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We propose a definition for topological order at nonzero temperature in analogy to the usual zero temperature definition that a state is topologically ordered, or "nontrivial", if it cannot be transformed into a product state (or a state close to a product state) using a local (or approximately local) quantum circuit.
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S. V. Isakov, M. B. Hastings, and R. G. Melko, arXiv:1102.1721
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S. Bravyi, unpublished, gives topologically trivial (according to the circuit definition) pure states with nonvanishing topological entanglement entropy for certain region choices; conversely a Chern insulator [ 23 ] is topologically ordered according to the circuit definition but has no topological entropy
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S. Bravyi and M. B. Hastings, arXiv:1001.4363, Commun. Math. Phys., in press
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Given an algorithm that measures the error syndrome and corrects errors using local unitaries, construct a unitary W W that does the same error correction while storing the state of the syndrome in some previously initialized auxiliary qubits; W † U x W W^{\dagger}U_{x}W acts as a unitary on the original system if the auxiliary qubits are properly initialized. Let V x V_{x} be the closest approximation to this unitary among all unitaries supported on sites within distance L / 8 L/8 of the support of U x U_{x} . Eqs. ( 9
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