Quantum algorithms for solving linear and nonlinear differential equations

Abstract

Solving linear and nonlinear differential equations with large degrees of freedom is an important task for scientific and industrial applications. In order to solve such differential equations on a quantum computer, it is necessary to embed classical variables into a quantum state. While the Carleman and Koopman embeddings have been investigated so far, the class of problems that can be mapped to the Schrödinger equation is not well understood.
In this work, we investigate the conditions for linear and nonlinear differential equations to be mapped to the Schrödinger equation and solved on a quantum computer.
In the case of a linear domain, as our first result, we determine the necessary and sufficient conditions for the Carleman and Koopman embeddings to map a given linear differential equation to the Schrödinger equation. The recently reported coupled harmonic oscillator model is a special case. In the general nonlinear case, the Carleman embedding does not yield the Schrödinger equation, and hence prior work employs discretization in the time direction and quantum linear system solver.
For our second result, we turned to the Koopman embedding, through which we are able to obtain the Schrödinger equation. However, the mapped Hamiltonian is not necessarily sparse even if the original nonlinear differential equation is sparse, due to the nature of the embedding. To address this issue, we identified a sufficient condition for the nonlinear differential equation whereby the mapped Hamiltonian becomes sparse. This enables us to estimate the computational complexity of the quantum algorithm used to solve such nonlinear differential equations.
These results are crucial for the development of quantum algorithms designed to solve differential equations with a large degree of freedom.