  In this work we study the single-qubit quantum state transfer in uniform
long-range spin XXZ systems in high-dimensional geometries. We consider
prototypical long-range spin exchanges that are relevant for experiments in
cold atomic platforms: Coulomb, dipolar and van der Waals-like interactions. We
find that in all these cases the fidelity increases with the dimensionality of
the lattice. This can be related to the emergence of a pair of bilocalized
states on the sender and receiver site due to the onset of an effective
weak-coupling Hamiltonian. The enhancement of the quantum state transfer
fidelity is more pronounced both with the increase of the couplings interaction
range and in going from a 1D to a 2D lattice. Finally, we test our predictions
in the presence of temperature-induced disorder introducing a model for the
thermal displacement of the lattice sites, considered as a set of local
adiabatic oscillators.
