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