Browsing by master's degree program "Magisterprogrammet i teoretiska och beräkningsmetoder"

The variational quantum eigensolver (VQE) is one of the most promising proposals for a hybrid quantum-classical algorithm made to take advantage of near-term quantum computers. With the VQE it is possible to find ground state properties of various of molecules, a task which many classical algorithms have been developed for, but either become too inaccurate or too resource-intensive especially for so called strongly correlated problems. The advantage of the VQE comes in the ability of a quantum computer to represent a complex system with fewer so-called qubits than a classical computer would with bits, thus making the simulation of large molecules possible. One of the major bottlenecks for the VQE to become viable for simulating large molecules however, is the scaling of the number of measurements necessary to estimate expectation values of operators. Numerous solutions have been proposed including the use of adaptive informationally complete positive operator-valued measures (IC-POVMs) by García-Pérez et al. (2021). Adaptive IC-POVMs have shown to improve the precision of estimations of expectation values on quantum computers with better scaling in the number of measurements compared to existing methods. The use of these adaptive IC-POVMs in a VQE allows for more precise energy estimations and additional expectation value estimations of separate operators without any further overhead on the quantum computer. We show that this approach improves upon existing measurement schemes and adds a layer of flexibility, as IC-POVMs represent a form of generalized measurements. In addition to a naive implementation of using IC-POVMs as part of the energy estimations in the VQE, we propose techniques to reduce the number of measurements by adapting the number of measurements necessary for a given energy estimation or through the estimation of the operator variance for a Hamiltonian. We present results for simulations using the former technique, showing that we are able to reduce the number of measurements while retaining the improvement in the measurement precision obtained from IC-POVMs.

Matrix product states provide an efficient parametrisation of low-entanglement many-body quantum states. In this thesis, the underlying theory is developed from scratch, requiring only basic notions of quantum mechanics and quantum information theory. A full introduction to matrix product state algebra and matrix product operators is given, culminating in the derivation of the density matrix renormalisation group algorithm. The latter provides a simple variational scheme to determine the ground state of arbitrary one-dimensional many-body quantum systems with supreme precision. As an application of matrix-product state technology, the kernel polynomial method is introduced in detail as a state-of-the art numerical tool to find the spectral function or the dynamical correlator of a given quantum system. This in turn gives access to the elementary excitations of the system, such that the locations of the low-energy eigenstates can be studied directly in real space. To illustrate those theoretical tools concretely, the ground state energy, the entanglement entropy and the elementary excitations of a simple interface model of a Heisenberg ferromagnet and a Heisenberg antiferromagnet are studied. By changing the location of the model in parameter space, the dependence of the above-mentioned quantities on the transverse field and the coupling strength is investigated. Most notably, we find that the entanglement entropy characteristic to the antiferromagnetic ground state stretches across the interface into the ferromagnetic half-chain. The dependence of the physics on the value of the coupling strength is, overall, small, with exception of the appearance of a boundary mode whose eigenenergy grows with the coupling. A comparison with a localised edge field shows however that the boundary mode is a true interaction effect of the two half-chains. Various algorithmic and physics extensions of the present project are discussed, such that the code written as part of this thesis could be turned into a state-of-the-art MPS library with managable effort. In particular, an application of the kernel polynomial method to calculate finite-temperature correlators is derived in detail.

Due to the unique properties of foams, they can be found in many different applications in a wide variety of fields. The study of foams is also useful for the many properties they share with other phenomena, like impurities in cooling metals, where the impurities coarsen similarly to bubbles in foams. For these and other reasons foams have been studied extensively for over a hundred years and continue being an interesting area of study today due to new insights in both experimental and theoretical work and new applications waiting to be used and realized in different industries. The most impactful early work in the study of the properties of foams was done in the late 1800s by Plateau. His work was extended in the early to mid-1900s by Lifshitz, Slyozov, Wagner and von Neumann and by many more authors in recent years. The early work was mostly experimental or theoretical in the sense of performing mathematical calculations on paper, while the modern methods of study have kept the experimental part -- with more refined methods of measurement of course -- but shifted towards the implementation of the theory as simulations instead of solving problems on paper. In the early 90s Durian proposed a new method for simulating the mechanics of wet foams, based on repulsive spring-like forces between neighboring bubbles. This model was later extended to allow for the coarsening of the foam, and a slightly changed version of this model has been implemented in the code presented in this thesis. As foams consist of a very large number of bubbles, it is important to be able to simulate sufficiently large systems to realistically study the physics of foams. Very large systems have traditionally been too slow to simulate on the individual bubble level in the past, but thanks to the popularity of computer games and the continuous demand for better graphics in games, the graphics processing units have become very powerful and can nowadays be used to do highly parallel general computing. In this thesis, a modified version of Durian's wet foam model that runs on the GPU is presented. The code has been implemented in modern C++ using Nvidia's CUDA on the GPU. Using this program first a typical two-dimensional foam is simulated with 100000 bubbles. It is found that the simulation code replicates the expected behaviour for this kind of foam. After this, a more detailed analysis is done of a novel phenomenon of the separation of liquid and gas phases in low gas fraction foams that arises only with sufficiently large system sizes. It is found that the phase separation causes the foam to evolve as would a foam of higher gas fraction until the phases have mixed back together. It is hypothesized that the reason causing the phase separation is related to uneven energy distribution in the foam, which itself is related to jamming and uneven distribution of the sizes of the bubbles in the foam.

We begin by discussing the essential concepts within the standard cosmology where the dark matter is "cold" and collisionless. We consider the structure formation in the dark matter component and present problems faced by the standard cosmology as well as some prospects for the solutions to those. The main problem considered in this work is the tension in the value of the Hubble constant measured with different procedures. We present the theories behind the procedures, and conclude the study of the tension by considering the most notable interpretations for the reason behind it. We then set up a proposal for an alternative model describing the dark sector. It is a hidden copy of the visible sector electromagnetism, allowing for a radiative cooling in virializing structures. By assuming first an asymmetric particle content, we study which scales of the dark matter halos are eligible to collapse into dense structure. Acquiring a mass function then allows to conclude how much from the total dark matter component is expected to collapse. If instead the dark matter particle content is taken to be symmetric, the collapsed fraction is assumed to annihilate into dark radiation. With certain modifications to the freely available Boltzmann code CAMB, we construct to the code a representation of the cosmology defined by our model. Lastly we use the modified cosmology to create a fit to the data defining the Hubble constant, and see for the relief of the tension. We find that our model provides a reasonable history for the energy content of the universe, and a notable relief to the Hubble tension, although the improvement is only a minor one compared to some more modest modifications to the cosmology.

Blurring is a common phenomenon during image formation due to various factors like motion between the camera and the object, or atmospheric turbulence, or when the camera fails to have the object in focus, which leads to degradation in the image formation process. This leads to the pixels interacting with the neighboring ones, and the captured image is blurry as a result. This interaction with the neighboring pixels, is the 'spread' which is represented by the Point Spread Function. Image deblurring has many applications, for example in Astronomy, medical imaging, where extracting the exact image required might not be possible due to various limiting factors, and what we get is a deformed image. In such cases, it is necessary to use an apt deblurring algorithm keeping all necessary factors like performance and time in mind. This thesis analyzes the performance of learning and analytical methods in Image deblurring Algorithm. Inverse problems would be discussed at first, and how ill posed inverse problems like image deblurring cannot be tackled by naive deconvolution. This is followed by looking at the need for regularization, and how it is necessary to control the fluctuations resulting from extreme sensitivity to noise. The Image reconstruction problem has the form of a convex variational problem, and its prior knowledge acting as the inequality constraints which creates a feasible region for the optimal solution. Interior point methods iterates over and over within this feasible region. This thesis uses the iRestNet Method, which uses the Forward Backward iterative approach for the Machine learning algorithm, and Total Variation approach implemented using the FlexBox tool for analytical method, which uses the Primal Dual approach. The performance is measured using SSIM indices for a range of kernels, the SSIM map is also analyzed for comparing the deblurring efficiency.

The Earth’s Bond albedo is the fraction of total reflected radiative flux emerging from the Earth’s Top of the Atmosphere (ToA) to the incident solar radiation. As such, it is a crucial component in modeling the Earth’s climate. This thesis presents a novel method for estimating the Earth’s Bond albedo, utilising the dynamical effects of Earth radiation pressure on satellite orbits that are directly related to the Bond albedo. Where current methods for estimating the outgoing reflected radiation are based on point measurements of the radiance reflected by the Earth taken in the proximity of the planet, the new method presented in this thesis makes use of the fact that Global Positioning Satellites (GPS) together view the entirety of the ToA surface. The theoretical groundwork is laid for this new method starting from the basic principles of light scattering, satellite dynamics, and Bayesian inference. The feasibility of the method is studied numerically using synthetic data generated from real measurements of GPS satellite orbital elements and the imaging data from the Earth Polychromatic Imaging Camera (EPIC) aboard the Deep Space Climate Observatory (DSCOVR) spacecraft. The numerical methods section introduces the methods used for forward modeling the ToA outgoing radiation, the Runge-Kutta method for integrating the satellite orbits and the virtual-observation Markov-chain Monte Carlo methods used for solving the inverse problem. The section also describes a simple clustering method used for classifying the ToA from EPIC images. The inverse problem was studied with very simple models for the ToA, the satellites, and the satellite dynamics. These initial results were promising as the inverse problem algorithm was able to accurately estimate the Bond albedo. Further study of the method is required to determine how the inverse problem algorithm works when more realism is added to the models.

The purpose of this work is to investigate the scaling of ’t Hooft-Polyakov monopoles in the early universe. These monopoles are a general prediction of a grand unified theory phase transition in the early universe. Understanding the behavior of monopoles in the early universe is thus important. We tentatively find a scaling for monopole separation which predicts that the fraction of the universe’s energy in monopoles remains constant in the radiation era, regardless of initial monopole density. We perform lattice simulations on an expanding lattice with a cosmological background. We use the simplest fields which produce ’t Hooft-Polyakov monopoles, namely the SU(2) gauge fields and a Higgs field in the adjoint representation. We initialize the fields such that we can control the initial monopole density. At the beginning of the simulations, a damping phase is performed to suppress nonphysical fluctuations in the fields, which are remnants from the initialization. The fields are then evolved according to the discretized field equations. Among other things, the number of monopoles is counted periodically during the simulation. To extend the dynamical range of the runs, the Press-Spergel-Ryden method is used to first grow the monopole size before the main evolution phase. There are different ways to estimate the average separation between monopoles in a monopole network, as well as to estimate the root mean square velocity of the monopoles. We use these estimators to find out how the average separation and velocity evolve during the runs. To find the scaling solution of the system, we fit the separation estimate on a function of conformal time. This way we find that the average separation ξ depends on conformal time η as ξ ∝ η^(1/3) , which indicates that the monopole density scales in conformal time the same way as the critical energy density of the universe. We additionally find that the velocity measured with the velocity estimators depends on the separation as approximately v ∝ dξ/dη. It’s been shown that a possible grand unified phase transition would produce an abundance of ’t Hooft-Polyakov monopoles and that some of these would survive to the present day and begin to dominate the energy density of the universe. Our result seemingly disagrees with this prediction, though there are several reasons why the predictions might not be compatible with the model we simulate. For one, in our model the monopoles do not move with thermal velocities, unlike what most of the predictions assume happens in the early universe. Thus future work of simulations with thermal velocities added would be needed. Additionally we ran simulations only in the radiation dominated era of the universe. During the matter domination era, the monopoles might behave differently.

Ga2O3 has been found to exhibit excellent radiation hardness properties, making it an ideal candidate for use in a variety of applications that involve exposure to ionizing radiation, such as in space exploration, nuclear power generation, and medical imaging. Understanding the behaviour of Ga2O3 under irradiation is therefore crucial for optimizing its performance in these applications and ensuring their safe and efficient operation. There are five commonly identified polymorphs of Ga2O3 , namely, β, α, γ, δ and structures, among these phases, β-Ga2O3 is the most stable crystal structure and has attracted majority of the recent attention. In this thesis, we used molecular dynamic simulations with the newly developed machine learned Gaussian approximation potentials to investigate the radiation damage in β-Ga2O3 . We inspected the gradual structural change in β-Ga2O3 lattice with increase doses of Frenkel pairs implantations. The results revealed that O-Frenkel pairs have a strong tendency to recombine and return to their original sublattice sites. When Ga- and O-Frenkel pairs are implanted to the same cell, the crystal structure was damaged and converted to an amorphous phase at low doses. However, the accumulation of pure Ga-Frenkel pairs in the simulation cells might induce a transition of β to γ-Ga, while O sublattice remains FCC crystal structure, which theoretically demonstrated the recent experiments finding that β- Ga2O3 transfers to the γ phase following ion implantation. To gain a better understanding of the natural behaviour of β-Ga2O3 under irradiation, we utilized collision cascade simulations. The results revealed that O sublattice in the β-Ga2O3 lattice is robust and less susceptible to damage, despite O atoms having higher mobility. The collision and recrystallization process resulted in a greater accumulation of Ga defects than O defects, regardless of PKA atom type. These further revealed that displaced Ga ion hard to recombine to β- Ga lattice, while the FCC stacking of the O sublattice has very strong tendency to recovery. Our theoretical models on the radiation damage of β-Ga2O3 provide insight into the mechanisms underlying defect generation and recovery during experiment ion implantation, which has significant implications for improving Ga2O3 radiation tolerance, as well as optimizing its electronic and optical properties.

In this thesis we model the term structure of zero-coupon bonds. Firstly, in the static setting by norm optimization Hilbert space techniques and starting from a set of benchmark fixed income instruments, we obtain a closed from expression for a smooth discount curve. Moving on to the dynamic setting, we describe the stochastic modeling of the fixed income market. Finally, we introduce the Heath-Jarrow-Morton (HJM) methodology. We derive the evolution of zero-coupon bond prices implied by the HJM methodology and prove the HJM drift condition for non arbitrage pricing in the fixed income market under a dynamic setting. Knowing the current discount curve is crucial for pricing and hedging fixed income securities as it is a basic input to the HJM valuation methodology. Starting from the non arbitrage prices of a set of benchmark fixed income instruments, we find a smooth discount curve which perfectly reproduces the current market quotes by minimizing a suitably defined norm related to the flatness of the forward curve. The regularity of the discount curve estimated makes it suitable for use as an input in the HJM methodlogy. This thesis includes a self-contained introduction to the mathematical modeling of the most commonly traded fixed income securities. In addition, we present the mathematical background necessary for modeling the fixed income market in a dynamic setting. Some familiarity with analysis, basic probability theory and functional analysis is assumed.

Topological defects are some of the more common phenomena of many extensions of the standard model of particle physics. In some sense, defects are a consequence of an unresolvable misalignment between different regions of the system, much like cracks in ice or kinks in an antiquated telephone cord. In our context, they present themselves as localised inhomogeneities of the fundamental fields, emerging at the boundaries of the misaligned regions at the cost of, potentially massive, trapped energy. Should the cosmological variety exist in nature, they are hypothesised to emerge from some currently unknown cosmological phase transition, leaving their characteristic mark on the evolution of the nascent universe. As of date, so called cosmic strings are perhaps the most promising type of cosmic defect, at least with respect to their observational prospects. Cosmic strings, as the name suggest, are linelike topological defects; exceedingly thin, yet highly energetic. Given the advent of gravitational wave astronomy, a substantial amount of research is devoted to detailed and expensive real-time computer simulations of various cosmic string models in hopes of extracting their effects on the gravitational wave background. In this thesis we discuss the Abelian-Higgs model, a toy model of a gauge theory of a complex scalar field and a real vector field. Through a choice of a symmetry-breaking scalar potential, this model permits line defects, so called local strings. We discuss some generalities of classical field theory as well as those of the interesting mathematical theory of topological defects. We apply these to our model and present the necessary numerical methods for writing our own cosmic string simulation. We use the newly written simulation to reproduce a number of contemporary results on the scaling properties of the string networks and present some preliminary results from a less investigated region of the model parameter space, attempting to compare the effects of different types of string-string interactions. Furthermore, preliminary results are presented on the thermodynamic evolution of the system and the effects a common computational trick, comoving string width, are discussed with respect to the evolution of the equation of state.