  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.
