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We address the question of whether symmetry-protected topological (SPT) ordercan persist at nonzero temperature, with a focus on understanding the thermalstability of several models studied in the theory of quantum computation. Wepresent three results in this direction. First, we prove that nontrivial SPTorder protected by a global on-site symmetry cannot persist at nonzerotemperature, demonstrating that several quantum computational structuresprotected by such on-site symmetries are not thermally stable. Second, we provethat the 3D cluster state model used in the formulation of topologicalmeasurement-based quantum computation possesses a nontrivial SPT-orderedthermal phase when protected by a global generalized (1-form) symmetry. The SPTorder in this model is detected by long-range localizable entanglement in thethermal state, which compares with related results characterizing SPT order atzero temperature in spin chains using localizable entanglement as an orderparameter. Our third result is to demonstrate that the high error tolerance ofthis 3D cluster state model for quantum computation, even without a protectingsymmetry, can be understood as an application of quantum error correction toeffectively enforce a 1-form symmetry.

Symmetry protected topological order at nonzero temperature