  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 $O(\log M)$ qubits with $M$ 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.
