Majorana bound states in topological insulators and nanowires

Quantum computers outperform classical computers by achieving exponential increases in calculation speed for certain types of problems and for that reason have great potential to revolutionize computing. Compared to their classical counterparts the elementary units of information in a quantum computer are not the classical bits, zero and one, but rather the so-called quantum bits (or qubits) which most generally are quantum mechanical superpositions of the zero and one state. Unfortunately, the quantum bits are highly sensitive to the effects of environmental noise and consequently storing the quantum information in a robust manner represents a major challenge. Historically, it was Kitaev in 2001 who first proposed that this problem can be circumvented by using Majorana bound states as the building block for robust, so-called topologically protected, qubits. Subsequently, it was Fu et al. in 2008 who proposed the first realistic setup for generating Majorana bound states, namely topological insulator-superconductor heterostructures where the Majorana bound states can emerge within vortex cores. Moreover, in 2010 Lutchyn et al. as well as Oreg et al. put forward that Majorana bound states can also appear at the ends of semiconductor Rashba nanowires which are proximity-coupled to an s-wave superconductor and subject to a magnetic field. Finally, in 2013 Klinovaja et al. found that Majorana bound states can arise in chains of magnetic atoms that are deposited on a superconducting substrate. Within the last years these theoretical proposals have all been implemented experimentally and the first signatures for Majorana bound states, such as zero-bias conductance peak measurements, were reported. However, despite these encouraging experimental results, there still exists a broad range of open questions and hurdles. In this thesis, we address some of the most important experimental challenges and present new theoretical solutions.
In the first part of this thesis, we introduce two new platforms for generating Majorana bound states based on proximity-induced Pi Josephson junctions in topological insulators and crossed-Andreev pairing between semiconductor Rashba nanowires. Unlike the current experimental setups, the proposed schemes require either low magnetic fields or no magnetic fields at all. The latter characteristic constitutes a compelling improvement over current experimental setups for two reasons: (1) The detrimental effects of the magnetic fields on the superconductivity are either reduced or completely avoided. (2) In current experimental schemes the proximity-induced superconducting gap, which assures the topological protection of the Majorana qubits, is well-defined only at low magnetic fields (``hard gap"). At strong magnetic fields, a finite subgap conductance arises (``soft gap") and destroys the topological protection . Hence, with regards to future experiments on quantum information procession with Majorana bound states, a setup operated at lower magnetic field is highly desirable.
In the second part of this thesis, we propose a new method for detecting Majorana bound states based on quantum dot Phi_0 Josephson junctions. Here, we are motivated by the search for new, more conclusive indicators for Majorana bound states which is one of the most urgent challenges following the experimental results mentioned above. In fact, the recent zero-bias conductance peak measurements only constitute a sufficient, but not a necessary condition for the emergence of Majorana bound states. That is to say, the zero-bias conductance peaks can be explained by a multitude of different physical effects which are completely unrelated to the presence or absence of Majorana bound states. Interestingly, in the case of quantum dot Phi_0 Josephson junctions, the required ingredients largely overlap with those necessary to obtain Majorana bound states in Rashba nanowire systems. This motivated us to compare both the trivial superconducting and the topologically superconducting regimes of quantum dot Phi_0 Josephson junction and work out qualitative differences that can serve as new indicators for Majorana bound states.
In the final part of the thesis, we put forward a scalable scheme for quantum computation based on both Majorana bound state qubits and conventional spin qubits. The motivation for this part is three-fold: (1) The topological Majorana qubits are not universal for quantum computation. That is to say, not every logical quantum gate necessary to perform a quantum computation can be executed using Majorana braiding alone. For that reason, we couple the Majorana qubit to another type of qubit, namely the spin qubit, which can supplement the logical quantum gates that cannot be carried out on the Majorana qubits. (2) Spin and Majorana qubits are complementary with regards to their strengths and weaknesses. For example, unlike spin qubits, the Majorana qubits are intrinsically robust against unwanted perturbations and noise. At the same time spin qubits allow for significantly faster operations times compared to Majorana qubits. The hybrid spin-Majorana qubit which we develop in this chapter allows us to combine the best features of both worlds. (3) To utilize the full power of a quantum computer, it is not enough to consider a single qubit alone. What we need is a collection of many qubits making up a so-called surface code architecture on which many operations can run in parallel. We thus show how to construct a scalable network of the spin-Majorana hybrid qubits that can readily be experimentally implemented based on recent breakthroughs in the lithographic fabrication of Majorana nanowires in InAs/Al heterostructures.