We find that the overlap 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 three-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 simulability remains an open problem. We complement the measurement-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.

• Received 16 November 2007

DOI:https://doi.org/10.1103/PhysRevA.77.042322