A variational quantum algorithm for the Feynman-Kac formula

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 Schrödinger 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.

References in corpus (31)

• Barren plateaus in quantum neural network training landscapes
• Synthesis of Quantum Logic Circuits
• Efficient variational quantum simulator incorporating active error minimisation
• Variational ansatz-based quantum simulation of imaginary time evolution
• Quantum Generative Adversarial Networks for Learning and Loading Random Distributions
• Theory of variational quantum simulation
• Quantum speedup of Monte Carlo methods
• Quantum Risk Analysis
• Quantum computational finance: Monte Carlo pricing of financial derivatives
• Creating superpositions that correspond to efficiently integrable probability distributions
• Amplitude estimation without phase estimation
• Variational quantum simulation of general processes
• Accelerated Variational Quantum Eigensolver
• Option Pricing using Quantum Computers
• Quantum Tomography
• Optimal Quantum Measurements of Expectation Values of Observables
• Iterative Quantum Amplitude Estimation
• A Threshold for Quantum Advantage in Derivative Pricing
• Reexamination of optimal quantum state estimation of pure states
• Quantum simulation of parity-time symmetry breaking with a superconducting quantum processor
• Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance
• Efficient State Preparation for Quantum Amplitude Estimation
• Quantum unary approach to option pricing
• Enhancing the Quantum Linear Systems Algorithm using Richardson Extrapolation
• Variational quantum simulations of stochastic differential equations
• Dirichlet Energy Constrained Learning for Deep Graph Neural Networks
• Error Bounds for Variational Quantum Time Evolution
• A Smoothness Energy without Boundary Distortion for Curved Surfaces
• Susy for non-Hermitian Hamiltonians, with a view to coherent states
• Quantum option pricing using Wick rotated imaginary time evolution
• Introduction to Stochastic Differential Equations (SDEs) for Finance

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