  We prove that the 2D Ising model is complete in the sense that the partition
function of any classical q-state spin model (on an arbitrary graph) can be
expressed as a special instance of the partition function of a 2D Ising model
with complex inhomogeneous couplings and external fields. In the case where the
original model is an Ising or Potts-type model, we find that the corresponding
2D square lattice requires only polynomially more spins w.r.t the original one,
and we give a constructive method to map such models to the 2D Ising model. For
more general models the overhead in system size may be exponential. The results
are established by connecting classical spin models with measurement-based
quantum computation and invoking the universality of the 2D cluster states.
