Majorana qubits in non-Abelian topological superconductors

Abstract

Non-Abelian superconductors are novel systems with exotic quasiparticle excitations, namely Majorana fermions, which obey non-Abelian quantum statistics. They are exploited as hardware platforms for fault-tolerant topological quantum computing. In this thesis, we primarily study the non-topological decoherence ef- fects existing in realistic systems and how they affect the stability of topological qubits and gates built from the Majorana quasiparticles. The main decoherence effects are the tunneling splitting of the topological degeneracy, thermal excitations and superconducting fluctuations which are not treated in the usual BCS mean-field theory. We calculate the tunneling splitting between non-Abelian vortices in both chiral p+ip superconductors and the superconductor/topological insulator het- erostructure, as a function of the inter-vortex distance, superconducting gap and the Fermi energy. It is shown that besides the well-known exponential suppression, the splitting also oscillates with the distance on the scale of Fermi wavelength. This implies that the fusion outcome of two non-Abelian particles depends strongly on microscopic details. We then investigate the robustness of topological qubits and

their braiding against thermal effects and non-adiabaticity, unavoidable in any re- alistic systems. We apply the formalism of density matrix and master equation and characterize the topological qubits in terms of physical observables. Based on this formulation, we show that the topological qubits are robust against both localized and extended fermionic excitations even when gapless bosonic modes are present. Finally, we explore the non-perturbative effect of strong fluctuations of superconducting order parameter, when the mean-field description in terms of Bogoliubov quasiparticles is invalidated. We consider a model of two-leg ladder of interacting fermions with only quasi-long-range superconducting order and derive the low-energy effective field theory using bosonization techniques. We find that although the whole spectrum is gapless, one can identify degeneracies of low-energy states resulting from Majorana edge modes. In the presence of certain impurity scatterings, we show that the splitting of the degeneracy has a power-law decay with the size of the system.