While quantum devices have seen major advancements in recent years, there are still significant
challenges to scaling up their computational power. Geometry optimization techniques pose a
useful tool for tackling these challenges and improving the characteristics of quantum dot devices.
These devices consist of metal gate electrodes on a semiconductor heterostructure. Within the
semiconductor heterostructure, the electron wavefunctions used as the qubits are ‘trapped’ by the
potential induced by these metal gates.

    In this work, we have modeled the potential induced by the gates by discretizing the corresponding
Poisson equation using the finite-volume method. The discretized linear system is
solved with factorization-based solvers, of which we make repeated calls more efficient by applying
the Woodbury identity. The potential is used to solve the Schrodinger equation, of which
the eigenstates are transformed to a maximally localized basis to obtain the dot wavefunctions.
The gate voltages of the device are tuned so the effective Hamiltonian of the dots approaches a
target Hamiltonian. We have modelled the disorder sensitivity of the devices by inducing changes
to the boundary of the gate electrodes, for which the model is evaluated efficiently by utilizing
perturbation theory.

    Using this device model, we have implemented a discrete geometry optimization algorithm to
optimize for the gate electrode shapes. This algorithm generates a range of random changes to the
geometry shape and evaluates which one has the best characteristics.
We have demonstrated that this technique is effective for optimizing devices to be less sensitive
to gate shape disorder, to have higher level spacing, and to have more local gate-dot interactions.
We have applied it to double dot devices, triple dot devices, and double dot devices with wires.
The algorithm does not converge to the global minimum of the optimization problem, as different
initial conditions lead to marginally different results.

    We have implemented several strategies for the sake of computational efficiency. The use of
the Woodbury identity, perturbation theory for loss function gradients, and linear corrections for
disordered geometries lead to an estimated speedup of more than 62 times.
Since the aim of this project was to be a proof-of-concept for geometry optimization techniques
for quantum devices, we simplified some of the dynamics for computational efficiency or coding
efficiency. We have not modelled the Coulomb repulsion between electrons in different dots,
nor the effects of strain on the system. Additionally, the square-grid discretization of the gate
electrodes has an impact on the resulting geometries.

    Nonetheless, we have established that it is possible to apply discrete geometry optimization
techniques to improve the characteristics of modelled quantum dot devices. Moreover, we have
successfully introduced various strategies to improve the computational efficiency of the model.

This dissertation centers around two topics: graphene nanoribbons (GNRs) and superconductors. The aim of this thesis is to work towards combine these two topics, in order to study how superconducting correlations interact with magnetic correlations within graphene nanoribbons, such as the magnetic edge states present in the zigzag edges of graphene nanoribbons. To introduce superconducting correlations into a graphene nanoribbon, it is important that there is a highly electrically transparent interface between a superconductor and the graphene nanoribbon. This is the main scope of this work.

This work presents research towards using molybdenum rhenium (MoRe) alloy as an electrical contact material for 9 atom wide armchair edge GNRs (9-AGNRs). MoRe electrodes with nanometer-size separations (30 nm and 6 nm) are made and compared with palladium electrodes. Experiments with contacting aerosol gold nanoparticles were performed to confirm that the MoRe electrodes are superconducting and capable of making a clean contact. Beside pure superconducting contacts, Palladium is considered as a contact material, which is made superconducting by the proximity effect. To study the proximity effect, electrical measurements at a base temperature of 30 mK were performed on variable thickness Nb-Au-Nb and Nb-Pd-Nb superconductor-normal metal-superconductor (SNS) junctions made by shadow mask evaporation. A constriction in the Pd layer allows for increasing the junction resistance by feedback-controlled electromigration until a tunnel contact is formed. As the final part of this research, a superconducting diode effect was identified and studied in these SNS junctions.

The detection of non-Abelian exchange statistics is an open challenge which holds important promises for the advent of topological quantum computation. A recent work proposes to rely on the edges to reveal the braiding statistics of nonAbelian anyons in the bulk, in an entirely deterministic dynamical process. A time-dependent gap in a Josephson junction couples two co-propagating Majorana fermions, and as the gap closes, a pair of edge-vortices is injected into the edges. Because these defects have the same non-Abelian statistics, they are braided with vortices in the bulk. Conveniently, the fusion of the edge-vortices results in a quantized unit of charge at the exit. However, this process is so far only predicted in the adiabatic limit. In this work, this assumption is relaxed by means of a full manybody evolution of the superconducting ground state in the Bogoliubov-de-Gennes formalism. Beyond revealing the collective nature of the edge-vortex excitation, we demonstrate that the quantization of charge still holds if the system does not return to the ground state. Furthermore, the effect of path length difference between the edge-vortices confirms the theoretical predictions done in another work on the subject. At fast injections, we reveal weak oscillations in current contributed by the bound states in the junction which average to zero and are removed in the short junction limit. This work is concluded with a preliminary evaluation of the manybody parity operator, which indicates that the edge-vortex may encode the parity of the bulk vortices. This opens the possibility for sequential qubit manipulations on the edge-vortex.

Random walks have been applied in a many different fields for a long time. More recently, classical random walks are being used in wide variety of computer algorithms used to solve complex computational problems like 2-SAT, 3-SAT and the estimation of the volume of complex bodies. In this thesis we look into the quantum version of the familiar random walk, distinguishing between the discrete- and continuous-time quantum randomwalk. Some general properties of random walks are studied throughout this thesis. First we look at the behaviour of both the classical and the random walk on a simple 1-dimensional lattice. Then, to investigate the speed and efficiency of algorithms that use quantumwalks, we analyse quantumrandom walks in graphs, as graphs do a good job of representing the decision trees of algorithms. The graph we look at specifically is the ndimensional hypercube. The criterion we use to see how fast a quantumwalk can traverse certain types of graph is the hitting time. The hitting time is the expected amount of time a random walk takes to reach a certain vertex in a graph for the first time. Literature has shown that classical random walks have a hitting time that scales exponentially with the dimension n of the hypercube. We find that quantum random walks offer a significant speed-up to its classical counterpart. Generally they have a polynomial hitting time, that depends on the Hamming distance between the start and sink vertex. Furthermore we find that quantum walks can have interesting properties such as a non-unitary hitting probability, meaning the quantum walker will never visit the sink vertex. Finally we see that, in some simple, symmetric cases, the hitting time of the quantum walk even is sub-linear, scaling almost like a square root, rather than a polynomial.

Conventional semiconductor diodes dissipate energy in the form of heat when current passes through them. This is unwanted in, for example, cryogenic environments. Using a superconducting diode could mitigate this problem. These have been made by using special materials or combining multiple different circuit elements. We provide a systematic method of designing a tunable superconducting diode using a circuit of solely Josephson tunnel junctions. We show that even for a small number of Josephson junctions a strong diode effect can be achieved and that this method is stable under manufacturing tolerances. This method involves solving computationally inexpensive linear least squares problems to tune the Josephson energies of the junctions used.

In this thesis we provide an elementary introduction in finite dimensional representation theory of the Lie groups SU(2) and SU(3) for undergraduate students in physics and mathematics. We will also give two application of representation theory of these two groups in physics: the spin and quark models. We begin with first discussing representation theory for finite groups to create intuition for representations. We will explain notions such as intertwining maps and complete reducibility and we will mention some application of representation theory of finite groups inquantum mechanics. Hereafter, we begin with representation theory for Lie groups and Lie algebras, especially the groups SO(3) and SU(2), as these groups will play an important role in the description of spin. One of the main results is that SU(2) is the universal cover of SO(3). Furthermore, we give a description of spin by means of representation theory of SO(3) and its Lie algebra so(3). We will show that half integer representations of the Lie algebra so(3) cannot be exponentiated to representations of the Lie group SO(3), but it can be exponentiated to its universal cover SU(2). Moreover, we study the irreducible representations of SO(3) inside the Hilbert space L₂(R³). We will argue that one of the simplest quantum Hilbert spaces of a particle L₂(R³), can be modified to the completion of the tensor product L₂(R³) ⊗V, where, V is a finite dimensional Hilbert space that incorporates the internal degrees of freedom: spin. V carries an irreducible projective representation of SO(3). We will also discuss the addition of angular momentum of two particles in quantum mechanics. For this, we show how the tensor product of irreducible representations V and W of so(3) decomposes into SO(3) invariant subspaces of L2(R3). Hereafter, we will turn to representation theory of the Lie group SU(3) for setting up the mathematical framework for analysing the quark model. We will proof that there is a one-to-one correspondence between the irreducible representations of sl(3;C)and SU(3). We will also proof the theorem of the highest weight by which we can classify all the irreducible representations of SU(3) and sl(3;C) by their highest weight. We will also introduce the notion of the Weyl group and show that the Weyl group is a symmetry of weights of the finite dimensional representation of sl(3;C). Other properties of these representation, such as the dimension of the irreducible representations of sl(3;C) will be provided. Lastly, the quark model is discussed by means of representation theory of SU(3). We will show how this model can be used to classify two type of particles which also interact by means of the strong force: baryons and mesons. We show that we can classify the lightest mesons and baryons in so-called multiplets by the irreducible representations of SU(3). However, we will also introduce a modification of the strong force which further refines this model. A topic for further study would be how the symmetry group SU(3) describing Quantum Chromo Dynamics (QCD) can be used for the description of mesons and baryons.

Superconducting quantum circuits came out as promising candidates for the exploration of topological phenomena that are currently inaccessible in condensed
matter systems. One such circuit is a Cooper pair transistor which has already
been widely studied in different regimes of operation due to its importance in
quantum computation. However, it has only recently been appreciated that
a Cooper pair transistor hosts a non-trivial Chern number and topologically
protected current switching behavior. We provide here a more detailed analysis
of Cooper pair transistor operation for different parameter regimes and explore
the quantized ac current.

In this work we investigate neural networks and subsequently physics-informed neural networks. Physicsinformed neural networks are away to solve physical models that are based on differential equations by using a neural network. The wave equation, Burgers’ equation, Euler’s equation, and the ideal magnetohydrodynamic equations are introduced and solved with physics-informed neural networks. The solutions to the first equations were captured well. The solution to the ideal magnetohydrodynamic equations contained some problems. These problems include transitions between different types of behaviour and exact values of constant sections. On the other hand, general shape and behaviour of the curve and locations of contact discontinuities were predicted well.

This thesis gives a thorough description of the mathematical tool called reflection positivity, which can be used to prove the occurrence of phase transitions in physical models. A major result, although already known, is a theorem that gives tractable conditions on the Hamiltonian such that the Boltzmann functional is reflection positive. In this thesis, the theorem is used to give conditions on free parameters in four different models, such that the model is reflection positive with respect to certain chosen reflections. The evaluated models are (a) the antiferromagnetic quantum Heisenberg model; (b) the spin ice model; (c) the 6-vertex model and (d) the 16-vertex model. For the Heisenberg model we found that for reflections in a reflection plane there are certain parameter values such that the Boltzmann functional is reflection positive, this is an already known and published result. For the spin ice model we found that for the spin invariant reflection there is no symmetry that yields a reflection positive Boltzmann functional, this is a new result. For both the 6-vertex and 16-vertex model we showed that, for certain energy values, the Boltzmann functional is reflection positive with respect to reflections in the diagonal, which are also new results. In the case of the 16-vertex model, this boils down to checking whether or not a matrix is positive semidefinite. Using this result we showed that energy values that allow for the existence of magnetic monopoles do not yield a reflection positive Boltzmann functional. A topic for further research is investigating the occurrence of phase transitions in the models that are shown to be reflection positive, for which chessboard estimates seem to be a promising approach. Furthermore, in this thesis it was not rigorously proved that the spin ice model with a spin inverting reflection gives a reflection positive Boltzmann functional for rotational symmetry or symmetry in a reflection plane. This is believed to be true, but does require further investigation.

This thesis investigates fundamental properties of Josephson junctions embedded in microwave circuits, and an application arising from this hybrid approach. We used the versatility of superconducting coplanar DC bias cavities to extract previously inaccessible information on phase coherent and subgap mechanisms of graphene Josephson junctions. Chapter 1 gives an introduction to the technology of Josephson field effect transistors, among which graphene junctions show promise for future improvements in quantum computation. Together with an overview of the Josephson effect in superconducting-semiconducting systems, we introduce the concept of coplanar DC bias cavities for probing Josephson junctions at gigahertz frequencies. In chapter 2, we describe the experimental methods developed for carrying out the subsequent measurements. We include details on fabrication, material properties and measurement setup. Results of graphene Josephson junctions embedded in DC bias microwave resonators are presented in chapters 3 and 4. By following the resonance frequency and losses of the circuit, we are able to extract the junctions’ Josephson inductance and subgap resistance. Studying the nonlinear power and bias current response reveals further information on the underlying loss mechanisms and current phase relation. We turn to an application of our hybrid bias cavity – Josephson junction devices to detect small, low-frequency currents in chapter 5. Our device is competitive with state-of-the-art techniques for microwave radiation detection and, with minor modifications, should be able to outperform existing technologies by orders of magnitude. Finally, we conclude the presented work in chapter 6 and provide an outlook on potential future research.

The aim of this research is to develop an N -dimensional adaptive sampling algorithm to efficiently sample functions, meaning that with fewer samples the same accuracy is achieved compared to what homogeneously spaced samples would achieve. This algorithm is based on an existing Python package called Adaptive. The developed algorithm is applied to find and plot the Fermi surface of crystals with a higher resolution than homogeneous sampling would with the same number of points.

As long as a wave has a large enough wavelength, it should reflect off of smooth surfaces specularly: that is what the law of reflection states. This phenomenon is widely known, and used in for instance sonars. Electrons that reflect off of boundaries within conductors should abide this law as well, due to the particle-wave duality. However, it was recently discovered that for disordered graphene boundaries electrons scatter diffusively even when their wavelengths should be long enough not to. Graphene is a 2-dimensional semiconductor with linear dispersion at low energies, making it a widely researched topic as its applications in future electronic parts are promising. Its bilayer counterpart is made by connecting two layers on top of each other, which in contrast has a quadratic dispersion at low energies. In this work, the specularity of electron reflections within bilayer graphene were studied. This was done by constructing a tight-binding model, where random potentials on the outermost atoms represented imperfections of the lattice. Subsequently the variance of the scattering angle and the scattering phase was analysed, which can be derived using the scattering matrix of themodel. This was done numerically for both, and analytically for the phase as well. It was discovered that the variance of the scattering angle increases cubically with the Fermi wavelength of the electrons. This happens no matter what the distance between the Fermi energy and the disorder mean is, as long as the disorder strength is nonzero. The behaviour of these reflections is the exact opposite of what the law of reflection states. Furthermore, the variance of the scattering phase remains constant when the Fermi energy is close to the disorder mean, and changes inversely proportional to the boundary periodicity if the Fermi energy is further away. Consequently, the variance of the scattering phase versus the Fermi wavelength and the variance of the scattering phase versus the boundary periodicity do not share the same trend in bilayer graphene. More research is needed to verify this phenomenon, which could be done using the continuum description or a magnetic focusing device.

Grading exams is a time-consuming activity for teachers. Zesje is an open-source tool created to aid teach-ers in exam grading and streamline the grading process. Zesje currently uses computer vision techniques torealign images, and automatically find student numbers. However, teachers can currently only use Zesje tograde questions manually. Moreover the computer vision capabilities of Zesje can be improved. To make iteasier to grade exams, it should be possible for teachers to have multiple choice questions graded automati-cally. This project describes various improvements for Zesje, most notably using computer vision for the auto-matic grading of multiple choice questions, improving the accuracy of aligning scanned submissions, andautomatically detecting blank solutions. The team had to make several choices regarding implementations and choice of technology. Design goalswere also created to serve as a guideline for the project. At the end of the project, with the features imple-mented by the team, Zesje can automatically grade multiple choice questions, identify blank solutions andhas the corresponding front-end changes that allow the user to create multiple choice checkboxes on theexam PDF. These features have been tested extensively. The use of Zesje also poses some ethical challenges. Using automated grading may result in the event thatsome submissions may never be seen by a grader. By using benchmarks to compare the performance of processing scans in Zesje, the team found out thatthe grading time has greatly been reduced.

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

Particle - particle interactions in graphene force charge carriers to behave as viscous liquid. In this report we model and study such a system in the framework of molecular dynamics. Two interaction regimes are studied, Dirac regime with negative interaction potential and Fermi regime with positive potential. We report on numerical stability condition in both regimes, qualitative liquid like behavior within the Dirac regime and quantitative conductance of typical experimental device within the Fermi regime.

The edge states in finite quantum Hall graphene have previously been shown to be valley polarised for zigzag and armchair edges. Assuming that the valley isospin is also conserved at a smooth normal-superconducting (NS) interface, theoretical research has previously predicted that plateaus in the longitudinal conductance are expected to occur in the lowest Landau level of the incoming edge modes, which depends on the angle difference between the isospins entering and leaving the superconductor. In this thesis, this prediction is verified with a tight-binding simulation of six different NS junctions: for both zigzag and armchair edge nanoribbons, the superconductor can cover a single edge, two adjacent edges or two opposite edges. The theoretical prediction could be confirmed successfully, suggesting that the edge states are valley polarised along a smooth NS interface. Some deviations from the theory could be observed for the armchair edge ribbon with opposite edges when the width of the ribbon is not a multiple of three hexagons. Two consecutive widths show a complementary behaviour in the conductance such that their average corresponds to the predicted value with a remarkable robustness. The reason for this complementarity was briefly conjectured by using the special Andreev reflection in graphene and the coupling between sublattice and valley degree of freedom for zigzag edges. The parameter regimes allowing for the existence of conductance plateaus were established, confirming that the plateaus emerge for a system size much larger than the magnetic length and the superconducting coherence length, and that a smooth chemical potential, magnetic field strength and superconducting order parameter are necessary at the NS interface. The robustness of the NS edge states was furthermore investigated with three methods: a Fermi energy mismatch between the bulk and the superconductor, and random normally distributed variation in the onsite electrostatic potential and a random potential landscape. All results could confirm that intervalley scattering is the reason for deviations from the plateaus predicted by the theory.

Grading can be a very time-consuming activity for teachers. For this reason, numerous tools exist to aid teachers in grading. One of these tools is Zesje, a web application that allows electronic grading of paper-based exams. A major drawback of Zesje was that teachers were required to create their exams with LaTeX, a popular document typesetting language. Having to use LaTeX meant two things: not being able to use other software to create exams and poor performance when compiling the LaTeX source. This report outlines the research, development, and evaluation of a project that involves the deprecation of the LaTeX template from the software stack of Zesje.

We study the flow of supercurrents between two superconducting contacts connected by a 2d layer of graphene. We use a Markov chain Monte Carlo method to find Andreev bound states for circular electron trajectories. Using sample trajectories we estimate the currents as function of the superconducting phase difference between the contacts and the magnetic field.

Majorana bound states (MBS) have non-Abelian exchange statistics, which means exchanging the position of two MBS changes the state of the system. This attracted attention for multiple reasons: It is a quantum effect without a classical analogue, it introduces topology to condensed matter physics and the non-Abelian exchange processes allow quantum computing with intrinsic error protection. For a long time research on MBS was purely theoretical, because there was no experimentally accessible idea how to create them. This changed with the appearance of a recipe combining conventional superconductivity, semiconductors and a magnetic field. Now such setups exist and their conductance indicates the successful creation of MBS, but challenges regarding their quality, conclusive detection and control remain. In this thesis I propose different strategies for avoiding open experimental problems when creating MBS, and identify new pathways for detecting them. Most proposals for creating MBS are similar in two regards: They confine electrons to lower dimensional systems, for example a 1-dimensional wire, and they use magnetic field to couple to the electron spin. I start this thesis by developing a system where the magnetic field forces electrons onto cyclotron orbits, thereby spatially confining them without relying on the system’s geometry. This leads to an increased resilience against imperfections. Imperfections in real systems are not the only technical problem. One example is the combination of superconductivity and a magnetic field. Superconductors expel weak magnetic fields, while strong magnetic fields induce vortices in the superconductor which have a nonsuperconducting region in their core. In order to avoid these problems I turn to a completely different regime and develop a system that does not require a magnetic field to create MBS. Instead the role of the magnetic field is taken by a combination of supercurrents and spin-orbit coupling in the semiconductor. Then I turn to the detection of MBS, which is always a competition between two goals: finding a unique signature only caused by MBS and relying on a simple setup. One well-established signature of MBS is single electron transport through a superconductor, because without MBS a superconductor only transports electrons in pairs. Of course single electron transport is not a unique signature at all: it happens in most conductors. That makes it hard to distinguish MBS from undesired side effects. I develop a setup that adds falsifiability to this signature. In addition to detecting single electron transport, my scheme allows to block it if it is due to MBS, therefore making MBS and a normal conduction channel distinguishable. In the last part of my thesis I do not propose a system, but instead I analyze and simulate an unexpected outcome of an experiment. My colleagues were studying a driven superconducting resonator coupled to a Josephson junction (two superconductors separated by a short barrier) with the goal of using it to detect MBS. Completely unexpectedly it turned out to be a high quality microwave laser. In a collaborative effort we explain this previously overlooked phenomenon.