This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrodingerization method-it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrodinger-type equations. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrodingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrodingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrodingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique. We discuss the relationship between operator splitting in the Schrodingerization method and its application directly to the original problem, illustrating how the Schrodingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibits greater efficiency than the latter.

翻译：本文研究了一种求解福克-普朗克方程的量子模拟技术。传统的半离散化方法往往无法保持潜在的哈密顿动力学特性，甚至可能在引入边界条件时改变哈密顿结构。我们通过采用薛定谔化方法解决了这一挑战——该方法将任何具有非厄米动力学的线性偏微分方程和常微分方程转化为薛定谔型方程组。我们探索了该方法在两种不同形式的福克-普朗克方程中的应用。对于守恒形式，我们表明基于半离散化的薛定谔化方法更优，尤其在处理非周期性边界条件时。此外，我们分析了薛定谔化方法在系数矩阵或微分算子实部具有正特征值的不稳定系统中的应用。我们的分析揭示，直接使用薛定谔化方法与稳定化过程具有相同效果。对于热方程形式，我们提出了一种基于时间分裂技术的量子模拟程序。我们讨论了薛定谔化方法中的算子分裂与其直接应用于原问题之间的关系，阐明了薛定谔化方法如何在每一步精确重现时间分裂解。此外，我们利用位移算子探索了热方程形式的有限差分离散化。通过傅里叶基，我们对位移算子进行对角化，从而能够在频率空间中进行高效模拟。在提供对角酉算子实现额外指导的同时，我们对贝尔基和傅里叶基下的对角化进行了比较分析，表明前者通常比后者具有更高的效率。