This paper covers the perfect indirect quantum measurement, specifically in the context of repeated measurement. The indirect measurement is useful as it allows information to be obtained from quantum systems without inflicting much disturbance on them. We restrict ourselves to cases with no evolution of the measured system between measurements and to perfect measurements, that is, measurements from which no outgoing information is missed and no extra information is added. In this case we can make use of the work by M. A. Nielsen (2005). It says that the expected amount of information following a perfect indirect measurement is larger than the information before the measurement. We make use of this result to show that the repeated indirect perfect measurement of a quantum state has two mutually exclusive outcomes. The first outcome is that the measured state becomes a pure state almost surely. The second is that the measurement eventually stops resulting in information being revealed. In the latter case, further measurements on the system result in the state switching through spaces with the same dimension, and thus it does not become a pure state. This paper builds on the work by Maassen and K¨ummerer from 2005, which already proved this, by expanding their proofs and adding additional theorems and proofs to create a more self-contained result. Further studies might look at the rate at which states become pure, and what might influence this rate. ... Read more

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. ... Read more

Chirality Induced Spin Selectivity is the phenomenon where the chirality of certain molecules favours the transmission of electrons based on their spin. Among many examples, this long-studied phenomenon appears in two-terminal transport experiments, where different magnetisations of the leads can give different current-voltage characteristics. In previous research by Rikken [20], the chiral geometry of the device was determined as a necessary condition for antisymmetric IV curves.

In this thesis, we implemented a Büttiker probe (BP) in a 6-helicene model based on the previous work of Geyer [9]. The probe mimics the decoherence in a two-terminal CISS experiment. Moreover, this enables us to magnetise both leads independently. By altering the magnetisation of the leads and the orientation of the Büttiker probe, we were able to analyse many possible experimental setups. This enabled us to express the current difference in terms of bias voltage, magnetisation and BP orientation, where the latter was the research objective.

Isotropic Büttiker probes lead to a CISS effect, which is absent in a coherent electron transport model. Further research is needed to determine the exact nature of the numerical errors in our isotropic BP experiments. The results of anisotropic BPs can be explained assuming that the current difference is linear in the anisotropy of the BP. Further research is needed to strengthen this conjecture. These results are supported for lead magnetisation along an axis, perpendicular to the helical axis of the molecule, as well as magnetisation along this helical axis. ... Read more

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.
... Read more

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. ... Read more

This thesis explores the quantum scarring in bilayer graphene, using the tight-binding model, in comparison to classical calculations. In particular the scarring for different geometries. We observed that integrability in systems with an anisotropic Fermi surface depends on the commensurability of the billiard shape and lattice symmetry. We simulated a circular, hexagonal, and stadium billiard. In the circular billiard, we found: 1) triangular scars, 2) scars along the diameter, and 3) chaotic trajectories. However, we did not find whispering-gallery scarring as predicted by classical theory. For the hexagonal and stadium billiard we found scarring when the billiard and lattice symmetry are commensurate. We also found that the scarring disappears when rotating the hexagonal and stadium. ... Read more

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. ... Read more

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. ... Read more

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. ... Read more