  In this paper we revisit a two-level discretization based on the Localized
Orthogonal Decomposition (LOD). It was originally proposed in [P.Henning,
A.M{\aa}lqvist, D.Peterseim. SIAM J. Numer. Anal.52-4:1525-1550, 2014] to
compute ground states of Bose-Einstein condensates by finding discrete
minimizers of the Gross-Pitaevskii energy functional. The established
convergence rates for the method appeared however suboptimal compared to
numerical observations and a proof of optimal rates in this setting remained
open. In this paper we shall close this gap by proving optimal order error
estimates for the $L^2$- and $H^1$-error between the exact ground state and
discrete minimizers, as well as error estimates for the ground state energy and
the ground state eigenvalue. In particular, the achieved convergence rates for
the energy and the eigenvalue are of $6$th order with respect to the mesh size
on which the discrete LOD space is based, without making any additional
regularity assumptions. These high rates justify the use of very coarse meshes,
which significantly reduces the computational effort for finding accurate
approximations of ground states. In addition, we include numerical experiments
that confirm the optimality of the new theoretical convergence rates, both for
smooth and discontinuous potentials.
