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 6th 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.
Highlights
Solving differential equations (DEs) is a ubiquitous task in the scientific and engineering community
Principle, the approach works for other higherorder explicit Runge-Kutta schemes since solely elementary arithmetic operations are required, and these operations can be directly mapped to quantum gates
As arbitrary connectivity between all qubits is not possible, the concrete connections available on the target annealing hardware have to be taken into account and the quadratic unconstrained binary optimization (QUBO) model has to be adapted to those hardware characteristics
Summary
Solving differential equations (DEs) is a ubiquitous task in the scientific and engineering community. Traditional numerical algorithms need to integrate the time steps sequentially due to their interdependence, which is at odds with the trend towards parallelization in modern highly-parallel high-performance computing (HPC) architectures. This trend already motivated research in various directions, such as parallelin-time algorithms [13]. We explore approaches of utilizing quantum computers for the purpose of solving DEs: (i) using basis encoding to describe the. To realize (i), we will first design quantum circuits for arithmetic operations in numerical fixed-point representation, and apply these to solve linear ordinary differential equations (ODEs) in two dimensions. We run the annealing task (based on a method of order six) on a D-Wave 2000Q system, and are able to demonstrate a good agreement with the reference solution obtained on a classical computer
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