Hybrid Quantum Genetic Algorithm under constrained search spaces: robustness, diversity, and convergence behavior

This work investigates the impact of constraints on the performance of the Hybrid Quantum Genetic Algorithm (HQGA) in portfolio optimization, in comparison with a classical Genetic Algorithm (GA). We consider both unconstrained and cardinality-constrained formulations and analyze not only convergence behavior but also population diversity as a key factor influencing algorithmic robustness. The results show that, in the unconstrained case, both algorithms exhibit similar convergence patterns, rapidly reaching high-quality solutions. However, in the presence of constraints, significant differences emerge: the HQGA consistently maintains higher levels of population diversity, while the GA rapidly loses diversity and becomes prone to premature convergence. This enhanced diversity preservation enables the HQGA to sustain exploration and achieve more robust performance in constrained scenarios. These findings highlight the importance of diversity in evolutionary optimization and suggest that hybrid quantum-classical approaches offer practical advantages in constrained combinatorial problems.