  The Kitaev honeycomb model is an approximate topological quantum error
correcting code in the same phase as the toric code, but requiring only a
2-body Hamiltonian. As a frustrated spin model, it is well outside the
commuting models of topological quantum codes that are typically studied, but
its exact solubility makes it more amenable to analysis of effects arising in
this noncommutative setting than a generic topologically ordered Hamiltonian.
Here we study quantum error correction in the honeycomb model using both
analytic and numerical techniques. We first prove explicit exponential bounds
on the approximate degeneracy, local indistinguishability, and correctability
of the code space. These bounds are tighter than can be achieved using known
general properties of topological phases. Our proofs are specialized to the
honeycomb model, but some of the methods may nonetheless be of broader
interest. Following this, we numerically study noise caused by thermalization
processes in the perturbative regime close to the toric code renormalization
group fixed point. The appearance of non-topological excitations in this
setting has no significant effect on the error correction properties of the
honeycomb model in the regimes we study. Although the behavior of this model is
found to be qualitatively similar to that of the standard toric code in most
regimes, we find numerical evidence of an interesting effect in the
low-temperature, finite-size regime where a preferred lattice direction emerges
and anyon diffusion is geometrically constrained. We expect this effect to
yield an improvement in the scaling of the lifetime with system size as
compared to the standard toric code.
