Majorana end states in disordered topological superconducting wires

Zeeman fields can drive semiconductor quantum wires with strong spin-orbit coupling and in proximity to s-wave superconductors into a topological phase. Such topological phases are characterized by the presence of Majorana Fermions which obey non-Abelian statistics, and provide a possible platform for a topological quantum computer. So far Majorana Fermions have never been observed in experiments. However, semiconducting wires with strong spin-orbit coupling offer a promising path towards this goal. A key question, both for theory and experiment, is whether the topological phase is robust against the unavoidable presence of disorder. Here, after briefly introducing the proposals for the realization and control of Majorana Fermions in quantum wires, I mainly address their robustness against disorder showing that Majorana fermions persist in disordered wires up to a critical disorder strength which, remarkably, depends sensitively and non-monotonously on the Zeeman field applied to the wire. In finite length disordered wires the Majorana states combine into fermionic excitation at finite energy with large sample-to sample fluctuations. I discuss the probability distribution of such low energy level and the consequences for the speed at which such topological quantum bits can be operated.