Particle swarm optimization algorithm (PSO) is an effective metaheuristic that can determine Pareto-optimal solutions. We propose an extended PSO by introducing quantum gates in order to ensure the diversity of particle populations that are looking for efficient alternatives. The quality of solutions was verified in the issue of assignment of resources in the computing cloud to improve the live migration of virtual machines. We consider the multi-criteria optimization problem of deep learning-based models embedded into virtual machines. Computing clouds with deep learning agents can support several areas of education, smart city or economy. Because deep learning agents require lots of computer resources, seven criteria are studied such as electric power of hosts, reliability of cloud, CPU workload of the bottleneck host, communication capacity of the critical node, a free RAM capacity of the most loaded memory, a free disc memory capacity of the most busy storage, and overall computer costs. Quantum gates modify an accepted position for the current location of a particle. To verify the above concept, various simulations have been carried out on the laboratory cloud based on the OpenStack platform. Numerical experiments have confirmed that multi-objective quantum-inspired particle swarm optimization algorithm provides better solutions than the other metaheuristics.