A set of lines passing through the origin in Euclidean space is called equiangular if the angle between any two lines is the same. The question of finding the maximum number of such lines, N(d) in any dimension d is an extensively studied problem. Closely related, is the problem of finding the maximum number of lines, N_α(d), such that the common angle between the lines is arccosα. In recent years, many progress has been made on this problem. We review some of these breakthrough results and the techniques they use to approach this problem. The first main result is a linear upper bound on N_α(d) which is found using a completely novel approach with respect to techniques used in previous works. Another main result that we discuss solves the problem of finding N_α(d) for high enough dimensions. Some classic results from some of the first studies on equiangular lines are also discussed. Finally, some suggestions are given for possible further research.

This thesis uses the method of interlacing polynomials to study the behaviour of eigenvalues of a matrix after a rank-one update. Specifically, interlacing polynomials, common interlacing and interlacing families are exhaustively studied. These are excellent tools to find bounds on the eigenvalues of updated matrices by keeping track of how they move.

This enables us to prove results in different mathematical fields. We investigate three of them. The first one is from spectral graph theory: we prove the existence of a sharp κ-approximation for any graph.

The second result is from linear algebra. It states that if a matrix has a high stable rank, it must contain a large column submatrix with large least singular value.  

Lastly, the proof of the Kadison-Singer Problem is discussed. Despite being a problem from analysis, it was solved with the interlacing method, which is originally a method from discrete mathematics.

This thesis shows how these three seemingly different problems are all connected by the same method, highlighting its advantages. The objective is to present a clear framework of the different facets of the interlacing method and provide an insight in the situations where one can expect the said method to be useful.

In the last decades there has been an increasing interest in computing the local strain at the atomic scale of materials. By knowing aspects of the local strain in a lattice, one has information about measurements of distortions of lattice parameters concerning shifts, deformations and defects computed with respect to a smooth, defect-free reference region. Multiple methods have been implemented so far in order to map the strain of two-dimensional lattice patterns, which are obtained through means of a High Resolution Electron Microscope (HRTEM). The functioning of a HRTEM is based on the same principles as an optical microscope, but it uses a beam of electrons instead of visible light. One of the computational methods which then processes the obtained two-dimensional images is called the Geometrical Phase Analysis (GPA) and makes use of a very important mathematical tool, the Fourier transform. The GPA method lies at the center of this project and consists of several steps. First, the Fourier transform of the lattice image is plotted and two Bragg peaks corresponding to two linearly independent frequency vectors in the power spectrum are chosen. Next, a mask is applied around these peaks, separately. In my project I have chosen to apply the Hann smoothing filter. Then, the inverse Fourier transform is applied to the masked image and the phase (also called the raw phase) is plotted. The next step is to compute the reduced phase, which is defined at a local pixel as being the raw phase from which the following product is subtracted: 2𝜋 ⃗𝑔 ⋅ ⃗𝑟, where ⃗𝑟 is the vector corresponding to a pixel in the real space and ⃗𝑔 the frequency vector corresponding to the Bragg peak around which one has applied the mask. At this point the reference region is computed by choosing a smooth, homogeneous area in the reduced phase image. In order to obtain the strain, one needs an optimal frequency vector ⃗𝑔 defined at every pixel of lattice. In order to do so, a minimization process defined in the context of a computer algorithm in the programming language Python has been implemented. These computations should lead to obtaining the lattice strain, which is calculated by taking the symmetric part of the derivation of the displacement obtained from the two linearly independent Fourier components, which is in turn called the distortion. The antisymmetric part of the distortion is the rotation component and serves as a check for the correctness of the computational method. The goal of my project is not only to provide a solid theoretical background for the GPA method and to discuss the strain at atomic level in several lattice patterns, but to also provide a rigorous computer algorithm that makes these computations reality. This algorithm, opposed to pre-existing software, facilitates the reader’s process immensely by the large amount of detail which is given at every step, detail which easily motivates and supports the reader in potential side-steps they would want to take in order to make the method their own.

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.

This thesis is divided into five parts (covering chapter 1–5), that together try to give the reader a basic understanding of the symmetries of and the mathematical structure behind the standard model. The first two chapters cover some Lie and representation theory of the Lie groups U(1), SU(2) and SU(3), which play a central role in the standard model. In the third chapter, the structure of the standard model is given, based purely on observed symmetries and (quantum numbers of) particles. In this chapter, the mathematics of the first two chapters is not used yet. In the fourth chapter, the Lie theory of chapter 1 is used to treat the gauge theory of U(1), SU(2) and SU(3), where for SU(2) also spontaneous symmetry breaking is incorporated. Finally, the fifth and last theoretical chapter gives examples of how representation theory and gauge theory can together structure the particles of the standard model and can dictate which particles and particle interactions are physically allowed. The thesis ends with some recommendations for further study of the topic at hand and for further research to symmetries beyond the standard model.

In this research, the implementations of quantum random walks in superconducting circuit-QED are studied. In particular, a walk that moves across the Fock states of a quantum harmonic oscillator by a Jaynes-Cummings model is investigated, which is difficult to implement because of different Rabi frequencies for different Fock states. Theoretically, the lower boundary vacuum state of the harmonic oscillator causes a reflection of the probability amplitudes in the distribution. A walk that moves across a grid of coherent states |nα+imα〉 in phase space is then investigated. A setup for a 1D and 2D quantum random walk is suggested, using controlled displacements of a resonator dispersively coupled to one or two superconducting transmon qubits in circuit-QED, followed by Hadamard gates. From numerical simulations it was observed that the 2D walk commutes for α·β = 0 mod π/2 for which the variance is proportional to the number of steps t squared. For other values of α·β the horizontal and vertical displacements do not commute, resulting in extra phase factors. The numerical simulations showed that for most values of α· β with a larger distance to 0 mod π/2 than 0.01, the probability distribution of the
walk collapses to a distribution centered around origin within t = 100 steps, similar to a classical random walk. Exceptions are the 2D walks for α·β = ±π/6 mod π/2 or ±
π/4 mod π/2, for which the variance is still proportional to the number of steps squared.

Vector spherical harmonics are a set of basis functions for vector fields derived from the spherical harmonic functions. They are commonly used in spectral methods in certain areas of applied mathematics. In most of the existing literature they are defined in a way that is heavily dependent on the used coördinate system. In this thesis we define vector spherical harmonics in a coördinate free manner. We first do this on a sphere, using two tensor operators: the exterior derivative and the Hodge-star. We then extend these functions to vector functions in 3-dimensional Euclidian space and define a last type of vector function to form a complete basis for vector fields. We do this in such a manner that the vector spherical harmonics inherit some of the nice properties of the spherical harmonic functions with respect to rotations and rotational symmetric operators.

This thesis gives a thorough description of the mathematical tool called reflection positivity, which can be used to prove the occurrence of phase transitions in physical models. A major result, although already known, is a theorem that gives tractable conditions on the Hamiltonian such that the Boltzmann functional is reflection positive. In this thesis, the theorem is used to give conditions on free parameters in four different models, such that the model is reflection positive with respect to certain chosen reflections. The evaluated models are (a) the antiferromagnetic quantum Heisenberg model; (b) the spin ice model; (c) the 6-vertex model and (d) the 16-vertex model. For the Heisenberg model we found that for reflections in a reflection plane there are certain parameter values such that the Boltzmann functional is reflection positive, this is an already known and published result. For the spin ice model we found that for the spin invariant reflection there is no symmetry that yields a reflection positive Boltzmann functional, this is a new result. For both the 6-vertex and 16-vertex model we showed that, for certain energy values, the Boltzmann functional is reflection positive with respect to reflections in the diagonal, which are also new results. In the case of the 16-vertex model, this boils down to checking whether or not a matrix is positive semidefinite. Using this result we showed that energy values that allow for the existence of magnetic monopoles do not yield a reflection positive Boltzmann functional. A topic for further research is investigating the occurrence of phase transitions in the models that are shown to be reflection positive, for which chessboard estimates seem to be a promising approach. Furthermore, in this thesis it was not rigorously proved that the spin ice model with a spin inverting reflection gives a reflection positive Boltzmann functional for rotational symmetry or symmetry in a reflection plane. This is believed to be true, but does require further investigation.

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.

In this thesis, networks of coupled quantum harmonic oscillators are studied. The dynamics of these networks are determined by single-frequency vibrations of the entire network called normal modes. We study the behavior of the nor- mal modes when the network is coupled to a thermodynamical heat bath by looking at the Lindblad Master Equation of the system. From this equation, we determine the rate at which the normal modes decay. Certain normal modes decay very slowly, and some do not decay at all. These normal modes are called quasi-noiseless and noiseless clusters respectively. We determine what happens to the noiseless clusters when the network pa- rameters are very slightly perturbed. We have found that two distinct types of noiseless clusters can be identified. The first type disappears with even the slightest perturbation, making it useless in practice. The second type instead be- comes quasi-noiseless, making it a viable candidate for applications. We show how to determine the degree to which these noiseless clusters become quasi- noiseless by looking at the other normal modes of the network. We also explain how a network of oscillators, including an optional heat bath, can be simulated with an optical setup as described in [3]. We suggest this setup can be used to verify our findings.

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.

Recent progress on the representation theory of certain infinite dimensional gauge groups has raised an interest in the strongly continuous unitary representations of groups of a specific form that satisfy a certain positive energy condition. An equivalent formulation of the positive energy condition is obtained, allowing for a geometrical interpretation of this condition and which yields necessary conditions for satisfying this condition. By the theory of the Mackey machine, the strongly continuous unitary representations of such groups that are of positive energy are classified by corresponding stabilizer subgroups. In a specific case, these are fully determined up to equivalence.\\ Finally, a method is developed that embeds homogeneous bundles as eigenspace subbundles of trivial bundles that in particular applies to the bundles obtained through the representation theory of groups mentioned above. The eigenspace subbundles thus obtained allow for a more detailed understanding of the induced representation and moreover resemble various theories in particle physics.

In deze scriptie volgen we de lijn die in W. van Est in “A group theoretic interpretation of area in the elementary geometries” heeft uitgezet, maar we gaan grondiger in op de stof en bewijzen de meeste claims die door Van Est worden gedaan. We kijken naar wat triviale en nontriviale centrale groepsuitbreidingen zijn, wat die te maken hebben met cocykels en coranden en cohomologieklassen. We zien dat iedere cocykel een corresponderende homogene cocykel heeft. Aan de hand van het oppervlaktebegrip in een tweedimensionale ruimte kunnen we een homogene cocykel definiëren en daarmee kunnen we dan een centrale groepsuitbreiding construeren van de symmetriegroepen op het tweedimensionale hyperbolische, euclidische en elliptische vlak. We zien dat die uitbreidingen niet triviaal zijn, maar dat wanneer we ieder van deze uitbreidingen verheffen naar hun universele overdekking, de uitbreiding op de symmetriegroep van het hyperbolische vlak triviaal wordt, in tegenstelling tot die van het euclidische vlak.