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-von Neumann embeddings have been investigated so far, the class of problems that can be mapped to the Schr\"{o}dinger equation is not well understood even for linear differential equations. In this work, we investigate the conditions for linear differential equations to be mapped to the Schr\"{o}dinger equation and solved on a quantum computer. Interestingly, we find that these conditions are identical for both Carleman and Koopman-von Neumann embeddings. We also compute the computational complexity associated with estimating the expected values of an observable. This is done by assuming a state preparation oracle, block encoding of the mapped Hamiltonian via either Carleman or Koopman-von Neumann embedding, and block encoding of the observable using qubits with is the mapped system size. Furthermore, we consider a general classical quadratic Hamiltonian dynamics and find a sufficient condition to map it into the Schr\"{o}dinger equation. As a special case, this includes the coupled harmonic oscillator model [Babbush et al., \cite{babbush_exponential_2023}]. We also find a concrete example that cannot be described as the coupled harmonic oscillator but can be mapped to the Schr\"{o}dinger equation in our framework. These results are important in the construction of quantum algorithms for solving differential equations of large-degree-of-freedom.