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 $N$.