Topological Quantum Computation - An Analysis of an Anyon Model Based on Quantum Double Symmetries

There exists various suggestions for building a functional and a fault-tolerant large-scale quantum computer. Topological quantum computation is a more exotic suggestion, which makes use of the properties of quasiparticles manifest only in certain two-dimensional systems. These so called anyons exhibit topological degrees of freedom, which, in principle, can be used to execute quantum computation with intrinsic fault-tolerance. This feature is the main incentive to study topological quantum computation. The objective of this thesis is to provide an accessible introduction to the theory. In this thesis one has considered the theory of anyons arising in two-dimensional quantum mechanical systems, which are described by gauge theories based on so called quantum double symmetries. The quasiparticles are shown to exhibit interactions and carry quantum numbers, which are both of topological nature. Particularly, it is found that the addition of the quantum numbers is not unique, but that the fusion of the quasiparticles is described by a non-trivial fusion algebra. It is discussed how this property can be used to encode quantum information in a manner which is intrinsically protected from decoherence and how one could, in principle, perform quantum computation by braiding the quasiparticles. As an example of the presented general discussion, the particle spectrum and the fusion algebra of an anyon model based on the gauge group S_3 are explicitly derived. The fusion algebra is found to branch into multiple proper subalgebras and the simplest one of them is chosen as a model for an illustrative demonstration. The different steps of a topological quantum computation are outlined and the computational power of the model is assessed. It turns out that the chosen model is not universal for quantum computation. However, because the objective was a demonstration of the theory with explicit calculations, none of the other more complicated fusion subalgebras were considered. Studying their applicability for quantum computation could be a topic of further research.

On linear order and computation : The expressiveness of interactive computations on linear orders and computations indexed by ordinals

We solve the Dynamic Ehrenfeucht-Fra\"iss\'e Game on linear orders for both players, yielding a normal form for quantifier-rank equivalence classes of linear orders in first-order logic, infinitary logic, and generalized-infinitary logics with linearly ordered clocks. We show that Scott Sentences can be manipulated quickly, classified into local information, and consistency can be decided effectively in the length of the Scott Sentence. We describe a finite set of linked automata moving continuously on a linear order. Running them on ordinals, we compute the ordinal truth predicate and compute truth in the constructible universe of set-theory. Among the corollaries are a study of semi-models as efficient database of both model-theoretic and formulaic information, and a new proof of the atomicity of the Boolean algebra of sentences consistent with the theory of linear order -- i.e., that the finitely axiomatized theories of linear order are dense.

Numerical computation of the module of a quadrilateral

The module of a quadrilateral is a positive real number which divides quadrilaterals into conformal equivalence classes. This is an introductory text to the module of a quadrilateral with some historical background and some numerical aspects. This work discusses the following topics: 1. Preliminaries 2. The module of a quadrilateral 3. The Schwarz-Christoffel Mapping 4. Symmetry properties of the module 5. Computational results 6. Other numerical methods Appendices include: Numerical evaluation of the elliptic integrals of the first kind. Matlab programs and scripts and possible topics for future research. Numerical results section covers additive quadrilaterals and the module of a quadrilateral under the movement of one of its vertex.

Storage and retrieval of individual genomes

A repetitive sequence collection is one where portions of a base sequence of length n are repeated many times with small variations, forming a collection of total length N. Examples of such collections are version control data and genome sequences of individuals, where the differences can be expressed by lists of basic edit operations. Flexible and efficient data analysis on a such typically huge collection is plausible using suffix trees. However, suffix tree occupies O(N log N) bits, which very soon inhibits in-memory analyses. Recent advances in full-text self-indexing reduce the space of suffix tree to O(N log σ) bits, where σ is the alphabet size. In practice, the space reduction is more than 10-fold, for example on suffix tree of Human Genome. However, this reduction factor remains constant when more sequences are added to the collection. We develop a new family of self-indexes suited for the repetitive sequence collection setting. Their expected space requirement depends only on the length n of the base sequence and the number s of variations in its repeated copies. That is, the space reduction factor is no longer constant, but depends on N / n. We believe the structures developed in this work will provide a fundamental basis for storage and retrieval of individual genomes as they become available due to rapid progress in the sequencing technologies.

Lyophilized disaccharides – the effect of water during storage

Nearly one fourth of new medicinal molecules are biopharmaceutical (protein, antibody or nucleic acid derivative) based. However, the administration of these compounds is not always that straightforward due to the fragile nature of aforementioned domains in GI-tract. In addition, these molecules often exhibit poor bioavailability when administered orally. As a result, parenteral administration is commonly preferred. In addition, shelf-life of these molecules in aqueous environments is poor, unless stored in low temperatures. Another approach is to bring these molecules to anhydrous form via lyophilization resulting in enhanced stability during storage. Proteins cannot most commonly be freeze dried by themselves so some kind of excipients are nearly always necessary. Disaccharides are commonly utilized excipients in freeze-dried formulations since they provide a rigid glassy matrix to maintain the native conformation of the protein domain. They also act as "sink"-agents, which basically mean that they can absorb some moisture from the environment and still help to protect the API itself to retain its activity and therefore offer a way to robust formulation. The aim of the present study was to investigate how four amorphous disaccharides (cellobiose, melibiose, sucrose and trehalose) behave when they are brought to different relative humidity levels. At first, solutions of each disaccharide were prepared, filled into scintillation vials and freeze dried. Initial information on how the moisture induced transformations take place, the lyophilized amorphous disaccharide cakes were placed in vacuum desiccators containing different relative humidity levels for defined period, after which selected analyzing methods were utilized to further examine the occurred transformations. Affinity to crystallization, water sorption of the disaccharides, the effect of moisture on glass transition and crystallization temperature were studied. In addition FT-IR microscopy was utilized to map the moisture distribution on a piece of lyophilized cake. Observations made during the experiments backed up the data mentioned in a previous study: melibiose and trehalose were shown to be superior over sucrose and cellobiose what comes to the ability to withstand elevated humidity and temperature, and to avoid crystallization with pharmaceutically relevant moisture contents. The difference was made evident with every utilized analyzing method. In addition, melibiose showed interesting anomalies during DVS runs, which were absent with other amorphous disaccharides. Particularly fascinating was the observation made with polarized light microscope, which revealed a possible small-scale crystallization that cannot be observed with XRPD. As a result, a suggestion can safely be made that a robust formulation is most likely obtained by utilizing either melibiose or trehalose as a stabilizing agent for biopharmaceutical freeze-dried formulations. On the other hand, more experiments should be conducted to obtain more accurate information on why these disaccharides have better tolerance for elevating humidities than others.