Self-correcting quantum memory with a boundary
Adrian Hutter, James R. Wootton, Daniel Loss
University of Basel
It is well-known that no 2D stabilizer Hamiltonian with local interactions may be used as a self-correcting quantum memory at finite temperature: the lifetime of the stored quantum information cannot be increased arbitrarily by making the memory larger. Consequently, different authors have studied the possibility of coupling a 2D toric code Hamiltonian with local interactions to an external field, thereby inducing long-range interactions between the anyonic excitations of the local Hamiltonian. Self-correcting quantum memories are usually discussed with periodic boundary conditions to avoid the complications that arise with the possibility of creating unpaired topological defects at the boundaries. One expects that the influence of the boundaries becomes neglible if L, the linear size of the memory, becomes large enough. However, here we study two proposals for effective Hamiltonians with long-range interactions between anyonic excitations and show that for these the influence of the boundary becomes in fact dominant for large enough L: the decay of the stored quantum information is driven by unpaired defects created at the boundaries. The influence of the boundaries can be both beneficial or detrimental. In particular, we study an effective Hamiltonian proposed by Pedrocchi et al. that describes repulsion between anyons and anyon holes. For this system, we find a lifetime of the stored quantum information that grows exponentially in L2, even for an archidecture with open boundaries. However, L is upper-bounded through the breakdown of the perturbative threatment of the underlying Hamiltonian.