论文标题
非线性普通和部分微分方程的线性表示量子算法的时间复杂性分析
Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations
论文作者
论文摘要
我们构建量子算法，以通过线性表示或非线性映射或非线性odes/hje和线性偏微分方程之间的精确映射来计算非线性普通微分方程（ODE）和非线性汉密尔顿 - 雅各比方程（HJE）的解决方案和/或物理可观察到（liouville等方程和koopman-von neumann-von von von neumann-von von。线性表示与原始非线性系统之间的连接是通过DIRAC DELTA函数或级别设置机制建立的。我们比较了基于量子线性算法的方法和由不同的数值近似值产生的量子模拟方法，包括有限的差异离散和两种不同的线性表示的傅立叶光谱离散化，结果表明量子模拟方法通常在时间上具有最佳性能。我们还提出了Schrödinger框架，以使用汉密尔顿的经典力学制定来解决HJE的Liouville方程，因为它可以作为Schrödinger方程的Wigner Transform的半经典限制进行重新铸造。还将建立Schrödinger和Liouville框架之间的赔偿。
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 Schrödinger 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 Schrödinger equation. Comparsion between the Schrödinger and the Liouville framework will also be made.
