To comprehensively address the challenges of aperture design, element spacing optimization, and sidelobe suppression in sparse radar array antennas, this paper proposes a hybrid particle swarm optimization (PSO) algorithm that integrates quantum-behavior mechanisms with genetic mutation. The algorithm enhances global search capability through the introduction of a quantum potential well model, while incorporating adaptive mutation operations to prevent premature convergence, thereby improving optimization accuracy during later iterations. The simulation results demonstrate that for sparse linear arrays, planar rectangular arrays, and multi-ring concentric circular arrays, the proposed algorithm achieves a sidelobe level (SLL) reduction exceeding 0.24 dB compared to conventional approaches, including the grey wolf optimizer (GWO), the whale optimization algorithm (WOA), and classical PSO. Furthermore, it exhibits superior global iterative search performance and demonstrates broader applicability across various array configurations.