  The ability to store quantum information without recourse to constant
feedback processes would yield a significant advantage for future
implementations of quantum information processing. In this paper, limitations
of the prototypical model, the Toric code in two dimensions, are elucidated
along with a sufficient condition for overcoming these limitations.
Specifically, the interplay between Hamiltonian perturbations and dynamically
occurring noise is considered as a system in its ground state is brought into
contact with a thermal reservoir. This proves that when utilizing the Toric
code on N^2 qubits in a 2D lattice as a quantum memory, the information cannot
be stored for a time O(N). In contrast, the 2D Ising model protects classical
information against the described noise model for exponentially long times. The
results also have implications for the robustness of braiding operations in
topological quantum computation.
