Quantum simulation of the Fokker-Planck equation via Schrodingerization

TL;DR

This work leverages Schrödingerization to transform the linear Fokker-Planck equation into Schrödinger-type Hamiltonian dynamics, enabling quantum simulation of FP evolution under two formulations: conservation and heat equation forms. It develops and analyzes two implementation routes for the conservation form and a time-splitting Schrödingerization approach for the heat form, highlighting stability considerations when real parts of the operator have positive eigenvalues and providing recovery guarantees for the original solution. The paper also compares diagonalization strategies for finite-difference discretizations, showing that Bell-basis diagonalization typically outperforms Fourier-based methods in efficiency. Collectively, these results offer practical, circuit-level guidance for deploying quantum PDE solvers on FP-type problems, including norm-preservation properties, recovery probabilities, and complexity scaling under various boundary and discretization choices.

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.

Figures (4)

• Figure 1: Schematic diagram for the computational domain of $p$
• Figure 2: The maximum positive eigenvalues vary with different spatial mesh sizes.
• Figure 3: Quantum circuit for the conservation form. The black box for the simulation of $\boldsymbol{H}\otimes D_\mu$ can be chosen as the method given in CJL23TimeSchr.
• Figure 4: Quantum circuit for solving \ref{['FPS']}. Top: circuit for the Schrödingerization method; Bottom: circuit for the operator $U_{\Delta t}$ at each time step.

Theorems & Definitions (14)

• Lemma 2.1
• proof
• Theorem 3.1
• Lemma 3.1
• Theorem 3.2
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof