  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].
