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 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.

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.