Sixty years after John Bell published his paper in which he showed that local hidden variables with statistical independence are incompatible with quantum mechanics, its implications for the nature of the universe are still hotly debated among physicists and philosophers. In his paper he showed that a universe with local hidden variables and statistical independence should satisfy Bell's inequality, but this is violated by quantum mechanics. Testing Bell's inequality turned out to be difficult, owing to a number of loopholes. These were closed in experiments in 2015 which violated Bell's inequality, putting local hidden variables with statistical independence to rest. In this paper we discuss two conceptually simple versions of Bell's inequality as a game (Maudlin and GHZ) and simulate them using Python. Using local means these games are unlikely to be won, but by utilizing the non-locality of quantum mechanics the game is expected to be won. In our simulations we find that the probability of winning decays exponentially as a function of the length of the game for local strategies in Maudlin's version and we prove that the memory loophole (also known as daily updating) will not help the participants to win. In simulating GHZ as a game we find that the game length follows a geometric distribution.

In this thesis the stored energy and its fluctuations of a central spin battery with nearest-
neighbour interactions between the battery spins are investigated. Using analytical ex-
pressions, it is shown that for 2 battery spins and equal strength in the flip-flop interaction
g and nearest-neighbour interaction J, the fluctuations are minimal whenever the bat-
tery is maximally charged when taking at least four charge spins. Similarly, whenever
the formed envelopes of the energy have a zero, the fluctuations have a global maximum.
In the same limit, it could also be seen that an increase of the charge spins Nc and
spin-ups m, resulted in a higher global maximum of the stored energy. Furthermore for
2 battery spins, taking the limit J ≫ g results in a situation where the battery cannot
be charged at all, whereas taking the limit g ≫ J results in a central spin battery where
no nearest-neighbour interactions are present; its stored energy as a function of time is
a single cosine function, that is always able to reach its theoretical maximum.
Similar results were found for systems with more than 2 battery spins. Increasing J with
constant g resulted in a decrease of the global maximum of the energy, dropping from its
theoretical maximum to its minimum. Opposite behaviour could be seen when increasing
g with constant J. Whenever the global maximum of the energy crossed the line E = 0,
the fluctuations at the same moment in time formed a peak.

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.

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.

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.

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.

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.