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. 

Quantum error correction is needed for future quantum computers. Classical error correcting codes are not suitable for this due to the nature of quantum mechanics. Therefore, new codes need to be developed. A promising candidate is the toric code, a surface code, because of its locality and its high error correcting capability and thresholds (the error probability below which increasing the size of the code decreases the failure rate). This thesis provides an introduction to quantum error correction and the stabilizer formalism, which is used to introduce the toric code. Several decoders are looked at, including the Minimum Weight Perfect Matching (MWPM) decoder and a recently developed decoder, the Union-Find (UF) decoder. The UF decoder is useful because of its almost-linear time complexity, while only reducing the threshold by a marginal amount compared to the MWPM decoder. In this thesis, the time complexity of the weighted growth version of the UF decoder is analysed. The toric code and the decoders have been implemented and simulated, and the thresholds have been determined. For the MWPM decoder, the threshold was determined to be 11.5±0.2%, which is not in agreement with the threshold in literature of 10.3%, and should not be possible according to the optimal threshold of about 11%. This probably is a result of the low grid sizes (up to a gridsize of 11) of the code used due to the time needed to simulate the MWPM decoder. For the UF decoder, the threshold found was 9.7 ± 0.9%, which is in agreement with the threshold of 9.9% found by N. Delfosse and N. Nickerson. The time complexity of the UF decoder has also been determined, and is indeed almost-linear as expected by the analysis.

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.

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.

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.

In this work, we first analyse a theoretical technique based on entropic inequalities, which in principle can be used to compare classical and quantum algorithms running on noisy quantum devices. The technique has been used in the past for depolarizing noise, and has shown that for NISQ variational quantum algorithms (VQAs) to give an advantage over purely classical methods, it is essential that the structure of the problem is close to the quantum hardware the VQA is run on. Here, we use this technique for relaxation and pure dephasing noise and examine whether we can deduce the same results for the Quantum Approximate Optimization Algorithm (QAOA) running on a noisy quantum device. In the later part of the thesis, we study the properties of the steady state of the Lindblad Master Equation for a 1D array of transmon qubits coupled by cross-resonance type interactions for relaxation and pure dephasing noise. For n=2,3 number of qubits we solve exactly the master equation, while for a general number of n qubits we approximate the solution by using two different approaches, namely perturbation theory and mean-field theory. Finally, we assess how well those approximation methods compare to the exact solution for n=2,3 qubits.

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.

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.

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.

The main investigation in this thesis is the research on a future quantum internet. The focus is laid on the network structure of this quantum network. The network is rst examined by the investigation of the Kuramoto model for classical oscillators. The main property of the network is the ability to synchronise all nodes such that all oscillate with the same frequency. Several parameters of the oscillators are varied to verify why and when synchronisation takes place. After modelling the calculated equations of motion, an interesting conclusion arises. For a symmetric ring network, a stable conguration is not always found but by introducing asymmetric oscillators in the network, the synchronous state can be found. This conclusion leads to the idea that a quantum network requires certain dierences in its structure in order to guarantee transmission to take place. Furthermore, the existence of synchronisation heavily depends on the parameters used for the oscillators. The analysis is then continued in the quantum domain where optomechanical systems are introduced. At rst, two systems are connected to each other by a gaseous interaction and an electric interaction via a Dung circuit. The main investigation is again into the synchronisation, which implies that the operators belonging to both systems behave the same as a function of time. Then several synchronisation measures are introduced to measure the ability of the systems to synchronise. As predicted, both systems synchronise in terms of the operators for each system. Then a quantum network is introduced, where a complex yet ecient network is created. This quantum network is a small world network, where a transmitter node is able to connect to the receiver node by only a few links. Furthermore, multiple transmissions can take place at the same time and links which are not connected do not synchronise with the transmitter node. After modelling this network, the results are in compliance with the theory available. Challenges for a practical quantum network include the experimental basis of being able to utilise optomechanical systems outside laboratory circumstances. Also, the network should be tested for practical use and expanded in a much larger size both in theoretical analysis as well as in experiments.

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.

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 ΔQ+ = 4949.25(2) kHz and ΔQ- = -4949.11(1) kHz.

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 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.

Quantum thermodynamics takes over from classical thermodynamics when systems are of the scale of single particles and quantum fluctuations have a noticeable effect. An interesting topic of research of this relatively new field is the quantum battery, which in this thesis consists of an array of N identical electron spin qubits. In an article by Binder et al. [4], it is proven that in theory, an N-times decrease in charging time of the battery is achieved when global operations on qubits are permitted. This thesis investigates if a similar advantage can be achieved by using a local qubitqubit interaction operator on a one-dimensional chain of exchange-coupled electron spin qubits that are driven by microwave radiation in the presence of decoherence. This system is described by a density matrix in order to include the presence of external influences. The time-evolution of the state of the system is calculated by solving the Von-Neumann equation both analytically and numerically, which is then used to calculate the extractable work. It is shown that exchange interaction does not have a direct effect on the extractable work, since it creates entanglement between two states of the same energy level and the operator commutes with the Hamiltonian of the system. The effect of the CNOT gate on the state of the system is then investigated. While it does have an effect on the extractable work, it did not achieve a decrease in charging time. These results are only relevant for the specific system used in this thesis. For other methods and systems, exchange interaction could lead to faster charging.

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.

Quantum Batteries (QBs) are quantum-mechanical devices for energy storage, gaining interest due to a potential quantum advantage in power. Recently, the first experimental implementations of QBs were realised. This study characterises the superconducting transmon qubits of Starmon-5 as QBs using the Quantum Inspire platform. In addition to direct charging of a QB, our focus is on charger-mediated energy transfer and parallel charging of an array of transmon qubits. The figures of merit include the average stored energy, the charging time and the charging power. The results from direct charging of the qubits align with existing literature. Charger-mediated energy transfer is demonstrated through the characterisation of the CNOT gate as an interaction gate, gaining the same amount of stored energy, but with a significant increase in the charging time, resulting in a lower charging power. Furthermore, our findings demonstrate that parallel charging of an array of qubits preserves the quality of direct charging of the individual qubits. To our knowledge, this work presents the first results of charger-mediated energy transfer in real quantum devices. Charger-mediated energy transfer can be interesting for specific applications such as quantum metrology, where preserving the quantum state is critical. Additionally, this is the first demonstration of parallel charging of superconducting transmon qubits in the QB context, giving promising results for the scalability of superconducting transmon qubits as QB. Our study paves the way forward to implementing quantum batteries for energy management in quantum technologies, a near-term future application of quantum batteries.

Originated from the electron’s intrinsic angular momentum, magnetism has endowed various manipulations in both macroscopic and microscopic setups with another degree of freedom. Beyond the traditionally developed usage such as storage and sensors, there are enormous applications based on engineering and integrating magnetism into heterostructures and their susceptibility to external stimuli. The emergent fields of nano-level spintronics and spin caloritronics with novel properties have been intensively studied both theoretically and experimentally. Within those developments, the interaction of atomic spins with electromagnetic waves (photons) and elastic dynamics (phonons) are of fundamental importance. This thesis is devoted to investigating the interplay of magnetism with electrodynamics and lattice elasticity in several hybrid systems. @en