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.

For many quantum applications we require high-fidelity entanglement between multiple pairs of solid state qubits at a distance. To achieve a high fidelity, we have to minimize the time during which the generated qubits need to stay coherent. Entanglement protocols often used in practice only generate one qubit at the same time. To generate multiple entangled pairs, the protocol is repeated. However during the time it takes for all pairs to be generated, the memory qubits will dephase. The required coherence time increases with the inverse transmission probability of the photons, which decreases exponentially with distance. This thesis is concerned with entanglement generation protocols that herald multiple entangled pairs simultaneously and in general herald N-dimensional entangled bipartite states. The main advantage of using more than 2 dimensions is that the qudits only dephase during the time in which the protocol executes. With simulations we show that the fidelity of the entangled pairs created with our protocols is higher than the fidelity of pairs created by protocols that heralds one entangled pair for distances L > 10 km. We also show a polynomial relation between the total success probability of the tailored protocol with dimension, which is an exponential improvement with respect to previous works.

In this thesis, gate set tomography (GST) has been conducted on the nitrogen vacancy center (NV). Gate set tomography is a protocol for characterization of logic operations (gates) on quantum computing processors. The NV’s electron served as a qubit. The quantum circuits were run both experimentally as well as on an NV-simulator. GST is different from its predecessors in the sense that it estimates all aspects of the processor simultaneously, without assuming any of its parts to be ideal. However, it does assume that the gates are Markovian. This allows to analyse the gate errors more precisely via their error generators. In addition to this, the diamond norm and a measure of the amount of model violation were used to examine the results. The electron qubit can couple to the nearby nitrogen nucleus, which can cause non-Markovian dynamics. The nitrogen nucleus (𝑆 = 1) can be initialised using nuclear spin polarization. The effects of different initialisation procedures on GST’s results were
researched. Furthermore, a dynamical decoupling XY-4 echo sequence could be employed to protect the quantum state of the qubit. How the presence of the echo affected the estimated gates and their errors was probed. It was discovered that the echo was extremely good at decreasing the amount of model violation, most likely by preventing the electron from coupling to the nitrogen, which can lead to non-Markovian dynamics. Without the echo, GST’s estimates were satisfactory only if the nitrogen nucleus was initialised with a high enough fidelity, at least 0.95. Another topic for further research would be to perform GST on the electron and the nitrogen system, by modelling the nitrogen nucleus as
a qutrit.

Quantum repeaters are critical in the development of the quantum internet because they enable quantum communication over long distances. Third-generation quantum repeaters or one-way quantum repeaters are the most advanced quantum repeaters that do not require two-way com- munication between nodes. Borregaard et al. (2019) proposed an architecture of one-way quantum repeaters based on photonic tree-cluster states. Tree-cluster states are highly entangled multi-qubit states that can protect quantum information from transmission losses. Creating these highly entangled multi-qubit states is a challenging and error-prone process. Conventional tree-cluster generation methods use an emitter that emits and entangles with photons. These methods, however, suffer from multiple photon emissions, which leads to errors. This paper presents a system for generating tree-cluster states that employ two different spin-cavity systems. The first spin-cavity system generates single photons using a cavity-assisted Raman scheme that prevents multiple photons from being emitted. The generated photons are scattered with a phase dependent on the spin state by the other spin-cavity system. The analytical model of the two spin-cavity systems for producing and scattering single-photons is presented. Furthermore, we model the errors that may occur during these processes.
We introduce a protocol for implementing a CZ gate between photons and spin. After that, this protocol is used to generate tree-cluster states. Furthermore, we optimize the entanglement between spin and photons by using the detuning between the two spin-cavity systems. Finally, we generate tree-cluster states using the spin-cavity system model and the photon-spin entanglement protocol. Following that, the effect of imperfect entanglement on the fidelity of the generated tree-cluster states is investigated. The fidelity of tree-cluster states with imperfect entanglement is then compared to the fidelity of tree-cluster states with single-qubit depolarising errors on photons. Unfortunately, we could not determine the relation between imperfect entanglement and the fidelity of the generated tree clusters in this study. Furthermore, we did not investigate the effects of imperfect entanglement on the encoding and decoding of information in tree-cluster states.

Quantum computing is an emerging field with many promising future applications.
These include, but are not limited to, quantum machine learning, quantum cryptography and quantum chemical engineering.

Before these can be realised, obstacles, which arise due to scaling, need to be overcome.
To accomplish this, quantum processors must be built with low noise and high target gate fidelity rates.
By characterising quantum systems, possible sources of noise can be identified, and consequently, systems can be designed which effectively suppress noise.

Gate Set Tomography (GST) is a protocol developed for the characterisation of quantum processors.
In this research, we apply GST to nitrogen-vacancy (NV) centre systems.
We also construct a widget meant to visualise GST results in an intuitive manner.

Our research is based on twelve models, varying in the number of gates used, the initial state of the nitrogen nucleus, and whether an XY4 echo was applied after gates.
We use simulated and experimental models, and analyse these using the error generator, diamond norm and Nσ metrics.

We conclude that our simulated models capture sources of noise, as the proportion of stochastic errors shifts to Hamiltonian errors when we change our target model from the ideal model to the simulated ones.
We also conclude that the XY4 echo significantly reduces non-Markovianity, which arises due to coupling.
Furthermore, evidence which points to calibration faults is discovered within certain models.
Finally, we present the visualisation widget and show how it can be used to interpret results.

High fidelity GHZ states among remote nodes is a precious commodity which can allow for non-local stabilizer measurements and thus pave the way for a modular fault-tolerant quantum computer. To this end, we extend the high fidelity intracavity gate introduced by Borregaard et al. (2015) to distributed paradigm, consisting of SnV-inspired atomic states in cavities connected by fibers. The adiabatic dynamics of this system can be solved efficiently using the effective operator formalism of Reiter and Sørensen (2012). We develop a Python framework that enables the analytical calculation of these effective dynamics reliably and swiftly, while being versatile and easily modifiable. The possibilities of this framework are showcased by obtaining results for a symmetric distributed setup and verifying its scalability. We present the ways that it can be optimized while taking into consideration experimentally inspired constraints, and proceed to optimize it for GHZ generation in color centers. These optimized gates are compared against an emission based protocol using the GHZ creation simulations of the Modicum protocol. As a byproduct of our investigation, we identify a specific set of Hamiltonians which, under certain conditions, can generate GHZ states with a single multi-qubit entangling gate.

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.

An improvement to the existing one-way quantum repeater network using the tree cluster state as the quantum error correcting code is presented. Namely, the [[5,1,3]] quantum error correcting code is introduced as an outer code while the tree code becomes the inner code. Using this approach, we present a novel hybrid one-way quantum repeater architecture with more than one type of quantum repeater in its network. We considered both the non-fault-tolerant and fault-tolerant variants of the [[5,1,3]] code in our study of the hybrid repeater network. A novel improvement to the [[5,1,3]] code is also presented, where we also consider using it for quantum erasure correction. Additionally, we introduced a novel method of extrapolating an approximate fidelity of a quantum state after arbitrary applications of a noisy quantum channel. With these, we see magnitudes of order of boost in the resulting secret key rate in our approach.

Quantum communication has been shown to be vastly superior to classical communication in many problems. However no general statements exist which tells us how much better quantum communication is to its classical counterpart. In this thesis it was studied the minimum amount of classical bits required to exactly simulate a quantum communication process. The quantum communication process specifically studied was a quantum prepare and measurement communication problem. It has been shown that the calculation of the amount of classical bits of communication required for simulation reduces to a minimization-maximization optimization problem. Several results have been presented for for solving this optimization problem and in addition a link was made between classical simulations of quantum communication and a recent debate on the reality of the quantum state.

As quantum computers are developing, they are beginning to become useful for practical applications, for example in the field of quantum metrology. In this work, a variational quantum algorithm is used to find an optimal probe state for measuring parameters in a noisy environment. This is achieved by optimizing a cost on a quantum computer, based on the Fisher information of the parameters to be estimated. These parameters are then estimated using maximum likelihood estimators. In a simulation, a probe state was found that performed better than the best possible state for noiseless measurements, although this could not be reproduced on an actual quantum computer.

The standard toolbox of modeling and characterizing quantum systems comes with a standard set of assumptions as well. The two-level approximation replaces a many-level system with a qubit, and the Markovian approximation assumes an environment with a short memory. In this thesis, these assumptions are relaxed, and the dynamics of a single qutrit are reconstructed from a complete set of measurement data, using maximum-likelihood estimation (MLE) to self-consistently infer a set of state-preparation and measurement parameters (SPAM), along with a time-dependent process map. The process map can then be used to quantify the non-Markovianity of the qutrit evolution. The SPAM parameters and process maps produced by the MLE framework are compared to ground-truth simulations, with good agreement found in all cases studied. A Markovian example, the amplitude and phase damping channel, and a non-Markovian example, two transmons with a static coupling, are investigated. With its ability to directly capture higher level effects such as leakage errors, and also to detect non-completely positive evolution due to entanglement with the environment, this framework improves upon existing characterization algorithms with the purpose of encouraging future experimental work with qutrits.

With this thesis project, we improve the classical simulation of quantum computers using stabilizers in the GSLC formalism. We do this in two ways: we present new algorithms that speed up their simulation and extend their applicability by defining new operations and subroutines for existing general circuit simulation using GSLC. To be precise: we present multiple new algorithms that speed up the simulation of the CZ gates, the most computationally expensive quantum operation in GSLC formalism. We define two new operations on GSLC that are useful when simulating stabilizer circuits: calculating fidelity (a measure of 'closeness' between two quantum states), and tracing out qubits (throwing away the information about the state contained in these qubits) from a GSLC. Finally, we present a new GSLC-based subroutine for a state of the art general quantum circuit simulation algorithm by Bravyi et al. that allows for the usage of the faster CZ algorithms. We show that the GSLC formalism can give a speedup in practical simulation tasks by evaluating the complexity of simulating an algorithm with possible applications on near-term quantum hardware: the quantum approximate optimization algorithm.