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 is an element of{0,1, . . . , N - 1}. All these algorithms work with the bit complexities bounded by a polynomial in N