Distributed quantum computers hold great promise in the realization of scalable and fault-tolerant quantum computers. They contain multiple nodes with small quantum devices that can generate inter-node entanglement. Next to realizing distributed architectures on a single chip, these systems can be extended to large-scale quantum networks. Connecting the nodes based on the topology of a quantum error-correction code forms an intuitive path toward fault tolerance.

Performing error-detection measurements in distributed error-correction codes requires the generation and consumption of entangled states. We focus on systems that are capable of generating remote two-qubit entanglement between pairs of connected nodes—i.e., Bell pairs. Entangled states of higher weight—the so-called Greenberger-Horne-Zeilinger (GHZ) states—can be generated by fusing Bell pairs. On top of this, the quality of the generated entangled states can be increased with entanglement distillation. We implement dynamic programming to generate high-quality GHZ states by fusing and distilling Bell pairs.

The dynamic program allows us to optimize the quality of error-detection measurements for a specific distributed error-correction code: the (toric) surface code. We numerically evaluate the performance of this code with noise models based on experimental characterization of diamond color center hardware, including the typical behavior of memory decoherence in these devices. This leads to the identification of a threshold in the ratio between entanglement generation and the decoherence rates.

For a two-dimensional error-correction code like the surface code, performing error-detection over multiple time steps can be reinterpreted as measuring the qubits of a three-dimensional cluster state. This equivalence enables considering more general three-dimensional cluster states as fault-tolerant channels that transform the logical qubits of the underlying error-correction code. We use this idea to investigate distributed logical memory channels for general types of circuit-level and entanglement noise.

Our results show that efficient generation of high-quality entanglement and strategic design of error-correction channels are important aspects for developing noise-resilient distributed quantum computers. @en

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.

Many applications of quantum communication require a high amount of entanglement between the states of two separated parties, for example quantum teleportation or superdense coding. However, when states are locally entangled and then separated, errors arise during transportation of the qubits. This results in a lower amount of entanglement between the states. Thus, we need protocols that apply local operations on the qubits of the two separate parties to achieve higher entanglement. This is called entanglement concentration. We analyze two protocols that classically can be used for the extraction of randomness and apply it to the concentration of entanglement: Von Neumann’s protocol and Elias’s protocol. These protocols are well-described in the literature. We analyze the performance of the protocols on quantum computers with varying error rates. We find that Von Neumann’s protocol extracts entanglement similar to the theory on the best 5-qubit processor (Athens): there is an average percentage change in the concurrence when the protocol succeeds of 49.0%, which is close to the theoretical value of 52.5%. Elias’s protocol requires many more qubits than Von Neumann’s protocol. The only quantum computer large enough to run the protocol (Melbourne) has high error rates. Also, Elias’s protocol requires more operations. This results in higher errors when we execute the protocol on Melbourne. We execute the protocol with 2 initial states and find an average percentage decrease of the concurrence when the protocol succeeds of 72.6%. Thus, we find that Elias’s protocol is not suited to use in practice and the Von Neumann protocol can be used to extract entanglement with the best quantum processors used in this work.

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.

This thesis investigates two types of classical capacities of both classical and quantum channels, giving rise to four different settings. The first type of classical capacity investigated is the ordinary capacity of a channel to transmit classical information with a probability of error which becomes arbitrarily small as the channel is used arbitrarily many times. The second type of classical capacity investigated is the capacity of a channel to transmit information with zero probability of error, called the zero-error classical capacity. The first setting which is studied is the ordinary capacity of a classical channel. The noisy channel coding theorem is proven in two different ways: one using the Markov inequality and the Law of Large Numbers and one using typical sets. The additivity of this capacity is also discussed. The second setting is the zero-error capacity of classical channel. Lower and upper bounds on this capacity are proven, and its superadditivity is discussed. The third setting is the ordinary classical capacity of a quantum channel. The Holevo-Schumacher-Westmoreland theorem is proven using typical subspaces and the packing lemma, and the superadditivity of the Holevo information is discussed in terms of entanglement at the encoder. The fourth and last setting investigated is that of the zero-error classical capacity of a quantum channel. It is shown that this capacity can be achieved using only pure input states and that this capacity never exceeds the ordinary classical capacity. Moreover, a detailed investigation of superactivation of the zero-error classical capacity is presented. A topic for further research would be an exposition of the analogous concepts in the case of the quantum capacity of quantum channels. Another topic for further research would be an explicit construction of two quantum channels whose zero-error classical capacity is superactivated.

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.

Bell inequalities are certain probabilistic inequalities that should hold in the context of quantum measurement under assumption of a local hidden variable model. These inequalities can be violated according to the theory of quantum mechanics and have also been violated experimentally. Bell inequalities were therefore historically used to disprove local hidden variable models as an interpretation of quantum mechanics. An interesting question is to what degree Bell inequalities can be violated according to the theory of quantum mechanics. The degree to which a particular Bell inequality can be violated is quantified by the largest violation of the Bell inequality. A central question we address in this thesis is how large the largest violation can become when considering all possible Bell inequalities. In particular the CHSH-inequality, a well-known Bell inequality, has a largest violation equal to the square root of 2 and we want to find a Bell inequality with largest violation exceeding the square root of 2. This thesis consists of two main parts. In the first part we formally define Bell inequalities and the largest violation and prove several theorems about Bell inequalities. In the second part we search for Bell inequalities with largest violation exceeding the square root of 2. We do this by first approximating the largest violation of a given Bell inequality using numerical optimization. Next, using this approximation, we use numerical optimization to maximize the largest violation. Due to run-time restrictions we were only able to consider Bell inequalities with a relatively small number of terms and were unable to find any among them with largest violation exceeding the square root of 2.

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.

We present a powerful approach for learning about uncomputability and undecidability in informationtheory. Our approach is to use automata from automata theory that have undecidable properties toconstruct channels for which an information-theoretic quantity is uncomputable or undecidable. Wedemonstrate this approach by showing that, for channels with memory, capacity is uncomputable andinformation-stability is undecidable.

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.

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.