Majorana fermions in mesoscopic topological superconductors. From quantum transport to topological quantum computation

In condensed-matter physics, Majorana fermions are realized as emergent quasi-particle excitations in the effective low-energy description of topological superconducting systems. Majorana bound states harbor much potential, both from a fundamental-physics viewpoint and for applications in quantum information processing, and their non-Abelian exchange statistics are fundamentally different from those of conventional fermions or bosons. We start by introducing Majorana systems that afford the topologically protected storage and manipulation of quantum information. A braiding of Majorana fermions may reveal their hallmark non-Abelian statistics, and forms the basic operation that is crucial for applications in quantum information processing. We discuss both an ideal braid scenario and corrections to this toy model view for interacting Kitaev chains. Next, the inclusion of charging energy effects allows for charge transport to access the non-local character of Majorana bound states in mesoscopic topological superconductors. The relevant physics of charge conservation are captured by a simple capacitor model, and we investigate how entanglement spreads between quantum dots tunnel-coupled by such Majorana boxes. Phase-coherent transport in coupled Majorana box devices also facilitates the formation of strongly-correlated low-energy states in simply-coupled islands contacted by normal leads. We here explain core solution strategies for quantum transport phenomena in Majorana networks while reviewing the topological Kondo effect, followed by an investigation of multi-junction geometries that go beyond the simple junctions considered before. In the Majorana box and loop qubit devices that comprise basic hardware units towards quantum computing applications, simple conductance or spectroscopic measurements can be used to characterize the ensuing Majorana-based qubits. We then discuss fundamental concepts and requirements for quantum information processing, starting from single- and two-qubit operations all the way to large-scale and fault-tolerant quantum error correcting codes. An extension of our basal hardware units to small networks allows for measurement-based protected quantum computations with Majoranas, up to and including Clifford-complete code networks that can run arbitrary quantum error-correction protocols. A promising example for Majorana-based quantum error-correction is the Majorana surface code shown in the last part of this thesis. Finally, we give an outlook of the current experimental progress on phase-coherent Majorana networks, and mention interesting directions of future research.