Topological crystalline insulators (TCI's) are materials that host robust gapless states protected by crystalline symmetries. In this thesis, SnTe is studied using a tight-binding model. We focus on the electronic and transport properties of nanowires with (100) and (110) surface terminations, in the mesoscopic regime. In these configurations, gapless states are characterized as robust (against finite-size effects, step edges, and hinge rounding) spin-polarized surface and hinge states with corner charge, demonstrating intrinsic higher-order-topological behavior. We also investigate a mixed nanowire configuration having both (001) and (101) surface terminations, which displays extrinsic topological behavior.

Transport simulations reveal distinct conductance signatures for each surface termination. Nanowires with (100) terminations host surface states extending along the nanowire's perimeter, showing Aharonov-Bohm oscillations in longitudinal transport. Nanowires with a (110) terminations host confined surface states, giving rise to resonant tunneling conductance signatures for transverse transport.

These findings contribute to the general understanding of TCI nanowires, specifically the relationship between surface termination, gapless states, and transport signatures, providing valuable insights for the future design of TCI-based electronic devices.

The Josephson effect is a quintessential topic of condensed matter physics. It has stimulated decades of fundamental research, leading to a plethora of applications from metrology to outer space. In addition, it is set to play a crucial role in the development of quantum computers, forming the dissipationless non-linear inductance that lies at the core of superconducting qubits.

While they are traditionally realized using oxide based tunnel barriers, in this thesis we construct Josephson junctions from non-insulating materials such as semiconducting nanowires and quantum dots. We investigate how their highly nontrivial interplay with superconductivity can lead to new effects, both of fundamental interest and of relevance for quantum applications. To study these effects we make use the exhaustive toolbox available for superconducting circuits, allowing us to probe the junction behavior to beyond what is possible with conventional transport techniques.

The first experimental chapter of this thesis examines the behaviour of a transmon that hosts a highly transparent semiconducting weak-link as the Josephson junction. In this system we find spectroscopic evidence for the predicted vanishing of Coulomb effects in open superconducting islands, in accordance with theoretical predictions from 1999.

In the second experiment we deterministically place a quantum dot inside the junction of a transmon circuit. We then demonstrate that by using microwave spectroscopy we are able to accurately probe the energy-phase relationship of the Josephson junction over a vast regime of parameter space. This reveals the remnants of a quantum phase transition, and allows us to probe the time dynamics of the junction parity.

We subsequently use the same type of device to reveal the predicted spin-splitting of the Andreev bound states in a quantum dot with superconducting leads, as brought about by the spin-orbit interaction. When combined with a magnetic field, this is shown to result in the anomalous Josephson effect. Furthermore, we demonstrate that transitions between the spin-split quantum dot states can be directly driven with microwaves.

This motivated the investigation of a novel superconducting spin qubit, performed in the fourth experiment. Here we demonstrate rapid, all-electric qubit manipulation in addition to detailed coherence characterization. We ultimately show signatures of strong coherent coupling between the superconducting spin qubit and the transmon into which it is embedded, setting the stage for future research of this nascent qubit platform.

In the fifth and final experiment, we utilize a different approach compared to the preceding chapters. While we once-more construct transmons based on semiconducting weak-links, we now do so to leverage the intrinsic magnetic field resilience of semiconducting nanowires. This allows us to use a single device to study the mitigation of phonon-induced quasiparticle losses by trapping the phonons using both super and normal-state conductors.

This thesis concludes by discussing several ideas and proposals that aim to leverage the alternative Josephson junctions studied in this thesis. Combined with the results of the preceding chapters, this shows that hybrid superconducting circuits can be used to obtain deep insights into the fundamental physics governing their constituent junctions, and opens avenues towards building better qubits.

Quantum technology is a developing field of sciencewhere devices possess novel and superior functionalities thanks to their quantum-mechanical behaviour at the nanometer scale. A typical example is a quantum computer, where information is stored in quantum states of its quantum bits. By manipulating entangled and superposition states of these qubits, quantumcomputers can achieve exponential speed-ups in calculation and therefore solve currently unsolvable problems within polynomial computational times. This powerful advantage of quantum computers is particularly difficult to achieve in practice, due to decoherence - a tendency of quantum objects to lose their quantummechanical properties when interacting with their environment. Obviously, qubit decoherence cannot be avoided because the control of a quantum computer inevitably causes couplings to the environment. To mitigate decoherence, fault-tolerant implementations of quantumcomputing need to be developed.

Topological quantum computing has been proposed to achieve fault-tolerance since its significant robustness to decoherence is inherent in the quantum-mechanical nature of topological qubits. Building units of a topological qubit are Majorana zero modes (MZMs) – zero-energy quasiparticles that possess the non-Abelian anyonic exchange statistics and are localized at the boundaries of a topological superconductor. In sufficiently large topological superconductors, MZMs exhibit no overlap and therefore can in pairs host non-local fermions. By braiding non-overlapping MZMs, the information stored in the non-local fermions is manipulated while being insensitive to local noise. In this way one can perform computation that is topologically protected against local sources of decoherence.

In 2010, III-V semiconductor nanowires proximitized by s-wave superconductorswere proposed as a suitable candidate platform for the realization of topological superconductors. Topological superconducting phase occurs in such a hybrid nanowire due to an interplay among the large spin-orbit interaction, s-wave superconductivity, controllable electron density and large Zeeman energy introduced by an externalmagnetic field. Consequently, the nanowire bulk undergoes a band inversion and two MZMs appear at the two nanowire ends. First signatures of MZMs were reported in 2012 and since then a lot of effort has been put in fully demonstrating them. Despite huge improvements in the materials and measurement techniques, conclusive evidence of MZMs in hybrid nanowires is still missing. This is because disorder in hybrid nanowires can also cause the observed signatures of MZMs and make the topological scenario indistinguishable from the trivial ones. Therefore, further improvements and more detailed studies are needed and this thesis shows some recent examples of these...

A minimal Kitaev chain can be realized by coupling two quantum dots to a superconductor on both sides. Andreev Bound States inside the superconductor mediate two types of interdot couplings: Cross Andreev Re ection (CAR) and Elastic Co-tunneling (ECT). Spin-orbit interaction (SOI) enables equal strengts of CAR and ECT, where Majorana Bound States with a quadratic protection emerge. By extending the chain to more dots, the protection is predicted to become stronger. In this project, semiconducting-superconducting InSb nanowires provide both SOI and superconductivity. We develop systematic procedures to tune a two-site device to hold Majorana Bound States. Next, quantum transport processes on a three-site device are studied. Sequential processes combining CAR and ECT are observed.

The electrostatics has effect on conductance of nanowire devices. In this model electrostatics are described by nonlinear coupling of the Poisson and the Schr¨odinger equations. In presence of magnetic and electric field and spin-orbit interaction, conductance develops a feature called the helical gap. This gap is characterised by a drop of conductance and is the main focus of the research. The solver is based around an Anderson mixing scheme, and specific class of points has been discovered for which the solver performs poorly. For those points, an temperature annealing subroutine has been put in place to speed up convergence. This subroutine efficiently solves the system for some small finite temperature. The solver has also been expanded to solve systems for magnetic field pointed in any direction of the y, z plane. As a result, it is now possible to perform simulations for different magnitudes and directions of magnetic field, which are a handy tool for understanding the behaviour of conductance as a function of VG in real nanowires. The relation between energy and conductance has been researched. The size of helical gap is found to scale linearly with the Zeeman energy EZ, while other features of conductance scale nonlinearly

In this thesis, the methods of Matrix Product States are examined. We give background in the origin of the methods and explain in detail how the method can be applied to a series of Josephson junctions. Numerical results are given for the ground state energy analysis of these series.

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.

A method (1SBA) is derived and used to calculate disordered band structures. The validity of the method is investigated and the disordered band structures are compared to transport calculations.