Every 2D translation-invariant topological CSS code is local-unitarily equivalent, at the maximal anyon-preserving translation symmetry, to stacks of toric codes and product states, with an explicit polynomial-time construction.
Statistical Mechanical Models and Topological Color Codes
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abstract
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.
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Decoupling 2D translation-invariant topological CSS codes
Every 2D translation-invariant topological CSS code is local-unitarily equivalent, at the maximal anyon-preserving translation symmetry, to stacks of toric codes and product states, with an explicit polynomial-time construction.