Browsing by master's degree program "Master 's Programme in Theoretical and Computational Methods"
Now showing items 2132 of 32

(2022)The nature of dense matter is one of the greatest mysteries in high energy physics. For example, we do not know how QCD matter behaves in neutron star densities as there the matter is strongly coupled. Thus auxiliary methods have to be applied. One of these methods is the AdS/CFTcorrespondence. This maps the strongly coupled field theory to weakly coupled gravity theory. The most well known example of this correspondence is the duality between N = 4 Super YangMills and type IIB supergravity in AdS 5 × S 5 . This duality at finite temperature and chemical potential is the one we invoke in our study. It has been hypothesized that the dense matter would be in a color superconducting phase, where pairs of quarks form a condensate. This has natural interpretation in the gravity theory. The AdS 5 × S 5 geometry is sourced by stack of N coincident D3branes. This N corresponds to the gauge group SU (N ) of N = 4 SYM. Then to study spontaneous breaking of this gauge group, one studies systems where D3branes have separated from the stack. In this work we present two methods of studying the possibility of separating these branes from the stack. First we present an effective potential for a probe brane, which covers the dynamics of a single D3brane in the bulk. We do this by using the action principle. Then we construct an effective potential for a shell constructed from multiple branes. We do this by using the Israel junction conditions. Single brane in the bulk corresponds to SU (N ) → SU (N − 1) × U (1) symmetry breaking and a shell of kbranes corresponds to SU (N ) → SU (N − k) × U (1) k symmetry breaking. Similar spontaneous breaking of the gauge group happens in QCD when we transition to a CSCphase and hence these phases are called color superconducting. We find that for sufficiently high chemical potential the system is susceptible to single brane nucleation. The phase with higher breaking of the gauge group, which corresponds to having shell made out of branes in the bulk, is metastable. This implies that we were able to construct CSCphases of N = 4 SYM, however, the exact details of the phase diagram structure is left for future research.

(2023)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 realtime computer simulations of various cosmic string models in hopes of extracting their effects on the gravitational wave background. In this thesis we discuss the AbelianHiggs model, a toy model of a gauge theory of a complex scalar field and a real vector field. Through a choice of a symmetrybreaking 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 stringstring 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.

(2023)In quantum field theory the objects of interest are the npoint vacuum expectations which can be calculated from the path integral. The path integral usually used in physics is not welldefined and the main motivation for this thesis is to give axioms that a welldefined path integral candidate has to at least satisfy for it to be physically relevant  we want the path integral to have properties which allow us to reconstruct the physically interesting objects from it. The axioms given in this thesis are called the OsterwalderSchrader axioms and the reconstruction of the physical objects from the path integral satisfying the axioms is called the OsterwalderSchrader reconstruction. The OsterwalderSchrader axioms are special in the sense that they are stated in terms of the Euclidean spacetime instead of the physically relevant Minkowski spacetime. As the physical objects live in Minkowski spacetime this means that when reconstructing the physically relevant objects we have to go back to Minkowski spacetime at some point. This thesis has three parts (and an introduction). In the first part we give a brief introduction to parts of functional analysis which we will need later  theory about distributions and about generators of families of operators. The second part is about the OsterwalderSchrader axioms and about the reconstruction of the physically relevant objects from the path integral. In the last part we check that the path integral for the free field of mass m satisfies the OsterwalderSchrader axioms.

(2023)Response Surface Models (RSM) are cheap, reduced complexity, and, usually, statistical models that are fit to the response of more complex models to approximate their outputs with higher computational efficiency. In atmospheric science, there has been a continuous push to reduce the amount of training data required to fit an RSM. With this reduction in costly data gathering, RSMs can be used more ad hoc and quickly adapted to new applications. However, with the decrease in diverse training data, the risk increases that the RSM is eventually used on inputs on which it cannot make a prediction. If there is no indication from the model that its outputs can no longer be trusted, trust in an entire RSM decreases. We present a framework for building prudent RSMs that always output predictions with confidence and uncertainty estimates. We show how confidence and uncertainty can be propagated through downstream analysis such that even predictions on inputs outside the training domain or in areas of high variance can be integrated. Specifically, we introduce the Icarus RSM architecture, which combines an outofdistribution detector, a prediction model, and an uncertainty quantifier. Icarusproduced predictions and their uncertainties are conditioned on the confidence that the inputs come from the same distribution that the RSM was trained on. We put particular focus on exploring outofdistribution detection, for which we conduct a broad literature review, design an intuitive evaluation procedure with three easilyvisualisable toy examples, and suggest two methodological improvements. We also explore and evaluate popular prediction models and uncertainty quantifiers. We use the onedimensional atmospheric chemistry transport model SOSAA as an example of a complex model for this thesis. We produce a dataset of model inputs and outputs from simulations of the atmospheric conditions along air parcel trajectories that arrived at the SMEAR II measurement station in Hyytiälä, Finland, in May 2018. We evaluate several prediction models and uncertainty quantification methods on this dataset and construct a proofofconcept SOSAA RSM using the Icarus RSM architecture. The SOSAA RSM is built on pairwisedifference regression using random forests and an autoassociative outofdistribution detector with a confidence scorer, which is trained with both the original training inputs and new synthetic outofdistribution samples. We also design a graphical user interface to configure the SOSAA model and trial the SOSAA RSM. We provide recommendations for outofdistribution detection, prediction models, and uncertainty quantification based on our exploration of these three systems. We also stresstest the proofofconcept SOSAA RSM implementation to reveal its limitations for predicting model perturbation outputs and show directions for valuable future research. Finally, our experiments affirm the importance of reporting predictions alongside wellcalibrated confidence scores and uncertainty levels so that the predictions can be used with confidence and certainty in scientific research applications.

(2023)Numerical techniques have become powerful tools for studying quantum systems. Eventually, quantum computers may enable novel ways to perform numerical simulations and conquer problems that arise in classical simulations of highly entangled matter. Simple one dimensional systems of low entanglement are efficiently simulatable on a classical computer using tensor networks. This kind of toy simulations also give us the opportunity to study the methods of quantum simulations, such as different transformation techniques and optimization algorithms that could be beneficial for the near term quantum technologies. In this thesis, we study a theoretical framework for a fermionic quantum simulation and simulate the realtime evolution of particles governed by the GrossNeveu model in onedimension. To simulate the GrossNeveu model classically, we use the Matrix Product State (MPS) method. Starting from the continuum case, we discretise the model by putting it on a lattice and encode the time evolution operator with the help of fermiontoqubit transformations, JordanWigner and BravyiKitaev. The simulation results are visualised as plots of probability density. The results indicate the expected flavour and spatial symmetry of the system. The comparison of the two transformations show better performance of the JordanWigner transformation before and after the gate reduction.

(2022)Quantum computers are one of the most prominent emerging technologies of the 21st century. While several practical implementations of the qubit—the elemental unit of information in quantum computers—exist, the family of superconducting qubits remains one of the most promising platforms for scaledup quantum computers. Lately, as the limiting factor of nonerrorcorrected quantum computers has began to shift from the number of qubits to gate fidelity, efficient control and readout parameter optimization has become a field of significant scientific interest. Since these procedures are multibranched and difficult to automate, a great deal of effort has gone into developing associated software, and even technologies such as machine learning are making an appearance in modern programs. In this thesis, we offer an extensive theoretical backround on superconducting transmon qubits, starting from the classical models of electronic circuits, and moving towards circuit quantum electrodynamics. We consider how the qubit is controlled, how its state is read out, and how the information contained in it can become corrupted by noise. We review theoretical models for characteristic parameters such as decoherence times, and see how control pulse parameters such as amplitude and rise time affect gate fidelity. We also discuss the procedure for experimentally obtaining characteristic qubit parameters, and the optimized randomized benchmarking for immediate tuneup (ORBIT) protocol for control pulse optimization, both in theory and alongside novel experimental results. The experiments are carried out with refactored characterization software and novel ORBIT software, using the premises and resources of the Quantum Computing and Devices (QCD) group at Aalto University. The refactoring project, together with the software used for the ORBIT protocol, aims to provide the QCD group with efficient and streamlined methods for finding characteristic qubit parameters and highfidelity control pulses. In the last parts of the thesis, we evaluate the success and shortcomings of the introduced projects, and discuss future perspectives for the software.

(2023)Lowfrequency $1/$ noise is ubiquitous, found in all electronic devices and other diverse areas such as as music, economics and biological systems. Despite valiant efforts, the source of $1/f$ noise remains one of the oldest unsolved mysteries in modern physics after nearly 100 years since its initial discovery in 1925. In metallic conductors resistance $1/f$ noise is commonly attributed to diffusion of mobile defects that alter the scattering cross section experienced by the charge carriers. Models based on twolevel tunneling systems (TLTS) are typically employed. However, a model based on the dynamics of mobile defects forming temporary clusters would naturally offer longterm correlations required by $1/f$ noise via the nearly limitless number of configurations among a group of defects. Resistance $1/f$ noise due to such motion of mobile defects was studied via Monte Carlo simulations of a simple resistor network resembling an atomic lattice. The defects migrate through the lattice via thermally activated hopping motion, causing fluctuations in the resistance due to varying scattering cross section. The power spectral density (PSD) $S(f)$ of the simulated resistance noise was then calculated and first compared to $S(f)=C/f^\alpha$ noise, where $C$ is a constant and $\alpha$ is ideally close to unity. The value of $\alpha$ was estimated via a linear fit of the noise PSD on a loglog scale. The resistor network was simulated with varying values of temperature, system size and the concentration of defects. The noise produced by the simulations did not yield pure $1/f^\alpha$ noise, instead the lowest frequencies displayed a white noise tail, changing to $1/f^\alpha$ noise between $10^{4}$ to $10^{2}$~Hz. In this way the spectrum of the simulated noise resembles a Lorentzian. The value of $\alpha$ was found to be the most sensitive to temperature $T$, which directly affects the motion of the defects. At high $T$ the value of $\alpha$ was closer to 1, whereas at low $T$ it was closer to $1,5$. Varying the size of the system was found to have little impact on $\alpha$ when the temperature and concentration of defects were kept fixed. Increasing the number of defects was found to have slightly more effect on $\alpha$ when the temperature and system size were kept fixed. The value of $\alpha$ was closer to unity when the concentration of defects was higher, but the effect was not nearly as pronounced compared to varying the temperature. In addition, the simulated noise was compared to a PSD of the form $S(f)\propto e^{\sqrt{N}/T}1/f$, where $N$ is the size of the system, according to recent theoretical proceedings. The $1/f^\alpha$ part of the simulated noise was found to roughly follow the above equation, but the results remain inconclusive. Although the simple toy model did not produce pure $1/f^\alpha$ noise, the dynamics of the mobile defects do seem to have an effect on the noise PSD, yielding noise closer to $1/f$ when there are more interactions between the defects due to either higher mobility or higher concentration of defects. However, this is disregarding the white noise tail. Recent experimental research on high quality graphene employing more rigorous kinetic Monte Carlo simulations have displayed more promising results. This indicates that the dynamics of temporary cluster formation of mobile defects is relevant to understand $1/f$ noise in metallic conductors, offering an objective for future work.

(2021)The goal of this work is to describe sheaves as an alternative to fiber bundles in geometric prequantization. We briefly go over geometric quantization of Euclidean space and make a connection with canonical quantization. After this, we look at the connections between covers of a topological space, Grothendieck topologies, and systems of local epimorphisms. Finally, we use these concepts to define sheaves and show how they can be used in prequantization in place of the more traditional fiber bundles to ensure the consistency of locally defined quantities.

(2022)This thesis is aimed to explore the topic of surface diffusion on copper and iron surfaces, using an accelerated molecular dynamics (MD) method known as collective variabledriven hyperdynamics (CVHD). The thesis is divided into six main sections: Introduction, Theory, Methods, Simulations, Results and Conclusion. The introduction briefly explains the main interest behind the topic and why diffusion is a difficult subject for classical MD simulations. In the theory section, the physical description of diffusion in metals is explained, as well as the important quantities that can be determined from these types of simulations. The following section dives into the basics concerning the molecular dynamics simulations method. It also gives a description of the theoretical basis of collective variabledriven hyperdynamics and how it is implemented alongside molecular dynamics. The simulations section more technically explains the system building methodology, discusses key parameters and gives reasoning for the chosen values of these parameters. Since, both copper and iron systems have been simulated, both sets of systems are explained independently. The results section displays the results for the copper and iron systems separately. In both sets of systems, the obtained activation energy of the dominant diffusion mechanisms remain the main point of focus. Lastly, the results are dissected and summarized.

(2022)One of the main ways of physically realizing quantum bits for the purposes of quantum technology is to manufacture them as superconducting circuits. These qubits are artiﬁcially built twolevel systems that act as carriers of quantum information. They come in a variety of types but one of the most common in use is the transmon qubit. The transmon is a more stable, improved version of the earlier types of superconducting qubits with longer coherence times. The qubit cannot function properly on its own, as it needs other circuit elements around it for control and readout of its state. Thus the qubit is only a small part of a larger superconducting circuit interacting with the qubit. Understanding this interaction, where it comes from and how it can be modiﬁed to our liking, allows researchers to design better quantum circuits and to improve the existing ones. Understanding how the noise, travelling through the qubit drive lines to the chip, affects the time evolution of the qubit is especially important. Reducing the amount of noise leads to longer coherence times but it is also possible to engineer the noise to our advantage to uncover novel ways of quantum control. In this thesis the effects of a variable temperature noise source on the qubit drive line is studied. A theoretical model describing the time evolution of the quantum state is built. The model starts from the basic elements of the quantum circuit and leads to a master equation describing the qubit dynamics. This allows us to understand how the different choices made in the manufacturing process of the quantum circuit affect the time evolution. As a proof of concept, the model is solved numerically using QuTiP in the speciﬁc case of a ﬁxedfrequency, dispersive transmon qubit. The solution shows a decohering qubit with no dissipation. The model is also solved in a temperature range 0K < T ≤ 1K to show how the decoherence times behave with respect to the temperature of the noise source.

(2023)Bayesian networks (BN) are models that map the mutual dependencies and independencies between a set of variables. The structure of the model can be represented as a directed acyclic graph (DAG), which is a graph where the nodes represent variables and the directed edges between variables represent a dependency. BNs can be either constructed by using knowledge of the system or derived computationally from observational data. Traditionally, BN structure discovery from observational data has been done through heuristic algorithms, but advances in deep learning have made it possible to train neural networks for this task in a supervised manner. This thesis provides an overview of BN structure discovery and discusses the strengths and weaknesses of the emerging supervised paradigm. One supervised method, the EQmodel, that uses neural networks for structure discovery using equivariant models, is also explored in further detail with empirical tests. Through a process of hyperparameter optimisation and moving to online training, the performance of the EQmodel is increased. The EQmodel is still observed to underperform in comparison to a competing scorebased model, NOTEARS, but offers convenient features, such as dramatically faster runtime, that compensate for the reduced performance. Several interesting lines of further study that could be used to further improve the performance of the EQmodel are also identified.

(2022)The Traveling Salesman Problem (TSP) is a wellknown optimization problem. The time needed to solve TSP classically grows exponentially with the size of the input, placing it into the NPhard computational complexity class–the class of problems that are at least as hard as any other problem solvable in nondeterministic polynomial time. Quantum computing gives us a new approach to searching through such a huge search space, using methods such as quantum annealing and phase estimation. Although the current state of quantum computers does not give us enough resources to solve TSP with a large input, we can use quantum computing methods to improve on existing classical algorithms. The thesis reviews existing methods to efficiently tackle TSP utilizing potential quantum resources, and discusses the augmentation of classical algorithms with quantum techniques to reduce the time complexity of solving this computationally challenging problem.
Now showing items 2132 of 32