Quantum Algorithms for Solving Ordinary Differential Equations via Classical Integration Methods

Abstract

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6${}^{\mathrm{th}}$ order Gauss-Legendre collocation method on a D-Wave 2000Q system, showing good agreement with the reference solution. We find that the quantum annealing approach exhibits the largest potential for high-order implicit integration methods. As promising future scenario, the digital arithmetic method could be employed as an "oracle" within quantum search algorithms for inverse problems.

References in corpus (21)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
• Quantum information with continuous variables
• Gaussian Quantum Information
• Quantum algorithm for solving linear systems of equations
• Quantum Annealing in the Transverse Ising Model
• The theory of variational hybrid quantum-classical algorithms
• Quantum computation over continuous variables
• Quantum Networks for Elementary Arithmetic Operations
• Quantum Computation by Adiabatic Evolution
• Quantum computers can search arbitrarily large databases by a single query
• Variational quantum algorithms for nonlinear problems
• Efficient quantum algorithm for dissipative nonlinear differential equations
• A practical heuristic for finding graph minors
• Addition on a Quantum Computer
• Quantum Algorithm for Simulating the Wave Equation
• Koopman-von Neumann Approach to Quantum Simulation of Nonlinear Classical Dynamics
• A Quantum Algorithm for Solving Linear Differential Equations: Theory and Experiment
• A Universal Training Algorithm for Quantum Deep Learning
• Quantum annealing for systems of polynomial equations
• Quantum algorithm for nonlinear differential equations

Cited by in corpus (9)

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• Quantum annealing-based computed tomography using variational approach for a real-number image reconstruction
• Solving the Gross-Pitaevskii equation on multiple different scales using the quantics tensor train representation
• Alternatives to a nonhomogeneous partial differential equation quantum algorithm
• Integrating Quantum Algorithms Into Classical Frameworks: A Predictor-corrector Approach Using HHL
• Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains
• Error and Resource Estimates of Variational Quantum Algorithms for Solving Differential Equations Based on Runge-Kutta Methods