A variational quantum algorithm for the Feynman-Kac formula

• URL: http://arxiv.org/abs/2108.10846v2
• Date: Thu, 2 Jun 2022 03:02:32 GMT
• Title: A variational quantum algorithm for the Feynman-Kac formula
• Authors: Hedayat Alghassi, Amol Deshmukh, Noelle Ibrahim, Nicolas Robles, Stefan Woerner, Christa Zoufal
• Abstract summary: We propose an algorithm based on variational quantum imaginary time evolution for solving the Feynman-Kac partial differential equation. We see a remarkable agreement between the classical methods and the quantum variational method for an illustrative example on six and eight qubits. Future research topics in the areas of quantitative finance and other types of PDEs are also discussed.
• Score: 0.6116681488656472
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose an algorithm based on variational quantum imaginary time evolution for solving the Feynman-Kac partial differential equation resulting from a multidimensional system of stochastic differential equations. We utilize the correspondence between the Feynman-Kac partial differential equation (PDE) and the Wick-rotated Schr\"{o}dinger equation for this purpose. The results for a $(2+1)$ dimensional Feynman-Kac system obtained through the variational quantum algorithm are then compared against classical ODE solvers and Monte Carlo simulation. We see a remarkable agreement between the classical methods and the quantum variational method for an illustrative example on six and eight qubits. In the non-trivial case of PDEs which are preserving probability distributions -- rather than preserving the $\ell_2$-norm -- we introduce a proxy norm which is efficient in keeping the solution approximately normalized throughout the evolution. The algorithmic complexity and costs associated to this methodology, in particular for the extraction of properties of the solution, are investigated. Future research topics in the areas of quantitative finance and other types of PDEs are also discussed.

Related papers

• A New Variational Quantum Algorithm Based on Lagrange Polynomial Encoding to Solve Partial Differential Equations [0.0]
  Partial Differential Equations (PDEs) serve as the cornerstone for a wide range of scientific endeavours. Finding solutions to PDEs often exceeds the capabilities of traditional computational approaches. Recent advances in quantum computing have triggered a growing interest from researchers for the design of quantum algorithms for solving PDEs.
  arXiv  Detail & Related papers  (2024-07-23T10:11:44Z)
• Quantum Circuits for the heat equation with physical boundary conditions via Schrodingerisation [33.76659022113328]
  This paper explores the explicit design of quantum circuits for quantum simulation of partial differential equations (PDEs) with physical boundary conditions. We present two methods for handling the inhomogeneous terms arising from time-dependent physical boundary conditions. We then apply the quantum simulation technique from [CJL23] to transform the resulting non-autonomous system to an autonomous system in one higher dimension.
  arXiv  Detail & Related papers  (2024-07-22T03:52:14Z)
• Solving the homogeneous Bethe-Salpeter equation with a quantum annealer [34.173566188833156]
  The homogeneous Bethe-Salpeter equation (hBSE) was solved for the first time by using a D-Wave quantum annealer. A broad numerical analysis of the proposed algorithms was carried out using both the proprietary simulated-anneaing package and the D-Wave Advantage 4.1 system.
  arXiv  Detail & Related papers  (2024-06-26T18:12:53Z)
• Quantum Variational Solving of Nonlinear and Multi-Dimensional Partial Differential Equations [1.2670268797931266]
  A variational quantum algorithm for numerically solving partial differential equations was proposed by Lubasch et al. We generalize the method to cover a broader class of nonlinear PDEs as well as multidimensional PDEs. We show via numerical simulations that the algorithm can solve instances of the Single-Asset Black-Scholes equation.
  arXiv  Detail & Related papers  (2023-11-02T18:29:31Z)
• Efficient Quantum Algorithms for Nonlinear Stochastic Dynamical Systems [2.707154152696381]
  We propose efficient quantum algorithms for solving nonlinear differential equations (SDE) via the associated Fokker-Planck equation (FPE) We discretize the FPE in space and time using two well-known numerical schemes, namely Chang-Cooper and implicit finite difference. We then compute the solution of the resulting system of linear equations using the quantum linear systems.
  arXiv  Detail & Related papers  (2023-03-04T17:40:23Z)
• Correspondence between open bosonic systems and stochastic differential equations [77.34726150561087]
  We show that there can also be an exact correspondence at finite $n$ when the bosonic system is generalized to include interactions with the environment. A particular system with the form of a discrete nonlinear Schr"odinger equation is analyzed in more detail.
  arXiv  Detail & Related papers  (2023-02-03T19:17:37Z)
• Quantum Kernel Methods for Solving Differential Equations [21.24186888129542]
  We propose several approaches for solving differential equations (DEs) with quantum kernel methods. We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented.
  arXiv  Detail & Related papers  (2022-03-16T18:56:35Z)
• Bernstein-Greene-Kruskal approach for the quantum Vlasov equation [91.3755431537592]
  The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed.
  arXiv  Detail & Related papers  (2021-02-18T20:55:04Z)
• Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance [1.128265591164748]
  We study quantum algorithms for differential equations (SDEs) We provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance.
  arXiv  Detail & Related papers  (2020-12-11T12:34:55Z)
• Quantitative Propagation of Chaos for SGD in Wide Neural Networks [39.35545193410871]
  In this paper, we investigate the limiting behavior of a continuous-time counterpart of the Gradient Descent (SGD) We show 'propagation of chaos' for the particle system defined by this continuous-time dynamics under different scenarios. We identify two under which different mean-field limits are obtained, one of them corresponding to an implicitly regularized version of the minimization problem at hand.
  arXiv  Detail & Related papers  (2020-07-13T12:55:21Z)
• Scalable Gradients for Stochastic Differential Equations [40.70998833051251]
  adjoint sensitivity method scalably computes gradients of ordinary differential equations. We generalize this method to differential equations, allowing time-efficient and constant-memory computation. We use our method to fit neural dynamics defined by networks, achieving competitive performance on a 50-dimensional motion capture dataset.
  arXiv  Detail & Related papers  (2020-01-05T23:05:55Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.