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Generating probability distributions using variational quantum circuits

Sampling from a probability distribution is a core task in many quantum and classical algorithms. Variational quantum circuits provide a natural approach to generating such distributions, as measurement outcomes directly define the probability values. However, designing circuits that train reliably while utilizing limited quantum resources remains largely a heuristic approach. In particular, the roles of expressibility, entanglement capability, and quantum resources in training performance and scalability are not well understood. In this work we present a systematic study of variational quantum circuits where we compare different ansatze family across multiple cost functions and classical optimization methods. We use expressibility and entanglement capability as circuit descriptors to explain convergence behaviors, optimizer sensitivity and robustness to noise. Our results provide a practical guidelines for designing resource aware, efficient and trainable quantum circuits, moving beyond heuristic methods for near term applications.