In recent years, we are witnessing the spread of many and various modern real-time applications implemented on computer networks such as video conferencing, distance education, online games, and video streaming. These applications require the high quality of different network resources such as bandwidth, delay, jitter, and packet loss rate. In this paper, we propose an improved quantum chaotic animal migration optimization algorithm to solve the multicast routing problem (Multi-Constrained Least Cost MCLC). We used a quantum representation of the solutions that allow the use of the original AMO version without discretization, as well as improving AMO by introducing chaotic map to determine the random numbers. These two contributions improve the diversification and intensification of the algorithm. The simulation results show that our proposed algorithm has a good scalability and efficiency compared with other existing algorithms in the literature.