Coupling a many-body system to a thermal environment typically destroys the quantum coherence of its state, leading to an effective classical dynamics at the longest time scales. We show that systems with anyon-like defects can exhibit universal late-time dynamics that is stochastic, but fundamentally non-classical, because some of the quantum information about the state is topologically protected from the environment. Our coarse-grained model describes one-dimensional systems with domain-wall defects carrying Majorana modes. These defects undergo Brownian motion due to coupling with a bath. Since the fermion parity of a given pair of defects is nonlocal, it cannot be measured by the bath until the two defects happen to come into contact. We examine how such a system anneals to zero temperature via the diffusion and pairwise annihilation of Majorana defects, and we characterize the nontrivial entanglement structure that arises in such stochastic processes. We also investigate simplified "quantum measurement circuits" in one or more dimensions, involving repeated pairwise measurement of fermion parities. We show these can be solved by mappings to classical loop models. The relaxation of the system of Majorana defects to its ground state is analogous to coarsening in a classical 1D Ising model via domain wall annihilation (the classical "$A+A \rightarrow \emptyset$" reaction-diffusion process). Here, however, configurations are labeled not only by the defect positions but by a nonlocal entanglement structure. This "$\gamma+ \gamma \rightarrow \emptyset$'' process is a new universality class for the coarsening of topological domain walls, whose universal properties can be obtained from an exact mapping.
Comment: 19+6 pages, 16+1 figures