Quantum mechanics is the physics of the very small. Quantum computers are devices that utilize the power of quantum mechanics for their computational primitives. Associated to each quantum system is an abstract space known as the Hilbert space. A subspace of the Hilbert space is known as a quantum code. Quantum codes allow to protect the computational state of a quantum computer against decoherence errors. The well-known classes of quantum codes are stabilizer or additive codes, non-additive codes and Clifford codes. This thesis aims at demonstrating a general approach to the construction of the various classes of quantum codes. The framework utilized is the Fourier transform over finite groups. The thesis is divided into four chapters. The first chapter is an introduction to basic quantum mechanics, quantum computation and quantum noise. It lays the foundation for an understanding of quantum error correction theory in the next chapter. The second chapter introduces the basic theory behind quantum error correction. Also, the various classes and constructions of active quantum error-control codes are introduced. The third chapter introduces the Fourier transform over finite groups, and shows how it may be used to construct all the known classes of quantum codes, as well as a class of quantum codes as yet unpublished in the literature. The transform domain approach was originally introduced in (Arvind et al., 2002). In that paper, not all the classes of quantum codes were introduced. We elaborate on this work to introduce the other classes of quantum codes, along with a new class of codes, codes from idempotents in the transform domain. The fourth chapter details the computer programs that were used to generate and test for the various code classes. Code was written in the GAP (Groups, Algorithms, Programming) computer algebra package. The fifth and final chapter concludes, with possible directions for future work. References cited in the thesis are attached at the end of the thesis.

There is a growing consensus among physicists that the classical notion of spacetime has to be drastically revised in order to nd a consistent formulation of quantum mechanics and gravity. One such nontrivial attempt comprises of replacing functions of continuous spacetime coordinates with functions over noncommutative algebra. Dynamics on such noncommutative spacetimes (noncommutative theories) are of great interest for a variety of reasons among the physicists. Additionally arguments combining quantum uncertain-ties with classical gravity provide an alternative motivation for their study, and it is hoped that these theories can provide a self-consistent deformation of ordinary quantum field theories at small distances, yielding non-locality, or create a framework for finite truncation of quantum field theories while preserving symmetries. In this thesis we study the gauge theories on noncommutative Moyal space. We nd new static solitons and instantons in terms of the so-called generalized Bose operators (GBO). GBOs are constructed to describe reducible representation of the oscillator algebra. They create/annihilate k-quanta, k being a positive integer. We start with giving an alternative description to the already found static magnetic flux tube solutions of the noncommutative gauge theories in terms of GBOs. The Nielsen-Olesen vortex solutions found in terms of these operators also reduce to the ones known in the literature. On the other hand, we nd a class of new instanton solutions which are unitarily inequivalent to the ones found from ADHM construction on noncommutative space. The charge of the instanton has a description in terms of the index representing the reducibility of the Fock space representation, i.e., k. After studying the static soliton solutions in noncommutative Minkowski space and the instanton solutions in noncommutative Euclidean space we go on to study the implications of the time-space noncommutativity in Minkowski space. To understand it properly we study the time-dependent transitions of a forced harmonic oscillator in noncommutative 1+1 dimensional spacetime. We also provide an interpretation of our results in the context of non-linear quantum optics. We then shift to the so-called DSR theories which are related to a different kind of noncommutative ( -Minkowski) space. DSR (Doubly/Deformed Special Relativity) aims to search for an alternate relativistic theory which keeps a length/energy scale (the Planck scale) and a velocity scale (the speed of light scale) invariant. We study thermodynamics of an ideal gas in such a scenario. In first chapter we introduce the subjects of the noncommutative quantum theories and the DSR. Chapter 2 starts with describing the GBOs. They correspond to reducible representations of the harmonic oscillator algebra. We demonstrate their relevance in the construction of topologically non-trivial solutions in noncommutative gauge theories, focusing our attention to flux tubes, vortices, and instantons. Our method provides a simple new relation between the topological charge and the number of times the basic irreducible representation occurs in the reducible representation underlying the GBO. When used in conjunction with the noncommutative ADHM construction, we nd that these new instantons are in general not unitarily equivalent to the ones currently known in literature. Chapter 3 studies the time dependent transitions of quantum forced harmonic oscillator (QFHO) in noncommutative R1;1 perturbatively to linear order in the noncommutativity . We show that the Poisson distribution gets modified, and that the vacuum state evolves into a \squeezed" state rather than a coherent state. The time evolutions of un-certainties in position and momentum in vacuum are also studied and imply interesting consequences for modelling nonlinear phenomena in quantum optics. In chapter 4 we study thermodynamics of an ideal gas in Doubly Special Relativity. We obtain a series solution for the partition function and derive thermodynamic quantities. We observe that DSR thermodynamics is non-perturbative in the SR and massless limits. A stiffer equation of state is found. We conclude our results in the last chapter.

Dans cette thèse nous étudions un système quantique, obtenu comme un analogue d'un système classique superintégrable perturbé au moyen de la quantification géométrique. Notre objectif est de mettre en évidence la présence des phénomènes analogues à ceux qui caractérisent la superintégrabilité classique, notamment la coexistence des mouvements réguliers et chaotiques liés aux effets des résonances ainsi que la régularité du régime non-résonant. L'analyse est effectuée par l'étude des distributions du Husimi des états quantiques sélectionnés, avec une attention particulière aux états stationnaires et à l'évolution des états cohérents. Les calculs sont effectués en utilisant les méthodes numériques et les méthodes perturbatives. Les calculs sont effectués en utilisant les méthodes numériques et les méthodes perturbatives. Bien que cette thèse devrait être considérée comme une étude préliminaire, dont l'objectif est de créer le socle des études futures, nos résultats donnent des indications intéressantes sur la dynamique quantique. Par exemple, il est démontré comment les résonancees classiques exercent une influence considérable sur le spectre du système quantique et comment il est possible, dans le comportement quantique, de trouver une trace de l'invariant adiabatique dans le régime de résonance. / The abundance, among physical models, of perturbations of superintegrable Hamiltonian systems makes the understanding of their long-term dynamics an important research topic. While from the classical standpoint the situation, at least in many important cases, is well understood through the use of Nekhoroshev stability theorem and of the adiabatic invariants theory, in the quantum framework there is, on the contrary, a lack of precise results. The purpose of this thesis is to study a perturbed superintegrable quantum system, obtained from a classical counterpart by means of geometric quantization, in order to highlight the presence of indicators of superintegrability analogues to the ones that characterize the classical system, such as the coexistence of regular motions with chaotic one, due to the effects of resonances, opposed to the regularity in the non resonant regime. The analysis is carried out by studying the Husimi distributions of chosen quantum states, with particular emphasis on stationary states and evolved coherent states. The computation are performed using both numerical methods and perturbative schemes. Although this should be considered a preliminary work, the purpose of which is to lay the fundations for future investigations, the results obtained here give interesting insights into quantum dynamics. For instance, it is shown how classical resonances exert a considerable influence on the spectrum of the quantum system and how it is possible, in the quantum behaviour, to find a trace of the classical adiabatic invariance in the resonance regime. / L'abbondanza, fra i modelli fisici, di perturbazioni di sistemi Hamiltoniani superintegrabili rende la comprensione della loro dinamica per tempi lunghi un importante argomento diricerca. Mentre dal punto di vista classico la situazione, perlomeno in molti case importanti, è ben compresa grazie all'uso del teorema di stabilità di Nekhoroshev e della teoria degli invariantiadiabatici, nel caso quantistico vi è, al contrario, una mancanza di risultati precisi. L'obiettivo di questa tesi è di studiare un sistema superintegrabile quantistico, ottenuto partendo da un corrispettivo classico tramite quantizzazione geometrica, al fine di evidenziare la presenza di indicatori di supertintegrabilità analoghi a quelliche caratterizzano il sistema classico, come la coesistenza di moti regolari e caotici, dovuta all'effetto delle risonanze, in contrapposizione con la regolarità nel regime non risonante. L'analisi è condotta studiando le distribuzioni di Husimi di stati quantistici scelti, con particolare enfasi posta sugli stati stazionari e sugli stati coerenti evoluti. I calcoli sono effettuati sia utilizzando tecniche numeriche che schemi perturbativi. Pur essendo da considerardi questo un lavoro preliminare, il cui compito è di porre le fondamenta per analisi future, i risultati qui ottenuti offrono interessanti spunti sulla dinamica quantistica. Per esempio è mostrato come le risonanze classiche abbiano un chiaro effeto sullo spettro del sistema quantistico, ed inoltre comesia possibile trovare una traccia, nel comportamento quantistico, dell'invarianza adiabatica classica nel regime risonante.

The development of the correlation consistent basis sets, cc-pVnZ (where n = D, T, Q, etc.) have allowed for the systematic elucidation of the intrinsic accuracy of ab initio quantum chemical methods. In density functional theory (DFT), where the cc-pVnZ basis sets are not necessarily optimal in their current form, the elucidation of the intrinsic accuracy of DFT methods cannot always be accomplished. This dissertation outlines investigations into the basis set requirements for DFT and how the intrinsic accuracy of DFT methods may be determined with a prescription involving recontraction of the cc-pVnZ basis sets for specific density functionals. Next, the development and benchmarks of a set of cc-pVnZ basis sets designed for the s-block atoms lithium, beryllium, sodium, and magnesium are presented. Computed atomic and molecular properties agree well with reliable experimental data, demonstrating the accuracy of these new s-block basis sets. In addition to the development of cc-pVnZ basis sets, the development of a new, efficient formulism of the correlation consistent Composite Approach (ccCA) using the resolution of the identity (RI) approximation is employed. The new formulism, denoted 'RI-ccCA,' has marked efficiency in terms of computational time and storage, compared with the ccCA formulism, without the introduction of significant error. Finally, this dissertation reports three separate investigations of the properties of FOOF-like, germanium arsenide, and silicon hydride/halide molecules using high accuracy ab initio methods and the cc-pVnZ basis sets.

Glycosaminoglycans (GAGs) play an important role in many biological processes in the extracellular matrix. In a theoretical approach, structures of monosaccharide building blocks of natural GAGs and their sulfated derivatives were optimized by a B3LYP6311ppdd//B3LYP/ 6-31+G(d) method. The dependence of the observed conformational properties on the applied methodology is described. NMR chemical shifts and proton-proton spin-spin coupling constants were calculated using the GIAO approach and analyzed in terms of the method's accuracy and sensitivity towards the influence of sulfation, O1-methylation, conformations of sugar ring, and ω dihedral angle. The net sulfation of the monosaccharides was found to be correlated with the 1H chemical shifts in the methyl group of the N-acetylated saccharides both theoretically and experimentally. The ω dihedral angle conformation populations of free monosaccharides and monosaccharide blocks within polymeric GAG molecules were calculated by a molecular dynamics approach using the GLYCAM06 force field and compared with the available NMR and quantum mechanical data. Qualitative trends for the impact of sulfation and ring conformation on the chemical shifts and proton-proton spin-spin coupling constants were obtained and discussed in terms of the potential and limitations of the computational methodology used to be complementary to NMR experiments and to assist in experimental data assignment.

Die Zeitverzögerung bei der Streuung in zwei Dimensionen ist eine Funktion von zwei unabhängigen Parametern. Wenn diese Funktion Sattelpunkte aufweist, so hat der entsprechende Funktionswert theoretisch ein unendlich großes Gewicht in der Wahrscheinlichkeitsverteilung der Zeitverzögerungen. Dieser Zusammenhang soll analytisch und numerisch nachgewiesen und detailliert beschrieben werden. Insbesondere soll die klassische und quantenmechanische Wahrscheinlichkeitsverteilung der Zeitverzögerung für ein Modellsystem aus mehreren nichtüberlappenden zentralsymmetrischen Potentialen berechnet werden. Erwartete Ergebnisse sind Aussagen über die Parameterwerte, bei denen der oben genannte Effekt zu beobachten ist sowie Näherungsformeln für die Verteilung der Zeitverzögerung in der Nähe der Singularitäten. Außerdem soll die quantenmechanisch zu erwartende Glättung der Verteilungsfunktion quantitativ beschrieben werden.:1 Einleitung 2 Zeitverzögerung in klassischen Streusystemen 2.1 Definition durch die Wirkung 2.2 Geometrisch motivierte Definitionen 2.2.1 Eigentliche Zeitverzögerung 2.2.2 Definition über retardierten Ort 2.2.3 Definition über Aufenthaltszeit 2.2.4 Numerische Bestimmung der Zeitverzögerung 2.3 Zeitverzögerungsfunktion und -verteilung 2.4 Rechenregeln 2.4.1 Koordinatensystemwechsel 2.4.2 Verkettung 3 Klassische Modellsysteme 3.1 Harte Scheibe 3.2 Verschobene harte Scheibe 3.2.1 Verhalten in der Umgebung von stationären Punkten 3.3 Weiches Scheibenpaar 3.3.1 Sattelpunkte 3.3.2 Extrempunkte 3.3.3 Zusammenfassung 4 Quantenmechanische Zeitverzögerung 4.1 Quantisierung der klassischen Definition 4.1.1 Definition über Aufenthaltszeit 4.1.2 Wigner-Smith-Matrix 4.1.3 Numerische Umsetzung 4.2 Einheitenlose Formulierung 4.3 Gegenüberstellung von Zeitentwicklungsmethoden 4.4 Split-Operator-Methode 4.4.1 Parameterwahl 4.4.2 Zur Abschätzung des systematischen Fehlers 4.5 Unterdrückung der periodischen Randbedingung 4.6 Harte Potentiale 5 Quantenmechanische Modellsysteme 5.1 Stationäre Punkte 5.2 Unschärfeeffekte 5.3 Numerische Ungenauigkeiten 5.3.1 Skalierungsverhalten der numerischen Methoden 5.4 Zusammenfassung der Ergebnisse 6 Zusammenfassung und Ausblick Anhang A Verhalten der Verteilung einer Funktion in der Nähe stationärer Punkte A.1 Umgebung eines Sattelpunktes A.2 Umgebung eines Extremums B Zeitverzögerung für das weiche Scheibenpaar / For scattering problems in two dimensions, time-delay is a function of two independent parameters. If this function features saddle points, the corresponding function value should theoretically have an infinite weight in the probability distribution of time-delays. This correlation shall be confirmed analytically and numerically and studied in-depth. In particular, the classical and quantum-mechanical probability distribution of time-delays shall be calculated for a model system consisting of multiple non-overlapping potentials with rotational symmetry. We expect to obtain information about the parameter values where the aforementioned effects can be observed, and analytical approximations for the time-delay distribution near the singularities. Furthermore, the smoothing of the distribution in the quantummechanical regime shall be quantified.:1 Einleitung 2 Zeitverzögerung in klassischen Streusystemen 2.1 Definition durch die Wirkung 2.2 Geometrisch motivierte Definitionen 2.2.1 Eigentliche Zeitverzögerung 2.2.2 Definition über retardierten Ort 2.2.3 Definition über Aufenthaltszeit 2.2.4 Numerische Bestimmung der Zeitverzögerung 2.3 Zeitverzögerungsfunktion und -verteilung 2.4 Rechenregeln 2.4.1 Koordinatensystemwechsel 2.4.2 Verkettung 3 Klassische Modellsysteme 3.1 Harte Scheibe 3.2 Verschobene harte Scheibe 3.2.1 Verhalten in der Umgebung von stationären Punkten 3.3 Weiches Scheibenpaar 3.3.1 Sattelpunkte 3.3.2 Extrempunkte 3.3.3 Zusammenfassung 4 Quantenmechanische Zeitverzögerung 4.1 Quantisierung der klassischen Definition 4.1.1 Definition über Aufenthaltszeit 4.1.2 Wigner-Smith-Matrix 4.1.3 Numerische Umsetzung 4.2 Einheitenlose Formulierung 4.3 Gegenüberstellung von Zeitentwicklungsmethoden 4.4 Split-Operator-Methode 4.4.1 Parameterwahl 4.4.2 Zur Abschätzung des systematischen Fehlers 4.5 Unterdrückung der periodischen Randbedingung 4.6 Harte Potentiale 5 Quantenmechanische Modellsysteme 5.1 Stationäre Punkte 5.2 Unschärfeeffekte 5.3 Numerische Ungenauigkeiten 5.3.1 Skalierungsverhalten der numerischen Methoden 5.4 Zusammenfassung der Ergebnisse 6 Zusammenfassung und Ausblick Anhang A Verhalten der Verteilung einer Funktion in der Nähe stationärer Punkte A.1 Umgebung eines Sattelpunktes A.2 Umgebung eines Extremums B Zeitverzögerung für das weiche Scheibenpaar

<div>We consider actions of the current Lie algebras $\gl_{n}[t]$ and $\gl_{k}[t]$ on the space $\mathfrak{P}_{kn}$ of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1}\lc z_{k})$ and $\bar{\alpha}=(\alpha_{1}\lc\alpha_{n})$, respectively.</div><div>We show that the images of the Bethe algebras $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}\subset U(\gl_{n}[t])$ and $\mathcal{B}_{\bar{z}}^{\langle k \rangle}\subset U(\gl_{k}[t])$ under these actions coincide.</div><div></div><div>To prove the statement, we use the Bethe ansatz description of eigenvectors of the Bethe algebras via spaces of quasi-exponentials. We establish an explicit correspondence between the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}$ and the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{z}}^{\langle k \rangle}$.</div><div></div><div>One particular aspect of the duality of the Bethe algebras is that the Gaudin Hamiltonians exchange with the Dynamical Hamiltonians. We study a similar relation between the trigonometric Gaudin and Dynamical Hamiltonians. In trigonometric Gaudin model, spaces of quasi-exponentials are replaced by spaces of quasi-polynomials. We establish an explicit correspondence between the spaces of quasi-polynomials describing eigenvectors of the trigonometric Gaudin Hamiltonians and the spaces of quasi-exponentials describing eigenvectors of the trigonometric Dynamical Hamiltonians.</div><div></div><div>We also establish the $(\gl_{k},\gl_{n})$-duality for the rational, trigonometric and difference versions of Knizhnik-Zamolodchikov and Dynamical equations.</div>

Influenza A is involved in hundreds of thousands of deaths globally every year resulting from viral infection-related complications. Previous efforts to subdue the virus by preventing proper function of wild-type (WT) neuraminidase (N), and M2 proteins using oseltamivir and amantadine (AMT) or rimantadine (RMT), respectively, exhibited success initially. Over time, these drugs began exhibiting mixed success as the virus developed drug resistance. M2 is a proton channel responsible for the acidification of the viral interior which facilitates release of the viral RNA into the host. M2 has a His37-tetrad that is the selective filter for protons. This protein has been demonstrated to be a feasible target for organic compounds. However, due to a mutation from serine to asparagine at residue 31 of M2, which is found in the majority of influenza strains circulating in humans, AMT and RMT block is insufficient. From simulations, it is unclear whether the insensitivity results from weak binding or incomplete block. The question of how the S31N mutation caused MT and RMT insensitivity in M2 is addressed here by analyzing the binding kinetics of AMT and RMT using the two-electrode voltage clamp electrophysiology method. The dissociation rate constant (k2) is dramatically increased compared to WT for both AMT and RMT, by 1500-fold and 17000-fold respectively. Testing of AMT at 10 mM demonstrates complete block, albeit weak, of the S31N M2 channel. At 10 mM, RMT does not reach complete block even though the binding site is saturated. When RMT is in the bound state, it is not blocking all the current, and is binding without block. These results motivated the development of novel M2 blockers using copper complexes focusing on the His37 complex in M2. I hypothesized that copper complexes would bind with the imidazole of a histidine in the His37 complex and prevent proton conductance. The His37 complex is highly conserved in the M2 channel and, therefore, would be important target for influenza therapeutics. By derivatizing the amines of known M2 blockers, AMT and cyclooctyalmine, to form the iminodiacetate or iminodiacetamide, we have synthesized Cu(II) containing complexes and characterized them by NMR, IR, MS, UV–vis, and inductively coupled plasma mass spectroscopy (ICP-MS). The copper complexes, but not the copper-free ligands, demonstrated H37-specific blocking of M2 channel currents and low micromolar anti-viral efficacies in both Amt-sensitive and Amt-resistant IAV strains with, for the best case, nearly 10-fold less cytotoxicity than CuCl2. Isothermal titration calorimetry was used to obtain enthalpies that showed the copper complexes bind to one imidazole and curve fitting to the electrophysiology data provided rate constants for binding in the M2 channel. Computational chemistry was used to obtain binding geometries and energies of the copper complexes to the His37-tetrad. The results show that the copper complexes do bind with the His37 complex and prevent proton conductance and influenza infection.

Tomando como punto de partida el proceso revisión de los fundamentos matemáticos llevado a cabo durante el siglo XIX, este estudio se centra en uno de los conceptos matemáticos más importantes: el infinito. Es innegable la importancia de este concepto en el avance de las Matemáticas y es fácil encontrar ejemplos matemáticos en los que interviene (definición de límite, definición de derivada, definición de integral de Riemann, entre otras). Debido a que algunas de las paradojas y contradicciones originadas por la falta de rigor en las Matemáticas están relacionadas con este concepto, se comienza con el estudio epistemológico del concepto matemático del infinito revisando la bipolaridad que presentan algunos conceptos semánticos, definidos de forma inseparable y conjunta, constituyendo un único concepto como si representaran los polos de un imán. En este estudio se concluye que la bipolaridad revela que una lógica conceptual que puede asumir la comprensión de la negación, debe ser una lógica dialéctica, es decir que admite como verdaderas algunas contradicciones. En el caso del concepto matemático de lo finito-infinito, nos encontramos de nuevo con una bipolaridad lógica. Por todo lo expuesto se presenta una teoría no cantoriana para el infinito potencial y actual, basada en la imprecisión lingüística del concepto de infinito, y utilizando el concepto de conjunto homógono, formado por una sucesión convergente y su límite, previamente introducido por Leibniz, que permite aunar los dos polos del concepto de infinito en un único conjunto. Esta nueva teoría de conjuntos permitirá presentar en lenguaje homogónico, algunos de los conceptos fundamentales del análisis tales como, la diferencial y la integral, así como algunas aplicaciones a la Óptica y a la Mecánica Cuántica. Posteriormente se presenta la categoría lógica de la oposición cualitativa a través de diferentes ejemplos de diversas áreas de la ciencia, y se define, a través de tres reglas o normas básicas, el paso de la lógica aristotélica o analítica a la lógica sintética, que incluye al neutro como parte de la oposición cualitativa. Con la aplicación de estas normas a la oposición cualitativa y, en particular, a su neutro, se demuestra que la lógica sintética permite la verdad de algunas contradicciones. Esta lógica sintética es dialéctica y multivaluada y da a cada proposición un valor de verdad en el intervalo [0,1], que coincide con el cuadrado del módulo de un número complejo. Esto marca una notable novedad respecto de la lógica aristotélica o analítica que otorga valores de verdad reales, o incluso a la lógica difusa que, a pesar de ser una lógica multivaluada otorga valores de verdad reales en el intervalo [0,1]. En esta lógica dialéctica, las contradicciones del neutro de una oposición pueden ser verdaderas. Finalmente se plantea la aplicación de la lógica dialéctica, a la Mecánica Cuántica, cuyo carácter es no determinista y en la que es posible encontrar ejemplos de situaciones contradictorias debido a la dualidad onda-corpúsculo. Para ello se establece un isomorfismo entre la lógica dialéctica y la teoría de la probabilidad, a la que se añade el concepto de fortuidad, precisamente para reflejar el carácter no determinista.

With the increase in speculation about the nature of our universe, there has been a growing need to find the truth about Dark Matter. Recent research shows that the Planck-Mass range could be a well-motivated space to probe for the detection of Dark Matter through gravitational coupling. This thesis dives into the possibility of doing the same in two parts. The first part lays out the analysis framework that would sense such an interaction, while the second part outlines a prototype experiment that when scaled up using quantum optomechanical sensors would serve as the skeleton to perform the analysis with.