  The ground state degeneracy of topologically ordered gapped Hamiltonians is
the bedrock for a quantum error-correcting code with macroscopic distance,
which is unfortunately not stable away from equilibrium. In this work, we show
that the presence of a bounded lightcone preserves topological order. As a
quantum code, the initial ground space will keep its macroscopic distance
during unitary time evolution. We also show how a bounded lightcone can emerge
through suitable perturbations in the two-dimensional toric code. Our results
suggest that topological quantum memory can be dynamically robust at zero
temperature.
