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.

Noncommutative Quantum Field Theories and UV/IR Mixing

Our present-day understanding of fundamental constituents of matter and their interactions is based on the Standard Model of particle physics, which relies on quantum gauge field theories. On the other hand, the large scale dynamical behaviour of spacetime is understood via the general theory of relativity of Einstein. The merging of these two complementary aspects of nature, quantum and gravity, is one of the greatest goals of modern fundamental physics, the achievement of which would help us understand the short-distance structure of spacetime, thus shedding light on the events in the singular states of general relativity, such as black holes and the Big Bang, where our current models of nature break down. The formulation of quantum field theories in noncommutative spacetime is an attempt to realize the idea of nonlocality at short distances, which our present understanding of these different aspects of Nature suggests, and consequently to find testable hints of the underlying quantum behaviour of spacetime. The formulation of noncommutative theories encounters various unprecedented problems, which derive from their peculiar inherent nonlocality. Arguably the most serious of these is the so-called UV/IR mixing, which makes the derivation of observable predictions especially hard by causing new tedious divergencies, to which our previous well-developed renormalization methods for quantum field theories do not apply. In the thesis I review the basic mathematical concepts of noncommutative spacetime, different formulations of quantum field theories in the context, and the theoretical understanding of UV/IR mixing. In particular, I put forward new results to be published, which show that also the theory of quantum electrodynamics in noncommutative spacetime defined via Seiberg-Witten map suffers from UV/IR mixing. Finally, I review some of the most promising ways to overcome the problem. The final solution remains a challenge for the future.

The Spin-Statistics Relation and Noncommutative Quantum Field Theory

The efforts of combining quantum theory with general relativity have been great and marked by several successes. One field where progress has lately been made is the study of noncommutative quantum field theories that arise as a low energy limit in certain string theories. The idea of noncommutativity comes naturally when combining these two extremes and has profound implications on results widely accepted in traditional, commutative, theories. In this work I review the status of one of the most important connections in physics, the spin-statistics relation. The relation is deeply ingrained in our reality in that it gives us the structure for the periodic table and is of crucial importance for the stability of all matter. The dramatic effects of noncommutativity of space-time coordinates, mainly the loss of Lorentz invariance, call the spin-statistics relation into question. The spin-statistics theorem is first presented in its traditional setting, giving a clarifying proof starting from minimal requirements. Next the notion of noncommutativity is introduced and its implications studied. The discussion is essentially based on twisted Poincaré symmetry, the space-time symmetry of noncommutative quantum field theory. The controversial issue of microcausality in noncommutative quantum field theory is settled by showing for the first time that the light wedge microcausality condition is compatible with the twisted Poincaré symmetry. The spin-statistics relation is considered both from the point of view of braided statistics, and in the traditional Lagrangian formulation of Pauli, with the conclusion that Pauli's age-old theorem stands even this test so dramatic for the whole structure of space-time.

Quantum Chemical Studies of Intermolecular Interactions

This thesis studies the intermolecular interactions in (i) boron-nitrogen based systems for hydrogen splitting and storage, (ii) endohedral complexes, A@C60, and (iii) aurophilic dimers. We first present an introduction of intermolecular interactions. The theoretical background is then described. The research results are summarized in the following sections. In the boron-nitrogen systems, the electrostatic interaction is found to be the leading contribution, as 'Coulomb Pays for Heitler and London' (CHL). For the endohedral complex, the intermolecular interaction is formulated by a one-center expansion of the Coulomb operator 1/rab. For the aurophilic attraction between two C2v monomers, a London-type formula was derived by fully accounting for the anisotropy and point-group symmetry of the monomers.

Studies of new inorganic species using relativistic quantum chemistry

In the present work the methods of relativistic quantum chemistry have been applied to a number of small systems containing heavy elements, for which relativistic effects are important. First, a thorough introduction of the methods used is presented. This includes some of the general methods of computational chemistry and a special section dealing with how to include the effects of relativity in quantum chemical calculations. Second, after this introduction the results obtained are presented. Investigations on high-valent mercury compounds are presented and new ways to synthesise such compounds are proposed. The methods described were applied to certain systems containing short Pt-Tl contacts. It was possible to explain the interesting bonding situation in these compounds. One of the most common actinide compounds, uranium hexafluoride was investigated and a new picture of the bonding was presented. Furthermore the rareness of uranium-cyanide compounds was discussed. In a foray into the chemistry of gold, well known for its strong relativistic effects, investigations on different gold systems were performed. Analogies between Au$^+$ and platinum on one hand and oxygen on the other were found. New systems with multiple bonds to gold were proposed to experimentalists. One of the proposed systems was spectroscopically observed shortly afterwards. A very interesting molecule, which was theoretically predicted a few years ago is WAu$_{12}$. Some of its properties were calculated and the bonding situation was discussed. In a further study on gold compounds it was possible to explain the substitution pattern in bis[phosphane-gold(I)] thiocyanate complexes. This is of some help to experimentalists as the systems could not be crystallised and the structure was therefore unknown. Finally, computations on one of the heaviest elements in the periodic table were performed. Calculation on compounds containing element 110, darmstadtium, showed that it behaves similarly as its lighter homologue platinum. The extreme importance of relativistic effects for these systems was also shown.

A quantum chemical investigation of the metal centres in cytochrome c oxidase

Quantum effects are often of key importance for the function of biological systems at molecular level. Cellular respiration, where energy is extracted from the reduction of molecular oxygen to water, is no exception. In this work, the end station of the electron transport chain in mitochondria, cytochrome c oxidase, is investigated using quantum chemical methodology. Cytochrome c oxidase contains two haems, haem a and haem a3. Haem a3, with its copper companion, CuB, is involved in the final reduction of oxygen into water. This binuclear centre receives the necessary electrons from haem a. Haem a, in turn, receives its electrons from a copper ion pair in the vicinity, called CuA. Density functional theory (DFT) has been used to clarify the charge and spin distributions of haem a, as well as changes in these during redox activity. Upon reduction, the added electron is shown to be evenly distributed over the entire haem structure, important for the accommodation of the prosthetic group within the protein. At the same time, the spin distribution of the open-shell oxidised state is more localised to the central iron. The exact spin density distribution has been disputed in the literature, however, different experiments indicating different distributions of the unpaired electron. The apparent contradiction is shown to be due to the false assumption of a unit amount of unpaired electron density; in fact, the oxidised state has about 1.3 unpaired electrons. The validity of the DFT results have been corroborated by wave function based coupled cluster calculations. Point charges, for use in classical force field based simulations, have been parameterised for the four metal centres, using a newly developed methodology. In the procedure, the subsystem for which point charges are to be obtained, is surrounded by an outer region, with the purpose of stabilising the inner region, both electronically and structurally. Finally, the possibility of vibrational promotion of the electron transfer step between haem a and a3 has been investigated. Calculating the full vibrational spectra, at DFT level, of a combined model of the two haems, revealed several normal modes that do shift electron density between the haems. The magnitude of the shift was found to be moderate, at most. The proposed mechanism could have an assisting role in the electron transfer, which still seems to be dominated by electron tunnelling.

Molecular Tuning of Rhodopsins for High Visual Sensitivity in Different Light Environments : Variation in Absorbance Spectrum and Opsin Sequence within and between Species

Visual pigments of different animal species must have evolved at some stage to match the prevailing light environments, since all visual functions depend on their ability to absorb available photons and transduce the event into a reliable neural signal. There is a large literature on correlation between the light environment and spectral sensitivity between different fish species. However, little work has been done on evolutionary adaptation between separated populations within species. More generally, little is known about the rate of evolutionary adaptation to changing spectral environments. The objective of this thesis is to illuminate the constraints under which the evolutionary tuning of visual pigments works as evident in: scope, tempo, available molecular routes, and signal/noise trade-offs. Aquatic environments offer Nature s own laboratories for research on visual pigment properties, as naturally occurring light environments offer an enormous range of variation in both spectral composition and intensity. The present thesis focuses on the visual pigments that serve dim-light vision in two groups of model species, teleost fishes and mysid crustaceans. The geographical emphasis is in the brackish Baltic Sea area with its well-known postglacial isolation history and its aquatic fauna of both marine and fresh-water origin. The absorbance spectrum of the (single) dim-light visual pigment were recorded by microspectrophotometry (MSP) in single rods of 26 fish species and single rhabdoms of 8 opossum shrimp populations of the genus Mysis inhabiting marine, brackish or freshwater environments. Additionally, spectral sensitivity was determined from six Mysis populations by electroretinogram (ERG) recording. The rod opsin gene was sequenced in individuals of four allopatric populations of the sand goby (Pomatoschistus minutus). Rod opsins of two other goby species were investigated as outgroups for comparison. Rod absorbance spectra of the Baltic subspecies or populations of the primarily marine species herring (Clupea harengus membras), sand goby (P. minutus), and flounder (Platichthys flesus) were long-wavelength-shifted compared to their marine populations. The spectral shifts are consistent with adaptation for improved quantum catch (QC) as well as improved signal-to-noise ratio (SNR) of vision in the Baltic light environment. Since the chromophore of the pigment was pure A1 in all cases, this has apparently been achieved by evolutionary tuning of the opsin visual pigment. By contrast, no opsin-based differences were evident between lake and sea populations of species of fresh-water origin, which can tune their pigment by varying chromophore ratios. A more detailed analysis of differences in absorbance spectra and opsin sequence between and within populations was conducted using the sand goby as model species. Four allopatric populations from the Baltic Sea (B), Swedish west coast (S), English Channel (E), and Adriatic Sea (A) were examined. Rod absorbance spectra, characterized by the wavelength of maximum absorbance (λmax), differed between populations and correlated with differences in the spectral light transmission of the respective water bodies. The greatest λmax shift as well as the greatest opsin sequence difference was between the Baltic and the Adriatic populations. The significant within-population variation of the Baltic λmax values (506-511 nm) was analyzed on the level of individuals and was shown to correlate well with opsin sequence substitutions. The sequences of individuals with λmax at shorter wavelengths were identical to that of the Swedish population, whereas those with λmax at longer wavelengths additionally had substitution F261F/Y in the sixth transmembrane helix of the protein. This substitution (Y261) was also present in the Baltic common gobies and is known to redshift spectra. The tuning mechanism of the long-wavelength type Baltic sand gobies is assumed to be the co-expression of F261 and Y261 in all rods to produce ≈ 5 nm redshift. The polymorphism of the Baltic sand goby population possibly indicates ambiguous selection pressures in the Baltic Sea. The visual pigments of all lake populations of the opossum shrimp (Mysis relicta) were red-shifted by 25 nm compared with all Baltic Sea populations. This is calculated to confer a significant advantage in both QC and SNR in many humus-rich lakes with reddish water. Since only A2 chromophore was present, the differences obviously reflect evolutionary tuning of the visual protein, the opsin. The changes have occurred within the ca. 9000 years that the lakes have been isolated from the Sea after the most recent glaciation. At present, it seems that the mechanism explaining the spectral differences between lake and sea populations is not an amino acid substitution at any other conventional tuning site, but the mechanism is yet to be found.

Plant guard cell anion channel SLAC1 regulates stomatal closure

Plants are rooted to their growth place; therefore it is important that they react adequately to changes in environmental conditions. Stomatal pores, which are formed of a pair of guard cells in leaf epidermis, regulate plant gas-exchange. Importantly, guard cells protect the plant from desiccation in drought conditions by reducing the aperture of the stomatal pore. They serve also as the first barrier against the major air pollutant ozone, but the behaviour of guard cells during ozone exposure has not been sufficiently addressed. Aperture of the stomatal pore is regulated by the influx and efflux of osmotically active ions via ion channels and transporters across the guard cell membrane, however the molecular identity of guard cell plasma membrane anion channel has remained unknown. In the frame of this study, guard cell behaviour during ozone exposure was studied using the newly constructed Arabidopsis whole-rosette gas-exchange system. Ozone induced a Rapid Transient Decrease (RTD) in stomatal conductance within 10 min from the start of exposure, which was followed by a recovery in the conductance within the next 40 min. The decrease in stomatal conductance was dependent on the applied ozone concentration. Three minutes of ozone exposure was sufficient to induce RTD and further ozone application during the closure-recovery process had no effect on RTD, demonstrating that the whole process is programmed within the first three minutes. To address the molecular components responsible for RTD, the ozone response was measured in 59 different Arabidopsis mutants involved in guard cell signalling. Four of the tested mutants slac1 (originally rcd3), ost1, abi1-1 and abi2-1 lacked RTD completely. As the ozone sensitive mutant slac1 lacked RTD, the next aim of this study was to identify and characterize SLAC1. SLAC1 was shown to be a central regulator in response to all major factors regulating guard cell aperture: CO2, light/darkness transitions, ozone, relative air humidity, ABA, NO, H2O2, and extracellular Ca2+. It encodes the first guard cell plasma membrane slow type anion channel to be identified at the molecular level. Interestingly, the rapid type anion conductance was intact in slac1 mutant plants. For activation, SLAC1 needs to be phosphorylated. Protein kinase OST1 was shown to phosphorylate several amino acids in the N-terminal tail of SLAC1, Ser120 was one of its main targets, which led to SLAC1 activation. The lack of RTD in type 2C protein phosphatase mutants abi1-1 and abi2-1, suggests that these proteins have a regulatory role in ozoneinduced activation of the slow type anion channel.

Epithelial sodium channel in airway epithelium of the newborn infant : Implications for postnatal pulmonary adaptation

The aim of the present thesis was to study the role of the epithelial sodium channel (ENaC) in clearance of fetal lung fluid in the newborn infant by measurement of airway epithelial expression of ENaC, of nasal transepithelial potential difference (N-PD), and of lung compliance (LC). In addition, the effect of postnatal dexamethasone on airway epithelial ENaC expression was measured in preterm infants with bronchopulmonary dysplasia (BPD). The patient population was formed of selected term newborn infants born in the Department of Obstetrics (Studies II-IV) and selected preterm newborn infants treated in the neonatal intensive care unit of the Hospital for Children and Adolescents (Studies I and IV) of the Helsinki University Central Hospital in Finland. A small population of preterm infants suffering from BPD was included in Study I. Studies I, III, and IV included airway epithelial measurement of ENaC and in Studies II and III, measurement of N-PD and LC. In Study I, ENaC expression analyses were performed in the Research Institute of the Hospital for Sick Children in Toronto, Ontario, Canada. In the following studies, analyses were performed in the Scientific Laboratory of the Hospital for Children and Adolescents. N-PD and LC measurements were performed at bedside in these hospitals. In term newborn infants, the percentage of amiloride-sensitive N-PD, a surrogate for ENaC activity, measured during the first 4 postnatal hours correlates positively with LC measured 1 to 2 days postnatally. Preterm infants with BPD had, after a therapeutic dose of dexamethasone, higher airway epithelial ENaC expression than before treatment. These patients were subsequently weaned from mechanical ventilation, probably as a result of the clearance of extra fluid from the alveolar spaces. In addition, we found that in preterm infants ENaC expression increases with gestational age (GA). In preterm infants, ENaC expression in the airway epithelium was lower than in term newborn infants. During the early postnatal period in those born both preterm and term airway epithelial βENaC expression decreased significantly. Term newborn infants delivered vaginally had a significantly smaller airway epithelial expression of αENaC after the first postnatal day than did those delivered by cesarean section. The functional studies showed no difference in N-PD between infants delivered vaginally and by cesarean section. We therefore conclude that the low airway epithelial expression of ENaC in the preterm infant and the correlation of N-PD with LC in the term infant indicate a role for ENaC in the pathogenesis of perinatal pulmonary adaptation and neonatal respiratory distress. Because dexamethasone raised ENaC expression in preterm infants with BPD, and infants were subsequently weaned from ventilator therapy, we suggest that studies on the treatment of respiratory distress in the preterm infant should include the induction of ENaC activity.

Ion channel gene variants predisposing to severe human ventricular arrhythmias

Congenital long QT syndrome (LQTS) with an estimated prevalence of 1:2000-1:10 000 manifests with prolonged QT interval on electrocardiogram and risk for ventricular arrhythmias and sudden death. Several ion channel genes and hundreds of mutations in these genes have been identified to underlie the disorder. In Finland, four LQTS founder mutations of potassium channel genes account for up to 40-70% of genetic spectrum of LQTS. Acquired LQTS has similar clinical manifestations, but often arises from usage of QT-prolonging medication or electrolyte disturbances. A prolonged QT interval is associated with increased morbidity and mortality not only in clinical LQTS but also in patients with ischemic heart disease and in the general population. The principal aim of this study was to estimate the actual prevalence of LQTS founder mutations in Finland and to calculate their effect on QT interval in the Finnish background population. Using a large population-based sample of over 6000 Finnish individuals from the Health 2000 Survey, we identified LQTS founder mutations KCNQ1 G589D (n=8), KCNQ1 IVS7-2A>G (n=1), KCNH2 L552S (n=2), and KCNH2 R176W (n=16) in 27 study participants. This resulted in a weighted prevalence estimate of 0.4% for LQTS in Finland. Using a linear regression model, the founder mutations resulted in a 22- to 50-ms prolongation of the age-, sex-, and heart rate-adjusted QT interval. Collectively, these data suggest that one of 250 individuals in Finland may be genetically predisposed to ventricular arrhythmias arising from the four LQTS founder mutations. A KCNE1 D85N minor allele with a frequency of 1.4% was associated with a 10-ms prolongation in adjusted QT interval and could thus identify individuals at increased risk of ventricular arrhythmias at the population level. In addition, the previously reported associations of KCNH2 K897T, KCNH2 rs3807375, and NOS1AP rs2880058 with QT interval duration were confirmed in the present study. In a separate study, LQTS founder mutations were identified in a subgroup of acquired LQTS, providing further evidence that congenital LQTS gene mutations may underlie acquired LQTS. Catecholaminergic polymorphic ventricular tachycardia (CPVT) is characterized by exercise-induced ventricular arrhythmias in a structurally normal heart and results from defects in the cardiac Ca2+ signaling proteins, mainly ryanodine receptor type 2 (RyR2). In a patient population of typical CPVT, RyR2 mutations were identifiable in 25% (4/16) of patients, implying that noncoding variants or other genes are involved in CPVT pathogenesis. A 1.1 kb RyR2 exon 3 deletion was identified in two patients independently, suggesting that this region may provide a new target for RyR2-related molecular genetic studies. Two novel RyR2 mutations showing a gain-of-function defect in vitro were identified in three victims of sudden cardiac death. Extended pedigree analyses revealed some surviving mutation carriers with mild structural abnormalities of the heart and resting ventricular arrhythmias suggesting that not all RyR2 mutations lead to a typical CPVT phenotype, underscoring the relevance of tailored risk stratification of a RyR2 mutation carrier.

Quantum Space-Time and Noncommutative Gauge Field Theories

Arguments arising from quantum mechanics and gravitation theory as well as from string theory, indicate that the description of space-time as a continuous manifold is not adequate at very short distances. An important candidate for the description of space-time at such scales is provided by noncommutative space-time where the coordinates are promoted to noncommuting operators. Thus, the study of quantum field theory in noncommutative space-time provides an interesting interface where ordinary field theoretic tools can be used to study the properties of quantum spacetime. The three original publications in this thesis encompass various aspects in the still developing area of noncommutative quantum field theory, ranging from fundamental concepts to model building. One of the key features of noncommutative space-time is the apparent loss of Lorentz invariance that has been addressed in different ways in the literature. One recently developed approach is to eliminate the Lorentz violating effects by integrating over the parameter of noncommutativity. Fundamental properties of such theories are investigated in this thesis. Another issue addressed is model building, which is difficult in the noncommutative setting due to severe restrictions on the possible gauge symmetries imposed by the noncommutativity of the space-time. Possible ways to relieve these restrictions are investigated and applied and a noncommutative version of the Minimal Supersymmetric Standard Model is presented. While putting the results obtained in the three original publications into their proper context, the introductory part of this thesis aims to provide an overview of the present situation in the field.