  Solitude verification is arguably one of the simplest fundamental problems in
distributed computing, where the goal is to verify that there is a unique
contender in a network. This paper devises a quantum algorithm that exactly
solves the problem on an anonymous network, which is known as a network model
with minimal assumptions [Angluin, STOC'80]. The algorithm runs in $O(N)$
rounds if every party initially has the common knowledge of an upper bound $N$
on the number of parties. This implies that all solvable problems can be solved
in $O(N)$ rounds on average without error (i.e., with zero-sided error) on the
network. As a generalization, a quantum algorithm that works in $O(N\log_2
(\max\{k,2\}))$ rounds is obtained for the problem of exactly computing any
symmetric Boolean function, over $n$ distributed input bits, which is constant
over all the $n$ bits whose sum is larger than $k$ for $k\in \{0,1,\dots,
N-1\}$. All these algorithms work with the bit complexities bounded by a
polynomial in $N$.
