We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schr\"odinger framework to solve the Liouville equation for the HJE with the Hamiltonian formulation of classical mechanics, since it can be recast as the semiclassical limit of the Wigner transform of the Schr\"odinger equation. Comparsion between the Schr\"odinger and the Liouville framework will also be made.

翻译：我们通过非线性常微分方程（ODEs）与非线性Hamilton-Jacobi方程（HJE）的线性表示或精确映射（即Liouville方程和Koopman-von Neumann方程），构建了计算其解和/或物理可观测量的量子算法。线性表示与原始非线性系统之间的连接通过Dirac delta函数或水平集机制建立。我们比较了基于量子线性系统算法的方法与不同数值近似（包括两种线性表示的有限差分离散和傅里叶谱离散）产生的量子模拟方法，结果表明量子模拟方法通常在时间复杂度上表现最优。我们还提出了求解HJE的Liouville方程的薛定谔框架，该框架基于经典力学的哈密顿表述，因该方程可重述为薛定谔方程Wigner变换的半经典极限。最后将对薛定谔框架与Liouville框架进行比较。