We explore the possibility of passive error correction in the toric code model. We first show that even coherent dynamics, stemming from spin interactions or the coupling to an external magnetic field,  lead to logical errors. We then argue that Anderson localization of the defects, arising from unavoidable fluctuations of the coupling constants, provides a remedy. This protection is demonstrated using general analytical arguments that are complemented with numerical results which demonstrate that self-correcting memory can in principle be achieved in the limit of a nonzero density of identical defects. The talk is mainly based on arXiv:1101.6028.

Zeeman fields can drive semiconductor quantum wires with strong spin-orbit coupling and in proximity to s-wave superconductors into a topological phase which supports end Majorana fermions and offers an attractive platform for realizing topological quantum information processing [1,2]. In this talk, I discuss how potential disorder affects the topological phase by a combination of analytical and numerical approaches. We find that the robustness of the topological phase against disorder depends sensitively and non-monotonously on the Zeeman field applied to the wire [3]. We also obtain the entire distribution function of the energy of the Majorana end states as well as of the lowest bulk state in wires of finite length, and discuss the implications for the speed at which a hypothetical topological quantum computer can be operated [4]. [1] Y. Oreg, G. Refael, F. von Oppen, Helical liquids and Majorana bound states in quantum wires, Phys. Rev. Lett. 105, 177002 (2010) [2] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, M.P.A. Fisher, Non-Abelian statistics and topological quantum computation in 1D wire networks, Nature Physics 7, 412 (2011) [3] P. W. Brouwer, M. Duckheim, A. Romito, F. von Oppen, Topological superconducting phases in disordered quantum wires with strong spin-orbit coupling, arXiv:1103.2746 [4] P. W. Brouwer, M. Duckheim, A. Romito, F. von Oppen, Probability distribution of Majorana end state energies in disordered wires, arXiv:1104.1531.

Robust edge states and non-Abelian excitations are the trademark of topological states of matter, with promising applications such as "topologically protected" quantum memory and computing. While so far topological phases have been exclusively discussed in a Hamiltonian context, we show that such phases and the associated topological protection and phenomena also emerge in open quantum systems with engineered dissipation. The specific system studied here is a quantum wire of spinless atomic fermions in an optical lattice coupled to a bath. The key feature of the dissipative dynamics described by a Lindblad master equation is the existence of Majorana edge modes, representing a non-local decoherence free subspace. The isolation of the edge states is enforced by a dissipative gap in the p-wave paired bulk of the wire. We describe dissipative non-Abelian braiding operations within the Majorana subspace, and we illustrate the insensitivity to imperfections. Topological protection is granted by a nontrivial winding number of the system density matrix. -Authors: S. Diehl, E. Rico, M.A. Baranov, P. Zoller -Reference: arXiv:1105.5947

TBA

Moore and Read's Pfaffian wave function remains one of the leading candidates for the description of the electronic ground state of the fractional quantum Hall plateaux at filling $\nu$ = 5/2. Much effort has been devoted to understanding the quasihole excitations but not much is known about the quasielectron excitations and about the neutral excitations (excitons), which can be viewed as combinations of quasiholes and quasielectrons. In this talk, we propose trial wave functions for quasielectron and exciton excitations for the Pfaffian fractional quantum Hall state and study these numerically. The trial wave functions are shown to have good overlaps with wave functions obtained by exact diagonalization of the model 3-body Hamiltonian for which the Pfaffian state is an exact zero energy ground state of maximal density.