Stochastic algorithms are powerful global optimizers in topology optimizations (TOs). However, the low convergence speed and high computational cost, due to the large dimensionality of a TO problem, of such an approach are still two open issues to be addressed. To address these two issues, a TO methodology based on a modified quantum evolutionary algorithm is proposed. In the proposed methodology, an adaptive updating mechanism for the rotation gate is introduced to balance the exploration and exploitation searches; and a redistribution mechanism of design variables is developed to reduce the computational cost of the algorithm. The numerical results as reported validate the effectiveness and feasibility of the proposed methodology and demonstrate its advantages over a traditional ON/OFF method and its ancestor.