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arXiv:2108.03190 · Advanced Quantum Technologies

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

Annie E. Paine, Vincent E. Elfving, Oleksandr Kyriienko

Published Aug 7, 2023• 18 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1002/qute.202300065

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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.
We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modeling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQC-based model, where data is encoded in a latent space with the proposed phase feature map of exponential capacity. This is followed by a trainable quantum circuit, forming the model. We then map the trained model to the bit basis using a fixed unitary transformation, in this case corresponding to a quantum Fourier transform circuit. It allows fast sampling from parametrized distributions using a single-shot readout. Importantly, latent space training provides models that are automatically differentiable, and we show how samples from solutions of stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Our approach opens a route to multidimensional generative modeling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubitwise training and feature map sparsification. Published by the American Physical Society 2024
Stochastic differential equations (SDEs), which model uncertain phenomena as the time evolution of random variables, are exploited in various fields of natural and social sciences such as finance. Since SDEs rarely admit analytical solutions and must usually be solved numerically with huge classical-computational resources in practical applications, there is strong motivation to use quantum computation to accelerate the calculation. Here, we propose a quantum-classical hybrid algorithm that solves SDEs based on variational quantum simulation. We first approximate the target SDE by a trinomial tree structure with discretization and then formulate it as the time-evolution of a quantum state embedding the probability distributions of the SDE variables. We embed the probability distribution directly in the amplitudes of the quantum state whereas the previous studies took the square-root of the probability distribution in the amplitudes. Our embedding enables us to construct simple quantum circuits that simulate the time-evolution of the state for general SDEs. We also develop a scheme to compute the expectation values of the SDE variables and discuss whether our scheme can achieve quantum speedup for the expectation-value evaluations of the SDE variables. Finally, we numerically validate our algorithm by simulating several types of stochastic processes. Our proposal provides a new direction for simulating SDEs on quantum computers.
Abstract Quantum generative adversarial networks (QGANs) have been investigated as a method for generating synthetic data with the goal of augmenting training data sets for neural networks. This is especially relevant for financial time series, since we only ever observe one realization of the process, namely the historical evolution of the market, which is further limited by data availability and the age of the market. However, for classical GANs it has been shown that generated data may (often) not exhibit desired properties (also called stylized facts ), such as matching a certain distribution or showing specific temporal correlations. Here, we investigate whether quantum correlations in quantum inspired models of QGANs can help in the generation of financial time series. We train QGANs, composed of a quantum generator and a classical discriminator, and investigate two approaches for simulating the quantum generator: a full simulation of the quantum circuits, and an approximate simulation using tensor network methods. We tested how the choice of hyperparameters, such as the circuit depth and bond dimensions, influenced the quality of the generated time series. The QGANs that we trained generate synthetic financial time series that not only match the target distribution but also exhibit the desired temporal correlations, with the quality of each property depending on the hyperparameters and simulation method.
Abstract An approach for building quantum models based on the exponentially growing orthonormal basis of Hartley kernel functions is developed. First, a differentiable Hartley feature map parameterized by real‐valued argument that enables quantum models suitable for solving stochastic differential equations and regression problems is designed. Unlike the naturally complex Fourier encoding, the proposed Hartley feature map circuit leads to quantum states with real‐valued amplitudes, introducing an inductive bias and natural regularization. Next, a quantum Hartley transform circuit is proposed as a map between computational and Hartley basis. The developed paradigm is applied to generative modeling from solutions of stochastic differential equations, and utilize the quantum Hartley transform for fine sampling from parameterized distributions through an extended register. Finally, the capability of multivariate quantum generative modeling is demonstrated for both correlated and uncorrelated distributions. As a result, the developed quantum Hartley‐based generative models (QHGMs) offer a distinct quantum approach to generative AI at increasing scale.
We develop and analyze generative models based on quantum solvers of stochastic differential equations. Unlike the common approach based on layered variational quantum circuits with parameterized rotations, we parameterize the potential of a Langevin process. The associated Fokker-Planck (FP) equations are Hermitianized and solved using a linear combination of cosine operators implemented through quantum singular value transformation. By encoding the square root of the stationary density, the model enables sampling through computational-basis measurements and is efficiently controlled by coefficients of the block-encoded diagonal potential. We train the proposed model to match empirical distributions within the family described by FP equations, demonstrating parameter recovery for multivariate probability distributions. Crucially, the model remains trainable at scale. Sample-efficient and convex optimization for in-distribution targets arises from a physics-informed inductive bias and holds regardless of the dimensionality of the problem. We provide circuit constructions and efficient block encodings of the main components of the algorithm and analyze resource requirements. Our approach frames quantum generative learning as quantum model discovery and provides a foundation for extending this framework to high-dimensional, strongly correlated scientific data.
We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations and compute density, temperature, and velocity profiles for the fluid flow in a convergent-divergent nozzle.