  In this paper, we propose a numerical method to approximate the solution of
the time-dependent Schr\"odinger 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.
