Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series

A quantum algorithm is proposed for sampling from a solution of stochastic differential equations (SDEs). Using differentiable quantum circuits (DQCs) with a feature map encoding of latent variables, the quantile function is represented for an underlying probability distribution and samples extracted as DQC expectation values. Using quantile mechanics the system is propagated in time, thereby allowing for time-series generation. The method is tested by simulating the Ornstein-Uhlenbeck process and sampling at times different from the initial point, as required in financial analysis and dataset augmentation. Additionally, continuous quantum generative adversarial networks (qGANs) are analyzed, and the authors show that they represent quantile functions with a modified (reordered) shape that impedes their efficient time-propagation. The results shed light on the connection between quantum quantile mechanics (QQM) and qGANs for SDE-based distributions, and point the importance of differential constraints for model training, analogously with the recent success of physics informed neural networks.