Quantum Linear Multistep Method for Using a Quantum Oracle with Differential Equations

TL;DR

This paper introduces a quantum linear multistep method (QLMM) that adapts classical numerical techniques for differential equations to quantum algorithms, enabling efficient solutions for initial value problems and their application in optimization.

Contribution

It proposes a novel quantum linear multistep method for solving IVPs, including an approach to optimize its form for specific problems, advancing quantum numerical methods.

Findings

QLMM can generate numerical solutions for IVPs in quantum algorithms.

Optimized QLMM solutions are applicable to optimization problems.

Simulation results demonstrate the effectiveness of the proposed method.

Abstract

Differential equations are a crucial mathematical tool used in a wide range of applications. If the solution to an initial value problem (IVP) can be transformed into an oracle, it can be utilized in various fields such as search and optimization, achieving quadratic speedup with respect to the number of candidates compared to its classical counterpart. In the past, attempts have been made to implement such an oracle using the Euler method. In this study, we propose a quantum linear multistep method (QLMM) that applies the linear multistep method, commonly used to numerically solve IVPs on classical computers, to generate a numerical solution of the IVP for use in a quantum oracle. We also propose a method to find the optimal form of QLMM for a given IVP. Finally, through computer simulations, we derive the QLMM formulation for an example IVP and show that the solution from the optimized QLMM can be used in an optimization problem.