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

Abstract

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.

v3, minor update

References in corpus (19)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Variational Quantum Algorithms
• Quantum computational advantage using photons
• The power of quantum neural networks
• Experimental Quantum Generative Adversarial Networks for Image Generation
• Solving nonlinear differential equations with differentiable quantum circuits
• Exploiting symmetry in variational quantum machine learning
• Theory of overparametrization in quantum neural networks
• Group-Invariant Quantum Machine Learning
• Stochastic population dynamics in spatially extended predator-prey systems
• Avoiding barren plateaus using classical shadows
• A Threshold for Quantum Advantage in Derivative Pricing
• Learning quantum data with the quantum Earth Mover's distance
• Learning temporal data with variational quantum recurrent neural network
• Variational quantum simulations of stochastic differential equations
• Dual-Parameterized Quantum Circuit GAN Model in High Energy Physics
• Quantum Generative Adversarial Networks For Anomaly Detection In High Energy Physics
• How to enhance quantum generative adversarial learning of noisy information
• Neural SDEs as Infinite-Dimensional GANs

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• Variational Quantum Algorithms for Differential Equations on a Noisy Quantum Computer
• Ground state-based quantum feature maps
• Quantum Iterative Methods for Solving Differential Equations with Application to Computational Fluid Dynamics
• Multidimensional Quantum Generative Modeling by Quantum Hartley Transform
• Vortex Detection from Quantum Data
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• Quantum Chebyshev Probabilistic Models for Fragmentation Functions