Numerical Study of the Hyperbolic Surface Code as a Gapped Phase of Matter

Hyperbolic surface codes are a generalization of the toric code on the hyperbolic surface. Due to the topological properties of the hyperbolic surface, these codes have linear encoding rate and their distance does not grow as a positive power of the system size. Thus, it is expected that hyperbolic surface codes are not stable against local perturbations, but the question of stability in these codes has not been investigated directly. Recently, it has been found that certain hyperbolic surface codes are equivalent to the gapped phases of a generalization of the Kitaev Honeycomb Model on the hyperbolic surface, allowing for numerical computation of the spectrum of these codes in the presence of perturbations. Exploiting this fact, we conduct an analysis to estimate the energy splitting of the ground states of the hyperbolic surface code with increasing system size. We find that for certain hyperbolic surface codes that have increasing distance with system size, the energy splitting of the ground states exponentially decreases as the codes grow in size. This suggests that some hyperbolic surface codes could in fact be stable against local perturbations.