Rank-1 lattices and higher-order exponential splitting for the time-dependent Schrödinger equation
This work addresses the curse of dimensionality for solving the Schrödinger equation, offering a method with dimension-independent convergence properties, which is significant for quantum simulations in high dimensions.
The paper proposes a numerical method for the time-dependent Schrödinger equation in high dimensions using rank-1 lattice points for spatial discretization and higher-order exponential operator splitting for time discretization, achieving higher-order time convergence independent of dimension. Numerical results from 2 to 8 dimensions demonstrate the practical feasibility.
In this paper, we propose a numerical method to approximate the solution of the time-dependent Schrödinger equation with periodic boundary condition in a high-dimensional setting. We discretize space by using the Fourier pseudo-spectral method on rank-$1$ lattice points, and then discretize time by using a higher-order exponential operator splitting method. In this scheme the convergence rate of the time discretization depends on properties of the spatial discretization. We prove that the proposed method, using rank-$1$ lattice points in space, allows to obtain higher-order time convergence, and, additionally, that the necessary condition on the space discretization can be independent of the problem dimension $d$. We illustrate our method by numerical results from 2 to 8 dimensions which show that such higher-order convergence can really be obtained in practice.