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 thesis, two topics are studied: mathematical inequalities and non-linear quantum entanglement witnesses. First, various inequalities, like the Cauchy-Schwarz inequality (on finite dimensional vector spaces) and Jensen's inequality, along with their extensions and generalisations, are proved and discussed. The intimate relationship between these inequalities is studied. Because this thesis was restricted to finite dimensional vector spaces, the consequences of generalising the results to infinite dimensional vector spaces are finally determined. Secondly, the topic of entanglement detection is discussed - specifically, non-linear entanglement witnesses are considered. A bipartite and multipartite entanglement criterion based on the previously discussed inequalities are introduced and assessed extensively by considering their optimality, how they relate to other criteria as well as their limitations.

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.

This document describes the design process the prototype of a measurement light bulb which is able to project spherical harmonics of orders 0-1-2. Photographs of these projections can be combined to represent many light distributions. The measurement light bulb can be used in the field of research focused on the computation of the illumination impact of lighting, more specifically, simulating different light sources. Our prototype consists of a laser beam rotating over two axes, which allows the device to project onto a sphere around itself. The device can be controlled wirelessly using Bluetooth. The lamp can project spherical harmonics in a resolution of 4 degree and 256 monochrome light levels. The time one projection takes is about one second, which allows for quick measurements. The dimensions and the weight of the lamp are such that it is portable. At this stage, the prototype can only operate in a dark environment, due to the use of a low powered laser. In this document, the focus will be on the design of control, communication and the PCB.

Throughout the work, the goal was to develop the physical layer of a quantum network stack. The layer of the network stack connected to the physical layer is the link layer. For entanglement-based quantum networks, we demonstrated the successful operation of a link layer and a physical layer. The physical layer’s entanglement generation procedure—implemented here using two NV center-based quantum network nodes—is abstracted by the link layer into a robust platform-independent service that can be utilized to run quantum networking applications. Other quantum network platforms using the approaches provided here (which are not unique to our diamond devices) will accelerate the development of large-scale and heterogeneous quantum networks. ---------------- A vision is developed on how successful university-industry collaborations can be established in the field of quantum technology development. By looking at the stakeholders, purpose and type of collaboration, and influencing factors, a three-step model is developed to move from awareness to acceptance to collaboration. By overcoming the existing barriers, university-industry collaborations can be successfully realized.

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 a promising means of satisfying the ever-increasing need for more and faster computations. While some specific applications can already benefit from quantum computing today, a large-scale general-purpose quantum computer is yet to be developed. Many different technologies are being explored to reach this goal, color centers in diamond being a promising candidate. This thesis focuses on a specific kind of color center: the nitrogen-vacancy center (NV center). Building on top of existing work, a quantum instruction set architecture (QISA) for a quantum computer based on NV centers is designed. A compiler targeting this QISA is designed and implemented using the OpenQL framework. In this process alternative approaches are also explored and contributions useful outside the context of NV centers are made. The final compiler is able to take a quantum circuit operating on logical qubits and transform it into a sequence of QISA instructions operating on physical qubits. This functionality integrates seamlessly with the existing passes in OpenQL, allowing for further extension and the possibility to make use of future developments. Additional functionality includes optimization, scheduling, and hardware-oriented features such as quantization of operands. Each step is easily configurable using a multitude of scripts and data files, and additional steps can be added in the future if needed. The functionality of the developed compiler is demonstrated by compiling several circuits, some of which are validated by passing their output through a simulator. The compiler configuration as used in these tests is able to transform logical circuit into simple physical circuits, but the compiler provides all the functionality to use more complicated logical qubits which incorporate quantum error correction.

The world of quantum computing is still quickly developing and growing as it is searching for the best technology to implement quantum bits. One of these technologies is a diamond-based quantum computer using nitrogen-vacancy (NV) centers. In order to sustain the mentioned growth, the limits of electronics as well as computer engineering need to be explored. Consequently, an Instruction Set Architecture (ISA) defining the functionality of the NV centers must be constructed that, in turn, controls the electronics needed to operate the qubits within the NV center. In this thesis, we created a simulator to verify the functionality of the ISA and, more importantly, utilize the simulator to explore the capabilities of a diamond-based quantum computer. The simulator mimics the interaction between components and qubits in a diamond-based quantum computer and maps these interactions onto quantum mechanical instructions, such as qubit rotations. The simulator is implemented by modeling the electronics based on literature and creating interaction between the components using an event scheduler. The event scheduler is part of the NetSquid framework in which this simulator is built. The framework has models implemented to represent qubits and perform operations on qubits as well. The qubit representation model represents the electron and carbon nuclei qubits. The quantum operations are used to perform quantum mechanical operations on the qubits. The simulator is tested for physical accuracy using single diamond tests from literature, namely magnetic biasing, rabi cycle checking, and charge resonance checks. The simulator returned an accurate value for the Larmor frequency and Rabi frequency meaning that it was close to the expected value. For example, setting the Larmor frequency to 1.72GHz and using the magnetic biasing sequence to determine the Larmor frequency resulted in the same 1.72GHz$value. After single diamond tests have been performed, the control of multiple diamonds and their qubits is tested using a quantum algorithm, the surface-7 code. The simulator accurately mimics the expected behavior of the noiseless execution of the surface-7 code. The results of the simulator follow the trend of the expected values. The final product of this thesis is a simulator that can accurately describe the interaction between an ISA and the components of the diamond-based system-architecture in the noiseless case. The simulator provides a setup for a future simulator with noise sources implemented, resulting in a simulator that can be used to determine the limits of electronic properties or to predict the outcome of lab experiments.

Building a working quantum computer brings many challenges to overcome. For quanum processors, promising qubits seem to be NV-centres in diamond. Such processor is divided into processing units (PUs), each comprising an NV-centre with several carbon spin qubits, and they are interconnected with each other by optical links. The qubits within a PU can be modelled as nodes in a star graph and the optical connections as connections between the centres of the star graphs, such that the centres with their interconnections form a complete graph. We call the overall graph a fully connected star graph. In order to execute a quantum circuit, multiple pairs of qubits should be brought together in PUs at different times, which requires planning. Sorting the qubits is done via swaps. A swap within a PU can be performed easily, however, swapping between different PUs is done by teleportation, which is much more costly. Therefore, we neglect the costs of swaps within PUs. Parallellisation of swaps between PUs is done in a step. The less steps are used, the faster the quantum circuit is executed. In practice, as we are interested in fast computations, we often want to minimize the number of steps. The following problem is addressed. Given a current distribution of the qubits over the nodes in the fully connected star graph, and given a wanted, permuted distribution, minimize the number of swaps and/or steps to route the qubits from the current to the wanted distribution. At the moment, no efficient algorithm exists to sort the qubits especially in fully connected star graphs. Efficient solutions are presented in the form of algorithms. It is shown that qubit moves in fully connected star graphs can be scheduled efficiently, such that the number of moves is minimized. Guide lines are given to minimize the execution time too.

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.

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.

There are still great hurdles to overcome in the construction of a practical quantum computer. Most significantly, there is still great need for less noisy operation devices and the ability to scale the computer into hundreds or thousands of qubits. In this work we study a approach to construct a practical quantum computer which is fundamentally scalable. We consider Nitrogen vacancy centers as nodes in a quantum network to model a distributed surface code. The analysis consists first in a study of how the surface code can be implemented over a quantum network and which error models we consider to get a realistic representation of the system. Thereafter, we calculate the error thresholds through the parameters of the most significant error sources in order to determine the code effectiveness. After the initial implementation we explore further designs which by utilizing additional resources in the network aim to reduce the impact of noise in the system. The results show a considerable improvement over the initial implementation and provide some insights into further developments that could produce better results.

In the field of quantum information theory, it is well-known that the purely quantum phenomenon called quantum entanglement can boost the capacity of a quantum channel, which is called the superadditivity of the capacity. Shor showed in his breakthrough paper on the equivalence of additivity conjectures that this superadditivity is equivalent to the subadditivity of the minimum output entropy of quantum channels, and Hastings gave a high-dimensional and probabilistic counterexample to the minimum output entropy additivity conjecture. Since then, researchers have endeavored to classify which quantum channels have subadditive minimum output entropy and to find deterministically constructed quantum channels with subadditive minimum output entropy. It is well-known that determining the minimum output entropy of an arbitrary quantum channel is a hard problem (in fact, it has been shown to be NP-complete), which in turn makes it difficult to determine whether a quantum channel has subadditive minimum output entropy. In this thesis, we utilize the representation theory of compact (partition) quantum groups to construct so-called covariant Clebsch-Gordan quantum channels and analyze their minimum output entropy. We review the work of Brannan and Collins in the case of the free orthogonal quantum groups $O_N^+$, and introduce a similar analysis for the quantum permutation group $S_N^+$. Afterwards, we specialize to the subfamily of lowest-weight Clebsch-Gordan channels, and we show that, in the case where one embeds the fundamental representation of $O_N^+$ in the tensor product of two irreducible representations of $O_N^+$, the associated lowest-weight quantum channels have sufficient additional structure to analytically compute their minimum output entropy in terms of a recurrence relation. Lastly, we present an analysis of certain numerical methods that can be utilized to bound the minimum output entropy from below and from above in low dimensions: we use epsilon-covers for lower bounds, and we use a modified version of the derivative-free optimization method called Particle Swarm Optimization over the unit sphere, hybridized with gradient descent, to find upper bounds. We show a proof-of-concept by applying the epsilon-cover to three $S_N^+$ channels with small input dimensions, but also note that the exponential scaling of epsilon-covers with the input dimension makes them intractable with higher input dimensions. We benchmark the Particle Swarm Optimization enriched with gradient-descent on the highest-weight Clebsch-Gordan channels for which it can be shown that the minimum output entropy is zero, and we see that the optimization technique performs adequately. We also apply the optimization scheme to the tensor product of two $S_N^+$-channels, but do not find any violation of the minimum output entropy additivity conjecture.

A network of devices capable of transmitting quantum information called a quantum network has promising applications with vast benefits. One of the most near term achievable application is the quantum key distribution. It could be used for sharing a secret key between two end-users through any insecure authenticated communication channel. Other than sharing a secret key, a quantum network can be used for connecting quantum computers. Quantum computers have the potential to solve computational problems that their classical counterparts would require significantly more time to do so. By connecting such devices, distributed computation problems such as leader election or distributed consensus between nodes can be solved securely. For such applications to be put into practice on a large scale, there are open practical questions still to be answered. In a quantum network, a data qubit containing quantum information is transmitted by an operation called quantum teleportation. For it, two nodes need to share entanglement. Quantum teleportation then ensures the safe transmission of a qubit by using the shared entanglement and the exchange of two classical bits. This operation, however, consumes entanglement between the nodes, changing the network topology with each served request. In quantum networks, certain nodes share entanglement between each other. Routing in quantum networks entails determining which of these shared entanglements are used to transmit quantum information. As this proves to be a difficult task, we study quantum networks and in particular consider the problem of routing entanglement in quantum networks. The average latency for routing entanglement in a quantum network has been studied so far using a distributed routing approach. Thus, we present numerical simulation results of centralised routing for the average latency of demands in two existing entanglement generation models. In the first, shared entanglement between nodes is created on-demand. In the second, certain nodes pre-share entanglement before the demand comes. Our results show the intuition that using a centralised routing approach in the on-demand model results in drastically more average latency than in the model with pre-shared entanglement. Since some nodes might not be informed about the change in topology, we study the effect of the propagation of information about the network topology to nodes in the network. Our observations based on simulation results show that the average latency is significantly higher if the information about the topology is not propagated well in the network. As propagating information to all nodes in a network would be a significant load for the network, we consider information propagation within a radius and still observe a considerable decrease in average latency. At last, we present a technique called link prediction which can be performed by any node without the need for information propagation and still achieve a considerable decrease in average latency. It can be effectively used to predict the change of topology in a quantum network by using knowledge about the network topology for a certain future point in time and previous knowledge about the network traffic.

The first part of this thesis provides a mathematical description for bipartite quantum correlations, aiming to analyze the geometry of several sets of correlations. We explain why quantum entanglement can be used to simulate shared randomness: Cloc(Γ) ⊆ Cqd(Γ) for a sufficiently large d. The known bound for this dimension d in the literature is d ≥ dim(Cloc(Γ))+1, but we improve this by showing that the inclusion is always true for d ≥ dim(Cloc(Γ)). For the proof of this bound, we show that the set Cprivate(Γ) of correlations using private randomness is connected, which allows the use of an improved version of Carathéodory’s Theorem. In the second part of this thesis, we define and analyze a see-saw method to determine the state and measurement operators that reconstruct both the correlation itself as its entanglement dimension, by solving consecutive semidefinite programs. One of the strengths of the algorithm is its generality: it applies to different dimensions, question sets, and answer sets. Some numerical experiments demonstrated that the method can indeed reconstruct quantum correlations, although some highly entangled correlations failed to be reconstructed due to the computationallimitations. The numerical experiments motivated several new theorems, for example the fact that every correlation with |A|= 1 or |B|= 1 has entanglement dimension 1, which means that it can be written as a private randomness correlation. The proof of this result is based on the earlier described improvement for the dimension d.

This report reviews two quantum key distribution (QKD) protocols: the BB84 protocol and the measurement device independent (MDI) QKD protocol. The goal of this report is to recreate the security proof of the BB84 protocol, to generate the secret key rate for a practical application of the BB84 protocol, both with and without decoy states, and to review the MDI-QKD protocol and look at the advantages it has for practical QKD. The proof security of the BB84 protocol is done by designing an equivalent theoretical protocol and proving its security by bounding the information of the eavesdropper to an exponentially small number. The best distance over which secret key rate can be generated with the practical model of the BB84 protocol that is presented in this report, is 165 km. This is achieved by employing decoy states and is more than three times as far as the model can achieve without decoy states, which gave a distance of 52 km at best. The zero distance key rate of the model with decoy states is R = 1.21 · 10-2 bits per pulse (bpp), at best. This is an order of magnitude larger than R =1.02 · 10-3 bpp, which is what the model without decoy states could produce. The decoy state method is therefore an improvement for practical QKD. By reviewing the MDI-QKD protocol it can be concluded that because it eliminates the measurement device side channels, it is very useful for practical QKD, since it makes security analysis simpler and more precise. It also necessarily employs decoy states, which improves the secret key rate. MDI-QKD therefore is a promising protocol to use for future quantum communications.