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.

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.

In the past few years, the search for good quantum low density parity check (qLDPC) codes suddenly took flight, and many different constructions of these codes have since been presented, including many product constructions. As these code constructions have a natural interpretation in the language of homology, this thesis studies the interplay between homological algebra and various recent product constructions of qLDPC codes. First, we provide an overview of the theory of singular homology, cellular homology, and homological algebra over vector spaces, and use this theory to analyse two product constructions: the hypergraph product construction and the distance balancing procedure. These constructions can be interpreted as tensor products of chain complexes over vector spaces. We present new proofs for results from the literature regarding the distance and the number of encoded qubits of these product codes. Secondly, we survey the theory of homology over ring modules, and use this theory to interpret and analyse another product code construction (the lifted product code construction), which is a generalisation of the hypergraph product construction. We prove a Künneth theorem for these codes, and use this theorem to prove a formula for the number of qubits such codes encode. Thirdly, we investigate the homology of fibre bundles. To this end, we provide an overview of the theory of covering spaces and of fibre bundles, as well as an overview of the theory of homology with local coefficients and of spectral sequences. We then calculate the homology of a specific class of fibre bundles using three different methods. After this discussion, we consider two product code constructions: the fibre bundle product construction and the balanced product construction. We explicate their mathematical foundations, and use these insights to prove two new results for the number of qubits that these product codes encode. Finally, we explain under which conditions these two code constructions and the lifted product construction coincide. Lastly, we consider a specific example of fibre bundle product codes: the twisted toric code. We determine an analytic expression for the distance of such a code and verify this expression using numerical simulations. Furthermore, we perform an extensive numerical study on these codes to determine how performing a twist alters the scaling of the logical error rate of such codes. We present a new analytic method to explain the scaling of these codes on a small domain, and verify the validity of these calculations by comparing them with the results of our numerical simulations.

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.

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.

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.

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.

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.

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 random walks are the quantum analogs of classical random walks and appear to be promising tools to design fast quantum algorithms. Therefore it is important to study their time-related features and see how these differ compared to the classical case. For the discrete-time quantum walk on the line it has been shown that the probability to be absorbed by an absorbing boundary equals 2/π in contrast to the classical case where this probability equals 1, hence a quantum walk may continue forever without getting absorbed. It is also shown that mixing times of discrete-time quantum walks on the hypercube scale with n, the dimension of the hypercube, which is faster than O(n log(n)) for the classical random walk. So the quantum walk might offer a slight speed-up compared to the classical case. Finally, it is shown that the mixing times of the continuous-time quantum walk on a 2-layer multiplex graph depend on the eigenvalue gaps of the corresponding Laplacian matrix L. When the strength of the connections between the layers of a multiplex graph becomes very large, the eigenvalues of the Laplacian matrix converge. Thus the mixing times of the continuous-time quantum walk on 2-layer multiplex graphs converge.

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, 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.

For a global quantum communication network, we need to have long distance links over which we can distribute entangled photon pairs. Due to exponential losses, using optical fibres alone is unfeasible. Quantum repeaters extend the range of quantum networks, but intercontinental links are still unfeasible. Because of this, research has gone towards exploring space based segments, where a satellite can be used to make long-distance links by making use of the lower losses of free space transmission compared to fibre transmission. Here, we develop a general analytical model to compute the rate and fidelity of setups consisting of two ground stations connected by a satellite in orbit. We consider three different schemes: a direct downlink and two memory assisted schemes where the satellite is used as a quantum repeater, in an uplink and a downlink. Combining orbital mechanics for the satellite with a detailed model for transmission probability, allows us to track what happens to the rate and fidelity at any point in time. The generality of the model allows for a broad applicability, fitting to many different setups. We apply our model to different setups by tuning the different parameters and identify that the rate of the protocol will benefit most from increasing the multimode capacity of our assisting quantum memory and that the fidelity is most benefited by a good entanglement swap in the memory assisted schemes. If we have small links, bad memories or a bad entanglement swap, the direct downlink protocol will be the best choice, but for an intercontinental link we will need our memory assisted schemes.

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.

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.