  We find that the overlapping of a topological quantum color code state,
representing a quantum memory, with a factorized state of qubits can be written
as the partition function of a 3-body classical Ising model on triangular or
Union Jack lattices. This mapping allows us to test that different
computational capabilities of color codes correspond to qualitatively different
universality classes of their associated classical spin models. By generalizing
these statistical mechanical models for arbitrary inhomogeneous and complex
couplings, it is possible to study a measurement-based quantum computation with
a color code state and we find that their classical simulatability remains an
open problem. We complement the meaurement-based computation with the
construction of a cluster state that yields the topological color code and this
also gives the possibility to represent statistical models with external
magnetic fields.
