Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series
This work addresses the challenge of time-series generation for fields such as finance and data augmentation, but it appears incremental as it builds on existing quantum and classical methods like physics-informed neural networks.
The authors tackled the problem of generating time-series from stochastic differential equations (SDEs) by proposing a quantum algorithm using differentiable quantum circuits to represent quantile functions, enabling efficient sampling and propagation over time, with testing on the Ornstein-Uhlenbeck process for applications like financial analysis.
We propose a quantum algorithm for sampling from a solution of stochastic differential equations (SDEs). Using differentiable quantum circuits (DQCs) with a feature map encoding of latent variables, we represent the quantile function for an underlying probability distribution and extract samples as DQC expectation values. Using quantile mechanics we propagate the system in time, thereby allowing for time-series generation. We test the method by simulating the Ornstein-Uhlenbeck process and sampling at times different from the initial point, as required in financial analysis and dataset augmentation. Additionally, we analyse continuous quantum generative adversarial networks (qGANs), and show that they represent quantile functions with a modified (reordered) shape that impedes their efficient time-propagation. Our 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.