Abstract Multi-scale Quantum Harmonic Oscillator Algorithm (MQHOA) is a population-based metaheuristic algorithm proposed recently. It has been proved effective and efficient to deal with unimodal and multimodal problems. Although the mechanism of replacing the worst particle with the fittest individual in MQHOA helps to fasten the iteration process, it can easily lead to premature convergence. Instead of direct replacement, several migration strategies are proposed to maintain the diversity of the population and help to obtain the global optima in difficult function evaluations. The impacts of the migration strategies and individual stabilization on the improvement of the algorithms in their effectiveness, reliability, accuracy and efficiency are well researched. A variety of multi-dimensional unimodal and multimodal benchmark functions are applied to illustrate the optimization performance of the proposed algorithms. Some of the best competitors in MQHOAs with migration strategies are selected to compare with several state-of-the-art stochastic algorithms. Experimental results presented suggest some conclusions: First, the individual stabilization mechanism does not significantly improve the performance of MQHOA. Second, random migration does not obviously help MQHOA perform much better. Third, migration strategies significantly affect the performance of MQHOA, and some of MQHOAs with migration strategies are very competitive to deal with numerical optimization problems.