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A diversity preserving cellular quantum evolutionary algorithm (DPCQEA) is proposed in which the quantum individuals are located in a specific topology and interact only with their neighbors. The proposed cellular structure aims to provide a better exploitation of local neighborhoods before moving towards a global best, hence it increases population diversity. This paper also proposes a new operator for diversity preservation in the population. In standard QEA the diversity in the population decreases across the generations. Decreasing the diversity of the population decreases the exploration performance of the algorithm and causes possible algorithm trapping in the local optima. In the proposed algorithm, only the fittest of converged q-individuals from among similar individuals are preserved, while others are reinitialized. A criterion is then proposed to measure convergence and similarity among individuals. Experimental results on knapsack problem, trap problem as well as 14 Numerical benchmark functions show that DPCQEA consistently exceeds the performance of QEA.