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.

Chirality-induced spin selectivity (CISS) is a general term denoting the interplay between the chiral structure of molecules and the electron spin. CISS has been studied for over two decades, leading to a consistent picture of experimental phenomena. In this thesis, we focus on transport experiments exhibiting CISS. In spite of the efforts of many scientists, there is no theoretical explanation for the high degrees of CISS that have been measured in electron transport experiments. In general, it is agreed upon by theorists that the effect is caused by the interplay between the electronic spin-orbit interaction and the helical molecule geometry, in combination with phase-breaking effects such as electron-electron or electron-phonon interactions. As of yet, no theoretical model has achieved realistic degrees of SOC without drastic inflation of the spin-orbit interaction strength.
In this thesis we explore the possibility of using matrix product states (MPS) to study spin-selectivity in boundary-driven electron transport through tight-binding models of chiral molecules, a novel approach in the field of CISS. To this end, we use a model proposed by Fransson in 2019, which considers interacting electrons in a Hubbard model with a spin-orbit interaction adapted from the Kane-Mele model. In this thesis work, the fermionic Hubbard model is mapped to a double spin chain using the Jordan-Wigner transformation. The state of the system is described by a matrix product density operator (MPDO) which is vectorised to a matrix product state (MPS). The system dynamics are described by a vectorised Lindblad equation. The advantage of this approach lies in the fact that it does not require the use of any systematic approximations to the Hamiltonian, in contrast to previous studies. The developed method is validated against the results of previous works studying the boundary-driven Heisenberg-XXZ model and the boundary-driven Hubbard model.
The method is shown to be capable of reproducing chirality-induced spin-selective effects for short chains. The results of this study show a finite magnetocurrent that is odd in bias voltage with an associated magnetoresistance of less than 1%. This is in line with previous studies of this model, but two orders of magnitude lower than experimentally measured values. However, these results are obtained using highly inflated values for the spin-orbit interaction strength. In multiple cases, the results do not satisfy the Onsager-Casimir and Büttiker reciprocity principles, which state that the magnetocurrent should vanish in the low-driving and in the non-interacting regimes. Moreover, the continuity of the current in the steady state was not fully satisfied.
We provide evidence that indicates these problems result from the time-integration error introduced by the Suzuki-Trotter decomposition. We expect that these can be mitigated using higher order time-integration schemes. From the results of this study we can conclude that matrix product states are a viable tool to study CISS in bound-electron transport. However, the method presented in this thesis suffer from numerical errors. We present several suggestions for improvement which address these shortcomings.

Efficiency in handling instructions within compilation and control processes is essential for scalability and fault-tolerant quantum computation. To mitigate the limited bandwidth for transmission of instructions and energy bottlenecks in cryogenic control architectures, this thesis aims to develop a compressed representation of quantum circuits. To achieve this goal, we study the concepts of algorithmic information theory and resource theory of computation. We focus on description complexity and establish compression as a useful estimate of algorithmic description complexity. With this motivation, we develop a generalized framework for the synthesis of quantum unitaries into a set of native gates and present a Huffman-encoded representation of the instruction stream that has a short code dictionary and offers a 60% compression over binary encoded representations. The developed framework offers 2 major contributions: an energy-efficient encoded representation of the quantum instruction stream and an estimate of the description complexity for quantum circuits. It qualifies as a successful algorithmic approach towards optimizing the QISA and aids the discovery of high-level quantum programming constructs.

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 research, the effect of Chiral Induced Spin Selectivity is studied by means of a transport calculation on a model of a chiral molecule between two gold contacts. The method consists of two main parts: optimizing the geometry of the entire system, being the molecule and the contacts, and performing the transport calculation on the system, which yields the density of states and the transmission over the energy range of -0.5 to 0.0 Hartree.

The geometry optimization is performed in two ways: in the first approach the entire system is optimized under the constraints that the y- and z-coordinates of the atoms of the contacts are frozen, in the second approach the atoms of the contacts are completely frozen on their initial positions. The first approach did not conserve the periodic structure of the gold lattice. The second approach yielded a geometrically optimized system with correct contacts.

The transport calculation is performed on the three systems, being the non-optimized system, the system optimized by the first method and the system optimized by the second method. There was no spin selectivity found: the density of states as well as the transmission are exact copies for the two spin orientations, which is the consequence of a spin-restricted transport calculation.

The density of states for the three systems are similar. The highest occupied molecular orbital as well as the lowest unoccupied molecular orbital were found to be situated just below and above the Fermi energy of the contacts, respectively, which is consistent with literature.
The transmission of the three systems show greater variation. The systems with optimized geometries have a constant transmission close to zero around the Fermi energy of the contacts. The non-optimized system has a fluctuating transmission above zero around this energy.

Based on these findings, a spin-unrestricted transport calculation including a spin-orbit ZORA-key is proposed. In order to speed up calculations, it is also recommended to apply the Wide Band Limit.

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.