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Solving linear and nonlinear differential equations with large degrees offreedom is an important task for scientific and industrial applications. Inorder to solve such differential equations on a quantum computer, it isnecessary to embed classical variables into a quantum state. While the Carlemanand Koopman-von Neumann embeddings have been investigated so far, the class ofproblems that can be mapped to the Schr\"{o}dinger equation is not wellunderstood even for linear differential equations. In this work, we investigatethe conditions for linear differential equations to be mapped to theSchr\"{o}dinger equation and solved on a quantum computer. Interestingly, wefind that these conditions are identical for both Carleman and Koopman-vonNeumann embeddings. We also compute the computational complexity associatedwith estimating the expected values of an observable. This is done by assuminga state preparation oracle, block encoding of the mapped Hamiltonian via eitherCarleman or Koopman-von Neumann embedding, and block encoding of the observableusing $O(\log M)$ qubits with $M$ is the mapped system size. Furthermore, weconsider a general classical quadratic Hamiltonian dynamics and find asufficient condition to map it into the Schr\"{o}dinger equation. As a specialcase, this includes the coupled harmonic oscillator model [Babbush et al.,\cite{babbush_exponential_2023}]. We also find a concrete example that cannotbe described as the coupled harmonic oscillator but can be mapped to theSchr\"{o}dinger equation in our framework. These results are important in theconstruction of quantum algorithms for solving differential equations oflarge-degree-of-freedom.

How to Map Linear Differential Equations to Schr\"{o}dinger Equations via Carleman and Koopman-von Neumann Embeddings for Quantum Algorithms