  Ising model is a widely studied class of models in quantum computation. In
this paper we investigate the computational characteristics of the random field
Ising model (RFIM) with long-range interactions that decays as an inverse
polynomial of distance, which can be achieved in current ion trap system. We
prove that for an RFIM with long-range interaction embedded on a 2-dimensional
plane, solving its ground state is NP-complete for all diminishing exponent,
and prove that the 1-dimensional RFIM with long-range interaction can be
efficiently approximated when the interaction decays fast enough.
