Statistics-informed parameterized quantum circuit: towards practical quantum state preparation and learning via maximum entropy principle

Quantum computing offers significant potential for tackling complex problems, yet preparing quantum states from real-world data remains a critical challenge. We introduce the statistics-informed parameterized quantum circuit (SI-PQC), an approach specifically designed to efficiently prepare arbitrary statistical distributions. By leveraging statistical symmetries in data through the maximum entropy principle, SI-PQC encodes prior information with a fixed-structure circuit and tunable parameters, eliminating extensive pre-processing. This method achieves exponential resource savings in preparing mixture models, crucial for applications in statistics and machine learning. SI-PQC also supports variational learning within an optimally dimensioned training space, enhancing generalization, trainability and statistical interpretability. Numerical experiments confirm that SI-PQC can effectively prepare diverse distributions and accurately learn Gaussian mixture models, aligning closely with theoretical expectations. Applications in financial derivatives pricing and online risk analysis showcase SI-PQC’s practical advantages, with substantial improvements in end-to-end quantum resource efficiency and applicability to empirical data. As a versatile and resource-efficient subroutine, SI-PQC broadens the scope of quantum algorithms, especially in real-time, data-driven fields such as finance, online machine learning, and medical diagnostics.