Symmetry-protected self-correcting quantum memory in three space dimensions

Whether self-correcting quantum memories can exist at nonzero temperature in a physically reasonable setting remains a great open problem. It has recently been argued [S. Roberts and S. D. Bartlett, Phys. Rev. X 10, 031041 (2020)] that symmetry-protected topological (SPT) systems in three space dimensions subject to a strong constraint---that the quantum dynamics respect a 1-form symmetry---realize such a quantum memory. We illustrate how this works in Walker-Wang codes, which provide a specific realization of these desiderata. In this setting we show that it is sufficient for the 1-form symmetry to be enforced on a subvolume of the system. This strongly suggests that the SPT character of the state is not essential. We confirm this by constructing an explicit example with a trivial (paramagnetic) bulk that realizes a self-correcting quantum memory. We therefore show that the enforcement of a 1-form symmetry on a measure-zero subvolume of a three-dimensional system can be sufficient to stabilize a self-correcting quantum memory at nonzero temperature.