Quasi-Majorana states in Majorana devices

Nowadays, quantum computers, a promising direction of computer hardware development, suffer too much from errors caused by disturbance of qubits to compete with the state-of-the-art classical computers. Topological phases in condensed matter physics offer a solution: the topological edge states emerging in such a phase are spatially separated by an insulating bulk, which makes a qubit constructed from such topological states much more resilient to local perturbations. Majorana states are examples of topological edge states, and form the main focus of research on topological quantum computing due to indications of successful creation and detection of Majorana states in one-dimensional superconductor-semiconductor hybrid devices. Quasi-Majorana states share most characteristics of Majorana states, but appear in the topologically trivial phase and appear at the same position, in contrast to topological, spatially separated Majorana states. For this reason, quasi-Majorana states are not protected from noise, and hence appear to be not useful for quantum computing. For a long while, quasi-Majorana states were considered a nuisance, because they mimic the local signatures of topological Majorana states, and therefore can make a false positive signature in the search for Majorana states. In trying to understand how Majorana devices work, I started this thesis with an investigation of electrostatics in Majorana devices. An applied gate voltage sets the chemical potential in a Majorana nanowire, relevant for the creation of Majorana states, but this is influenced by other electrostatic components in the environment. I found that these electrostatic effects introduce non-universal, geometry-dependent behaviour of two Majorana characteristics: the shape of the topological phase boundary and the oscillations of the Majorana splitting energy. In addition to controlling the band structure, gate electrodes alter the transport properties of electrons by creating a tunnel barrier or a constriction in the potential. Studying such constrictions, I have demonstrated that the confinement potential barriers are smooth, allowing to measure the helical gap in the band structure, which agrees with experimental observations. Since quasi-Majorana states appear at the slope of a smooth confinement potential, the results of my simulations of the electrostatics motivated to continue with an investigation of these states. I showed that quasi-Majorana states not only have an exponentially suppressed energy as a function of magnetic field, but also have an exponentially different tunnel coupling across the barrier where they are located. This realization allowed me to strengthen the recent observations of similarity between Majorona states and quasi-Majorana states, and conclude that tunneling measurements can not distinguish Majorana states from quasi-Majorana states as a matter of principle. Because of this extreme similarity, I turned to study a possible alternative strategy to distinguish topological Majorana states from quasi-Majorana states. This strategy ix x SUMMARY focusses on rectifying behaviour in the nonlocal conductance through a Majorana wire connected to two normal leads. This phenomenon measures a global topological phase transition, rather than a local measure of the density of states, and therefore it is not influenced by the presence of quasi-Majorana states. The similarity of quasi-Majorana and topological Majorana states also leads to an unexpected consequence. Braiding (an exchange of two Majorana states), a building block of a topological quantum computer, can also be done with quasi-Majorana states. Although quasi-Majorana states appear next to each other, their couplings are exponentially different, which allows to control them individually. Braiding quasi-Majorana states can even be advantageous, because it requires less precise control over system parameters. I therefore conclude that braiding of quasi-Majorana states is within experimental reach, and opens an alternative route on realizing a quantum computer.