In recent years, germanium hole spin qubits have become a research hotspot for semiconductor quantum dot quantum computing. Their strong spin-orbit coupling gives rise to an anisotropic exchange interaction. The exchange interaction plays an important role in implementing two-qubit gates for electron/hole spin qubits. Currently, the modeling, control, and optimization of exchange interaction primarily focus on the isotropic type between electron spins. Strategies for making full use of the anisotropic exchange interaction to implement high-fidelity two-qubit gates remain immature. In this thesis, we focus on germanium hole spin qubits and utilize the anisotropic exchange interaction between them to implement two-qubit gates protected against errors by means of composite pulse technique.

The rotational velocities of stars at the edges of galaxies are higher than predicted by Newtonian dynamics. One theory that is capable of explaining this anomaly is called MOdified Newtonian Dynamics
(MOND). To further test this MOND theory, the wide binary test has been suggested. In the wide binary test, the stability of wide binary systems is tested. The MOND theory deviates significantly from the
Newtonian theory when accelerations are smaller than a0 = 1.2·10−10 m/s2 , which is present in wide binary systems. The MOND theory is non-linear, which introduces a non-trivial effect of the external field
(EF) of the Milky Way on the wide binary orbit. To perform the wide binary test, an iterative particle mesh N-body code was used with initial conditions taken from the Gaia mission database. Additionally, the EF was modeled in three ways, where the Newton potential adjustment model was found to be the most realistic. The results include nine simulations, split up into three models and three different regimes. The models include the Newton model, the MOND without EF model, and the MOND with EF model. The regimes consist of the Newtonian regime (ainternal > a0), the transition regime (ainternal ≈ a0), and the deep MOND regime (ainternal < a0). In the Newton model, only the Newtonian regime gave stable orbits. The MOND without EF model gave stable orbits for all regimes, and the MOND with EF model gave stable orbits only for the Newtonian regime. The MOND without EF gave stable wide binary orbits, whereas the other models did not. If additional measurements detect these wide binary systems, then the MOND without EF model gains credibility. The orbits of the Newton model and the MOND with EF model were similar, with only a small difference. This is because the strength of the external field is 1.6 · a0, which makes the external field dominant over the internal accelerations, making the simulation Newtonian-like. To determine which model is correct, additional long-term measurements of Gaia are required. Further research recommendations include improving the performance and accuracy of the code. This is done by better implementation of the fast Fourier transform, translating the code into a faster programming language than Python, using adaptive time steps, and reducing self-interaction. The most important extension to the model is to introduce the galactic tidal effect.

The distribution of multiple entangled pairs plays a vital role in various applications, such as quantum metrology, blind quantum computing, quantum key distribution, and quantum teleportation. While qubit-based protocols exist, they are often limited by short coherence times in quantum memories and the need for individual pair generation, leading to inefficiencies and reduced fidelity. To address these challenges, this thesis focuses on the development and analysis of a comprehensive theoretical framework for satellite-based entanglement distribution using high-dimensional qudits. By employing high-dimensional qudit encoding, which enables the simultaneous entanglement of multiple pairs, we aim to enhance the capacity and efficiency of entanglement distribution schemes.

The objectives of this research are twofold, first, to compare the performance of the qudit-based protocol with a similar qubit-based protocol, particularly in a satellite-based context, and second, to investigate the feasibility and potential advantages of employing high-dimensional qudit encoding for near-term long-range entanglement distribution. The theoretical framework encompasses various elements, including a single photon pair source, lossy transmission, mapping of qudit states to qubit memories, and heralding measurements.

Through various simulations, we evaluate the impact of different parameters on the performance of the proposed protocol. Our findings reveal that high-dimensional encoding significantly alleviates memory requirements, making it suitable for current technologies with low memory coherence times. Multiplexing techniques further enhance the achievable distances and transmission rates, highlighting their importance for realistic implementations. However, we note that higher-dimensional protocols introduce additional experimental challenges and errors, warranting further research and optimisation. Furthermore, we explore the effect of reducing dark count rates in detectors and analyse its influence on the overall performance. Our investigations underscore the need for substantial detector improvements, as current dark count probabilities are on the same order of magnitude as channel losses.

By advancing our understanding of satellite-based entanglement distribution using qudits, this research contributes to the development of practical and secure quantum communication networks. The results presented herein lay the groundwork for future advancements in quantum communication technologies, fostering the realisation of global-scale quantum networks.

Quantum computers can solve certain problems faster than classical computers, but they require many qubits to solve valuable problems. Distributed quantum computing provides a scalable approach to increasing the number of qubits by interconnecting small-capacity quantum devices, or quantum nodes. Entangled states shared between nodes, so-called entangled links, can serve as a resource for implementing nonlocal operations. A better understanding of distributing links in a network of quantum nodes can guide the design of hardware and protocols for distributed quantum computing systems. We used two metrics to measure the performance of such entanglement distribution protocols considering the network’s objectives. Specifically, the virtual node degree reflects the requirement for many links to perform nonlocal operations, while the virtual neighbourhood size reflects the need for links between remote nodes to increase the number of qubits available for computation. Contrary to most prior research, these metrics explicitly consider the time-dependent fidelity of entangled links. We used discrete-time simulations to investigate the performance of a protocol that continuously distributes entanglement in a quantum network with a regular topology. The number of entangled links in the network evolves as quantum nodes create new links through entanglement generation and entanglement swaps, and remove low-fidelity links. The nodes probabilistically attempt swaps and can maximise the performance metrics by varying this probability.

We found that the performance metrics exhibit qualitatively similar behaviour for various network parameters, such as coherence time and entanglement generation fidelity. However, the network parameters shift the swap attempt probability that maximises the virtual neighbourhood size differently. The effect of network boundaries on performance metrics depends on the network topology.

In this thesis Modified Newtonian Dynamics (MOND) is explored in galaxy clusters similar to the Virgo cluster. MOND is a theory proposed to explain the flat rotation curves of galaxies and the velocities of galaxies within galaxy clusters, as an alternative to the Dark Matter (DM) model. MOND states that Newton’s law of gravitation is incorrect at accelerations of the order of and smaller than Milgrom’s constant a_0 = 1.2 · 10^−10m/s^2 [1]. 
The MOND potential φ_M created by a certain mass distribution ρ satisfies the MOND equation, a non-linear partial differential equation. For accelerations much smaller than a0 this equation gives a quadratic relation between the gradient of the potential (∇φ_M) and the mass distribution ρ, this is called deep MOND. This is much different from the Poisson equation, that infers a linear relation between ∇φ_M and the mass sources, and which still holds for accelerations much larger than a_0 [1], referred to as Newtonian Dynamics (ND). For accelerations around a0 an interpolation of deep MOND and ND is used. It appears that the potential and acceleration in Virgo-like clusters is according to ND at the center, and approaches deep MOND at the edge. Therefore an interpolation function µ is necessary to model such clusters accurately.
When the MOND potential φ_M is substituted into the Poisson equation, a new mass distribution is found, the apparent mass distribution ρ_AM, which would need to match the actual mass distribution in DM models, which use the Poisson equation. This apparent mass distribution ρ_AM is the sum of the actual mass distribution ρ extracted from optical observations and the apparent dark mass distribution ρ_ADM, a distribution that is interpreted as a theoretical DM halo. This allows us to compare MOND and DM. With our method, realistic mass configurations of galaxy clusters that are Virgo-like, generate apparent mass distributions ρ_AM with regions containing negative mass. The existence, shapes and locations of these regions are in agreement with what Milgrom found [2]. The total mass of the actual mass distribution is M = 10^15M_sun, while the sum of the negative mass is M_negative ≈ −0.09 · 10^15M_sun = −0.09M is approximately 9% of the total mass. Since negative mass is not acceptable, this gives us the opportunity to create conditions to falsify either the MOND model or the DM model. 

In this thesis two topics are discussed. The covariant formulation ofMaxwell’s equations of electromagnetism and the formulation of said equations in the context of a rotating frame of reference. Through the development of the necessary theories of Differential Geometry and Special Relativity we show how to formulateMaxwell’s equations in terms of the covariant derivative and in terms of the hodge operator and differential operator. Using this formulation we study the transformation properties of these equations under Lorentz transformations and we conclude that they remain invariant under said transformations. Secondly,we discuss literature concerning the problem of electromagnetism in the context of a rotating frame of reference. We show that the method of direct transformation to a rotating frame results in an unobservable frame of reference and we show that it is impossible to reconstruct Coriolis like forces in an observable frame of reference.

Generation of high fidelity entanglement between quantum nodes is a key component of a future quantum internet. Heralded entanglement generation of two spatially separated qubit nodes can be established by interference and measurement of two photons, each entangled with one qubit state. The two-node entanglement fidelity is limited by the degree of indistinguishability of the photons, which can be measured in a Two-Photon Quantum Interference (TPQI) experiment. In this thesis, a TPQI experiment has been performed with photons emitted by a single Nitrogen-Vacancy (NV) center. This self-interference experiment shows a visibility of V=0.91±0.02 and a photon indistinguishability of J=0.945 in the Monte-Carlo method obtained 1σ-confidence interval of [0.920, 0.966] after correction for system imperfections, demonstrating near-perfect indistinguishability of zero-phonon line photons emitted by a single NV center. Furthermore, an extensive TPQI model was developed that includes possible arrival time- and frequency differences of the photons. This model predicts a dark- and noise count limited V=0.79±0.06 for a future two-node NV TPQI experiment with quantum frequency-converted photons, at a distinguishable photon coincidence rate of 1.2mHz, allowing for an experimentally feasible double-click two-node entanglement fidelity of 0.89±0.03.

Synchronisation is the remarkable phenomenon of in-phase movement of coupled oscillators during a prolonged period of time, which even occurs when these have different natural frequencies. This property is used in many fields like physics, biology and chemistry, to model various system behaviours, for instance: circadian rhythms in the chemistry of the eyes \cite{Rompala2007}.
\newline\noindent This thesis focuses on synchronisation and entanglement in systems consisting of quantum Van der Pol oscillators. After looking at the exemplary properties of the classical VdP oscillator, it follows the methods of \cite{EshaqiSani2020} to explore the behaviour of two coupled QVdPOs, quantum Van der Pol oscillators, using Monte Carlo simulations for trajectories of this system. The system is then expanded to three-oscillator systems with all-to-all and chain coupling. The validity of the extension of the properties found in two QVdPOs in \cite{EshaqiSani2020} with regard to synchronisation and entanglement is tested. Does an Arnold tongue, the region of parameters for which a system shows synchronisation, still exist and if so has its shape changed? Do synchronisation and strong entanglement still show a positive correlation? \newline\noindent Simulations of the 2-oscillator system gave results which were just slightly different from those obtained in\cite{EshaqiSani2020}, validating the occurrence of synchronisation within the Arnold tongue and showing a positive correlation between synchronisation and entanglement of the system. Both the 3-oscillator systems showed synchronisation as well in their respective Arnold tongues, which were increasingly smaller for the all-to-all coupled and the chain coupled system when compared to the 2-oscillator system. Overall, the amount of synchronisation, shown in three oscillators was very close to the amount shown in two oscillators when strongly coupled and just a little less for weak coupling. The correlation between synchronisation and strong entanglement was also found for three oscillators, though for weaker coupling the chain coupled system showed a little more entanglement than before.

Topos theory and quantum mechanics are both known for having a logic that is different from ordinary logic. With this in mind, much work has been done on unifying these two fields. Loveridge, Dridi and Raussendorf apply this unification to measurement-based quantum computation [14], revealing links between computation, contextuality and the failure of the law of excluded middle in the topoi associated with each computation. We review their work, fill in gaps, follow their research suggestion and have some criticism. Our main original finding is a formula, in the formal language of the topos associated with a computation, that expresses that the computation is deterministic.

This thesis deals with n-to-1 entanglement distillation protocols for bipartite quantum systems in a group theoretical setting. The local operations in the protocols are restricted to Clifford operations. An efficient representation of the elements of the Clifford group in terms of binary matrices is presented. Moreover, a characterization of the subgroup of the Clifford group which leaves the fidelity and the success probability invariant is given. Based on this characterization, a novel approach for the optimization of distillation protocols is presented: instead of checking every element of the Clifford group individually, it is sufficient to consider one element of every right coset of this subgroup. These new insights are used to find optimal protocols for n = 2, 3, 4 and 5.

In this project, quantum cloning machines are analyzed that take in N quantum systems in the same unknown pure state and output M quantum systems with M > N, such that the output best resembles the ideal, but impossible output of an M-fold tensor product of the pure input state. The proof of Keyl and Werner is reviewed in which a unique optimal solution is constructed, both in the case where the quality of the cloning is determined by the entire output, and in the case where the quality is determined by measurements on a single clone. Furthermore, it is shown that previous work on qubit cloning machines is a special case of the presented optimal solution.

Report about how Matrix Product States can be used to analyze a sub set of problems in quantum mechanics. A exact derivation is given of the Ising Model in a Transverse field and with these solutions the TEBD algorithm using Matrix Product states is validated.

The Hamiltonian of the NV center contains unknown parameters, such as the zero-field splitting and the magnetic field. Knowing how the frequency measurements depend on the parameters will give greater understanding
of the NV center. Including the nitrogen nuclear spin gives us the opportunity to find more parameters such as the quadrupole splitting. Analytically finding expressions for all of these parameters is very time consuming to do, and therefore the method of second order perturbation is used in this report
to find approximations. With this method, estimations of the eigenvalue differences and parameters were found, including new estimations for the quadrupole splitting. As we don't exactly know whether the quadrupole splitting is positive or negative, there are two estimates found given by Δ_Q⁺ = 4949.25(2) kHz and Δ_Q^- = -4949.11(1) kHz.

Quantum networks play an important role in the fields of quantum information and quantum computation. One of the current problems for these networks concerns nonlocality. Characterizing and detecting nonlocality is relevant to the implementation of quantum networks and quantum repeaters, where Bell inequalities can be used to test if configurations are prepared correctly.
This report contains an overview of an iterative method to find Bell inequalities for networks. Starting from a given network, this method constructs a new Bell inequality for a network containing one additional source and one additional party. We use this procedure to find new Bell inequalities for specific network structures and analyse how these can be used to detect nonlocality within a network.
In the first part the Bell inequalities are considered from a more theoretical point of view. We focus on star-shaped networks and discuss violations predicted by quantum mechanics. We look for quantitative bounds describing a set of states that lead to violation, giving an indication of the required quality of the sources.
Finally this method is applied to a setup similar to the one described by Bernien et al. We consider a network consisting of three parties and two sources. The effect of errors during preparation of an entangled pair of photons on the ability to detect nonlocality is evaluated. The same is done for the effect of measurement errors. Using numerical computations we show that violation of the Bell inequality can be improved by choosing different measurement angles.

In this thesis I take a look at various aspects of the brachistochrone problem. In the first half I consider the classical brachistochrone problem, without friction, after which I generalize to also consider drag; friction proportional to the squared speed. I do this numerically, with the shooting method, and following some papers, which Euler wrote. Then I will look at how Euler's results compare to modern results and discuss how they are different. In the second half I consider the quantum brachistochrone problem, following a paper by A. Carlini, and afterwards I look at a related problem, where there is not enough time to change an electron spin between two predetermined states. This problem has been numerically solved in an example.