  We study mappings between distinct classical spin systems that leave the
partition function invariant. As recently shown in [Phys. Rev. Lett. 100,
110501 (2008)], the partition function of the 2D square lattice Ising model in
the presence of an inhomogeneous magnetic field, can specialize to the
partition function of any Ising system on an arbitrary graph. In this sense the
2D Ising model is said to be "complete". However, in order to obtain the above
result, the coupling strengths on the 2D lattice must assume complex values,
and thus do not allow for a physical interpretation. Here we show how a
complete model with real -and, hence, "physical"- couplings can be obtained if
the 3D Ising model is considered. We furthermore show how to map general
q-state systems with possibly many-body interactions to the 2D Ising model with
complex parameters, and give completeness results for these models with real
parameters. We also demonstrate that the computational overhead in these
constructions is in all relevant cases polynomial. These results are proved by
invoking a recently found cross-connection between statistical mechanics and
quantum information theory, where partition functions are expressed as quantum
mechanical amplitudes. Within this framework, there exists a natural
correspondence between many-body quantum states that allow universal quantum
computation via local measurements only, and complete classical spin systems.
