showing quantum speedups and lower bounds compared to classical algorithms.

Findings

Quantum algorithms achieve $(1+o(1))$-approximation with sublinear rounds.

Quantum advantage over classical algorithms for weighted diameter/radius.

Lower bounds indicate inherent difficulty in quantum approximation for small diameter graphs.

Abstract

This paper studies the round complexity of computing the weighted diameter and radius of a graph in the quantum CONGEST model. We present a quantum algorithm that $(1+o(1))$-approximates the diameter and radius with round complexity $\widetilde O\left(\min\left\{n^{9/10}D^{3/10},n\right\}\right)$, where $D$ denotes the unweighted diameter. This exhibits the advantages of quantum communication over classical communication since computing a $(3/2-\varepsilon)$-approximation of the diameter and radius in a classical CONGEST network takes $\widetilde\Omega(n)$ rounds, even if $D$ is constant [Abboud, Censor-Hillel, and Khoury, DISC '16]. We also prove a lower bound of $\widetilde\Omega(n^{2/3})$ for $(3/2-\varepsilon)$-approximating the weighted diameter/radius in quantum CONGEST networks, even if $D=\Theta(\log n)$. Thus, in quantum CONGEST networks, computing weighted diameter and weighted radius of graphs with small $D$ is strictly harder than unweighted ones due to Le Gall and Magniez's $\widetilde O\left(\sqrt{nD}\right)$-round algorithm for unweighted diameter/radius [PODC '18].