We present an analysis of the relaxation dynamics of finite-size topological
qubits in contact with a thermal bath. Using a continuous-time Monte Carlo
method, we explicitly compute the low-temperature nonequilibrium dynamics of
the toric code on finite lattices. In contrast to the size-independent bound
predicted for the toric code in the thermodynamic limit, we identify a
low-temperature regime on finite lattices below a size-dependent crossover
temperature with nontrivial finite-size and temperature scaling of the
relaxation time. We demonstrate how this nontrivial finite-size scaling is
governed by the scaling of topologically nontrivial two-dimensional classical
random walks. The transition out of this low-temperature regime defines a
dynamical finite-size crossover temperature that scales inversely with the log
of the system size, in agreement with a crossover temperature defined from
equilibrium properties. We find that both the finite-size and
finite-temperature scaling are stronger in the low-temperature regime than
above the crossover temperature. Since this finite-temperature scaling competes
with the scaling of the robustness to unitary perturbations, this analysis may
elucidate the scaling of memory lifetimes of possible physical realizations of
topological qubits.Comment: 14 Pages, 13 figure
