Skip to main content
Login | Suomeksi | På svenska | In English

Browsing by master's degree program "Teoreettisten ja laskennallisten menetelmien maisteriohjelma (Theoretical Calculation Methods)"

Sort by: Order: Results:

  • Mäkelä, Noora (2022)
    Sum-product networks (SPN) are graphical models capable of handling large amount of multi- dimensional data. Unlike many other graphical models, SPNs are tractable if certain structural requirements are fulfilled; a model is called tractable if probabilistic inference can be performed in a polynomial time with respect to the size of the model. The learning of SPNs can be separated into two modes, parameter and structure learning. Many earlier approaches to SPN learning have treated the two modes as separate, but it has been found that by alternating between these two modes, good results can be achieved. One example of this kind of algorithm was presented by Trapp et al. in an article Bayesian Learning of Sum-Product Networks (NeurIPS, 2019). This thesis discusses SPNs and a Bayesian learning algorithm developed based on the earlier men- tioned algorithm, differing in some of the used methods. The algorithm by Trapp et al. uses Gibbs sampling in the parameter learning phase, whereas here Metropolis-Hasting MCMC is used. The algorithm developed for this thesis was used in two experiments, with a small and simple SPN and with a larger and more complex SPN. Also, the effect of the data set size and the complexity of the data was explored. The results were compared to the results got from running the original algorithm developed by Trapp et al. The results show that having more data in the learning phase makes the results more accurate as it is easier for the model to spot patterns from a larger set of data. It was also shown that the model was able to learn the parameters in the experiments if the data were simple enough, in other words, if the dimensions of the data contained only one distribution per dimension. In the case of more complex data, where there were multiple distributions per dimension, the struggle of the computation was seen from the results.
  • Kähärä, Jaakko (2022)
    We study the properties of flat band states of bosons and their potential for all-optical switching. Flat bands are dispersionless energy bands found in certain lattice structures. The corresponding eigenstates, called flat band states, have the unique property of being localized to a small region of the lattice. High sensitivity of flat band lattices to the effects of interactions could make them suitable for fast, energy efficient switching. We use the Bose-Hubbard model and computational methods to study multi-boson systems by simulating the time-evolution of the particle states and computing the particle currents. As the systems were small, fewer than ten bosons, the results could be computed exactly. This was done by solving the eigenstates of the system Hamiltonian using exact diagonalization. We focus on a finite-length sawtooth lattice, first simulating weakly interacting bosons initially in a flat band state. Particle current is shown to typically increase linearly with interaction strength. However, fine-tuning the hopping amplitudes and boundary potentials, particle current through the lattice is highly suppressed. We use this property to construct a switch which is turned on by pumping the input with control photons. Inclusion of particle interactions disrupts the system, resulting in a large non-linear increase in particle current. We find that certain flat band lattices could be used as medium for an optical switch capable of controlling the transport of individual photons. In practice, highly optically nonlinear materials are required to reduce the switching time which is found to be inversely proportional to the interaction strength.
  • Nurmela, Mika (2022)
    We study a system of cold high-density matter consisting purely of quarks and gluons. The mathematical construction of Quantum Chromodynamics (QCD) introduces interactions between the fields, which modify the thermodynamic properties of the system. In the presence of interactions, we can not solve the thermodynamic properties of the system analytically. The method is to expand the result in a series in terms of the QCD coupling constant. This is referred to as the perturbation theory in the context of thermal field theory (TFT). The coupling constant describes the strength of the interaction. We introduce the basic calculation methods used in the QCD and the TFTs in general. We will also include the chemical potential associated with the number of quarks in the system in the calculation. In the case of zero temperature, quarks form a Fermi-sphere such that energy states lower than the chemical potential will be Pauli blocked. The resulting fermionic momentum integrals are modified as a consequence. We can split these integrals into two parts, referred to as the vacuum and matter parts. We can split the calculation of the pressure into two distinct contributions: one from skeleton diagrams and one from ring diagrams. The ring diagrams have unphysical IR divergences that we can not cancel using the counterterms. This is why hard thermal loop (HTL) effective field theory (EFT) is introduced. We will discuss this HTL framework, which requires the computation of the matter part of the gluon polarization tensor, which we will also evaluate in this thesis.
  • Vihko, Sami Vihko (2022)
    We will review techniques of perturbative thermal quantum chromodynamics (QCD) in the imaginary-time formalism (ITF). The Infrared (IR)-problems arising from the perturbative treatment of equilibrium thermodynamics of QCD and their phenomenological causes will be investigated in detail. We will also discuss the construction of two effective field theory (EFT) frameworks most often used in modern high precision calculations to overcome these. The EFTs are the dimensionally reduced theories EQCD and MQCD and Hard thermal loop effective theory (HTL). EQCD is three-dimensional Euclidean Yang-Mills theory coupled to an adjoint scalar field and MQCD is three-dimensional Euclidean pure Yang-Mills theory. The effective parameters in these theories are determined through matching calculations. HTL is based on resummation of hard thermal loops and uses effective propagators and vertex functions. We will also discuss the determination of the pressure of QCD perturbatively. In general, this thesis details calculations and the methodology.
  • Korhonen, Keijo (2022)
    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.
  • Seppänen, Kaapo (2021)
    We determine the leading thermal contributions to various self-energies in finite-temperature and -density quantum chromodynamics (QCD). The so-called hard thermal loop (HTL) self-energies are calculated for the quark and gluon fields at one-loop order and for the photon field at two-loop order using the real-time formulation of thermal field theory. In-medium screening effects arising at long wavelengths necessitate the reorganization of perturbative series of thermodynamic quantities. Our results may be directly applied in a reorganization called the HTL resummation, which applies an effective theory for the long-wavelength modes in the medium. The photonic result provides a partial next-to-leading order correction to the current leading-order result and can be later extended to pure QCD with the techniques we develop. The thesis is organized as follows. First, by considering a complex scalar field, we review the main aspects of the equilibrium real-time formalism to build a solid foundation for our thermal field theoretic calculations. Then, these concepts are generalized to QCD, and the properties of the QCD self-energies are thoroughly studied. We discuss the long-wavelength collective behavior of thermal QCD and introduce the HTL theory, outlining also the main motivations for our calculations. The explicit computations of self-energies are presented in extensive detail to highlight the computational techniques we employ.
  • Korhonen, Teo Ilmari (2022)
    Flares are short, high-energy magnetic events on stars, including the Sun. Observations of young stars and red dwarfs regularly show the occurrence of flare events multiple orders of magnitude more energetic than even the fiercest solar storms ever recorded. As our technology remains vulnerable to disruptions due to space weather, the study of flares and other stellar magnetic activity is crucial. Until recently, the detection of extrasolar flares has required much manual work and observation resources. This work presents a mostly automatic pipeline to detect and estimate the energies of extrasolar flare events from optical light curves. To model and remove the star's background radiation in spite of complex periodicity, short windows of nonlinear support vector regression are used to form a multi-model consensus. Outliers above the background are flagged as likely flare events, and a template model is fitted to the flux residual to estimate the energy. This approach is tested on light curves collected from the stars AB Doradus and EK Draconis by the Transiting Exoplanet Survey Satellite, and dozens of flare events are found. The results are consistent with recent literature, and the method is generalizable for further observations with different telescopes and different stars. Challenges remain regarding edge cases, uncertainties, and reliance on user input.
  • Hippeläinen, Antti (2022)
    This thesis reviews state-of-the-art top-down holographic methods used for modeling dense matter in neutron stars. This is done with the help of the Witten-Sakai-Sugimoto (WSS) model, which attempts to construct a holographic version of quantum chromodynamics (QCD) to mimic its features. As a starting chapter, string theory is reviewed in a quick fashion for the reader to understand some of the (historical) developments behind this construction. Bosonic and superstrings are reviewed along conformal field theory, and focus is put on Dp-branes and compactifications of spacetime. This chapter will also explain much of the jargon used in the thesis, which otherwise easily obstructs the main message. After a sufficient understanding of string theory has been achieved, we will move on to holography and holographic dualities in the next chapter, focusing on AdS/CFT and actual computations using holography. Matching of theories is discussed to set up a holographic dictionary. After this, we need to choose either a top-down or a bottom-up approach, from which we will use the former since we are going to use the WSS model. After this comes a brief review of QCD and its central features to be reproduced in holographic QCD. Immediately following this, we will review the Witten-Sakai-Sugimoto model, which is qualitatively and sometimes also quantitatively a reasonable holographic version of QCD. We will discuss WSS’s successes and room for improvement, especially in places that might affect the analysis that we are about to perform on neutron stars. Finally, after all this theoretical development, we will delve into the world of neutron stars. A quick review of the basic features and astrophysical constraints of neutron stars, along with difficulties in modeling them, is given. After this, we will discuss two models of neutron stars, the first one being a toy model with simplified physics and the other a more realistic one. The basic workflow that is required to get to the equation of state data and other relevant observables from a string theoretic action is given step-by-step, and many recent results using this model are reviewed. In the end, the future of the development of the holographic duality, constructing models with it, and modeling of neutron stars is discussed.
  • Piispa, Aleksi (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/CFT-correspondence. 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 Yang-Mills 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 D3-branes. 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 D3-branes 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 D3-brane 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 k-branes 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 CSC-phase 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 CSC-phases of N = 4 SYM, however, the exact details of the phase diagram structure is left for future research.
  • Suominen, Heikki (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 scaled-up quantum computers. Lately, as the limiting factor of non-error-corrected 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 tune-up (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 high-fidelity 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.
  • Heinonen, Arvo Arnoldas (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.
  • Lindblad, Victor (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 variable-driven 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 variable-driven 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.
  • Vaaranta, Antti (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 two-level 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 fixed-frequency, 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.
  • Rychkova, Kseniya (2022)
    The Traveling Salesman Problem (TSP) is a well-known optimization problem. The time needed to solve TSP classically grows exponentially with the size of the input, placing it into the NP-hard 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.