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：本文研究了一种用于求解福克-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底层哈密顿动力学特性，甚至在引入边界条件时可能改变哈密顿结构。我们通过采用薛定谔化方法应对这一挑战——该方法可将任意具有非厄米动力学的线性偏微分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，研究表明基于半离散化的薛定谔化方法更为可取，尤其当处理非周期边界条件时。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存在正特征值的不稳定系统中的应用。分析表明，直接使用薛定谔化方法与稳定化程序具有相同效果。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的关系，阐明了薛定谔化方法如何精确复现每个时间步的时间分裂解。进一步，我们利用位移算子对热方程形式进行有限差分离散化。通过傅里叶基对角化位移算子，实现了频域空间的高效模拟。在提供对角化酉算子实现指引的同时，我们对贝尔基与傅里叶基的对角化进行了比较分析，结果表明前者通常比后者具有更高效率。