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

Abstract

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.

Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes

“…Quantum machine learning was shown to be applicable to solving differential equations [44][45][46]. Here, we introduce a hybrid quantum neural network called HQPINN and compare it against its classical counterpart, classical PINN.…”

“…in image processing [36][37][38][39] and natural language processing [40][41][42][43]. Solving nonlinear differential equations is also an application area for quantum algorithms that use differentiable quantum circuits [44,45] and quantum kernels [46].…”

Protocols for classically training quantum generative models on probability distributions

“…In the context of QGM, QCBM is an excellent example of implicit training where typically a MMD like loss-function is used. On the other hand, recent work showcases how explicit quantum models such as DQGM [38] and quantum quantile mechanics [11] benefit from a functional access to the model probability distributions, allowing input-differentiable quantum models [39,40] To enable the classical training, we propose a strategy described in a schematic shown in Fig. 2.…”

“…Note that in the context of solving SDEs [11,38,56], it can be shown that differentials with respect to x such as d p model (x)/dx and higher-order derivatives can also be estimated efficiently classically.…”

“…QGM exploits the inherent superposition properties of a quantum state generated from a parametrized unitary, along with the probabilistic nature of quantum measurements, to efficiently sample from a trainable model. QGMs have potential applications in generating samples from solutions of stochastic differential equations (SDEs) for simulating financial and diffusive physical processes [11,12], scrambling data using a quantum embedding for anonymization [13][14][15], and generating solutions to graph-based problems like maximum independent set or maximum clique [16,17], among many others. Given the successes of quantum sampling this makes QGM a promising contender for achieving a quantum advantage.…”