Quantum simulation of the Fokker-Planck equation via Schrodingerization

TL;DR

This paper introduces a quantum simulation technique called Schrodingerization for solving the Fokker-Planck equation, preserving Hamiltonian structure and enabling efficient simulation of both conservation and heat equation forms.

Contribution

It applies Schrodingerization to the Fokker-Planck equation, analyzing its stability, boundary condition handling, and efficiency improvements via Fourier basis diagonalization.

Findings

Schrodingerization preserves Hamiltonian structure in semi-discretization.

The method stabilizes systems with positive eigenvalues.

Fourier basis diagonalization enhances simulation efficiency.

Abstract

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.