Self-correcting quantum memory with a boundary

We study the two-dimensional toric-code Hamiltonian with effective long-range interactions between its anyonic excitations induced by coupling the toric code to external fields. It has been shown that such interactions allow an arbitrary increase in the lifetime of the stored quantum information by making L, the linear size of the memory, larger Chesi et al. , Phys. Rev. A 82, 022305 (2010). We show that for these systems the choice of boundary conditions (open boundaries as opposed to periodic boundary conditions) is not a mere technicality; the influence of anyons produced at the boundaries becomes in fact dominant for large enough L. This influence can be either beneficial or detrimental. In particular, we study an effective Hamiltonian proposed by Pedrocchi et al. Phys. Rev. B 83, 115415 (2011) that describes repulsion between anyons and anyon holes. For this system, we find a lifetime of the stored quantum information that grows exponentially in L^2 for both periodic and open boundary conditions, although the exponent in the latter case is found to be less favorable. However, L is upper bounded through the breakdown of the perturbative treatment of the underlying Hamiltonian.