  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.
