Finding an optimal routing path which satisfies different kinds of metrics of the QoS requirement is a very important issue in the research areas of networks and distributed systems. The key problem of QoS multicast routing optimization algorithm, known as the constrained minimum Steiner tree, has been proved to be a NP-complete problem. In this paper, a new genetic algorithm for solving the problem called Quantum Genetic Algorithm (QGA) is mainly investigated and an adaptive quantum multicast routing optimization algorithm is proposed to solve the problem of the large computational complexity of an exhaustive search over all the paths in QoS routing. The simulation results have demonstrated the superiority of our algorithm in terms of robustness, success ratio, convergence and global search capability.