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Atoms in optical lattices are promising candidates to implement quantum information processing. Their behaviour is well understood on a microscopic level, they exhibit excellent coherence properties, and they can be easily manipulated using external fields. In very deep optical lattices, each atom is restricted to a single lattice site and can be used as a qubit. If the lattice is shallow enough such that the atoms can move, their properties can be used to simulate certain condensed matter phenomena such as superconductivity. In this thesis, we show how technical problems of optical lattices such as restricted decoherence times, or fundamental shortcomings such as the lack of phonons or strong spin interactions, can be overcome by using current or near-future experimental techniques. We introduce a scheme that makes it possible to simulate model Hamiltonians known from high-temperature superconductivity. For this purpose, previous simulation schemes to realise the spin interaction terms are extended. We especially overcome the condition of a filling factor of exactly one, which otherwise would restrict the phase of the simulated system to a Mott-insulator. This scheme makes a large range of parameters accessible, which is difficult to cover with a condensed matter setup. We also investigate the properties of optical lattices submerged into a Bose-Einstein condensate (BEC). A weak-coupling expansion in the BEC-impurity interaction strength is used to derive a model that describes the lattice atoms in terms of polarons, i.e.~atoms dressed by Bogoliubov phonons. This is analogous to the description of electrons in solids, and we observe similar effects such as a crossover from coherent to incoherent transport for increasing temperatures. Moreover, the condensate mediates an attractive off-site interaction, which leads to macroscopic clusters at experimentally realistic parameters. Since the atoms in the lattice can also be used as a quantum register with the BEC mediating a two-qubit gate, we derive a quantum master equation to examine the coherence properties of the atomic qubits. We show that the system exhibits sub- and superdecoherence and that a fast implementation of the two-qubit gate competes with dephasing. Finally, we show how to realise the encoding of qubits in a decoherence-free subspace (DFS) using optical lattices. We develop methods for implementing robust gate operations on qubits encoded in a DFS exploiting collisional interactions between the atoms. We also give a detailed analysis of the performance and stability of the gate operations and show that a robust implementation of quantum repeaters can be achieved using our setup. We compare the robust repeater scheme to one that makes use of conventional qubits only, and show the conditions under which one outperforms the other.

The conjugacy length function of a group Γ determines, for a given a pair of conjugate elements u,v ∈ Γ, an upper bound for the shortest γ in Γ such that uγ = γv, relative to the lengths of u and v. This thesis focuses on estimating the conjugacy length function in certain finitely generated groups. We first look at a collection of solvable groups. We see how the lamplighter groups have a linear conjugacy length function; we find a cubic upper bound for free solvable groups; for solvable Baumslag--Solitar groups it is linear, while for a larger family of abelian-by-cyclic groups we get either a linear or exponential upper bound; also we show that for certain polycyclic metabelian groups it is at most exponential. We also investigate how taking a wreath product effects conjugacy length, as well as other group extensions. The Magnus embedding is an important tool in the study of free solvable groups. It embeds a free solvable group into a wreath product of a free abelian group and a free solvable group of shorter derived length. Within this thesis we show that the Magnus embedding is a quasi-isometric embedding. This result is not only used for obtaining an upper bound on the conjugacy length function of free solvable groups, but also for giving a lower bound for their L_p compression exponents. Conjugacy length is also studied between certain types of elements in lattices of higher-rank semisimple real Lie groups. In particular we obtain linear upper bounds for the length of a conjugator from the ambient Lie group within certain families of real hyperbolic elements and unipotent elements. For the former we use the geometry of the associated symmetric space, while for the latter algebraic techniques are employed.

Le but des modèles traditionnels en classification (comme les partitions et les hiérarchies de parties) est de permettre de discriminer sans ambiguïté et donc de produire des classes non empiétantes (i.e. l’intersection de deux classes est vide ou une classe est incluse dans l'autre). Cependant, cette exigence de non ambiguïté peut conduire à occulter de l’information. Dans le cas des plantes hybrides en biologie par exemple ou encore de textes appartenant à plusieurs genres en analyse textuelle. Les modèles généraux comme les hypergraphes ou les treillis permettent de prendre en compte l’empiétance entre les classes. Plus précisément, les modèles dits "totalement équilibrés" autorisent l'empiétance tout en conservant certaines contraintes utiles en classification.En apprentissage automatique, les arbres de décision, très utilisés pour leur simplicité d'utilisation et de compréhension réalisent à chaque étape un partitionnement d'un ensemble en deux sous-ensembles.Nous montrons dans ce travail différents liens entre la classification traditionnelle et l'apprentissage automatique supervisé et montrons certains apports que chacun des deux mondes peut faire à l'autre.Nous proposons deux méthodes de classification mêlant les deux univers puis étendons la notion de binarité, très utilisée dans le cas des arbres, aux hypergraphes et aux treillis. Nous montrons alors l'équivalence entre les systèmes binarisables et les systèmes totalement équilibrés, faisant de ces derniers de parfaits candidats à la réalisation de modèles de classification en classes empiétantes. Nous proposons également diverses approximations de systèmes par des systèmes totalement équilibrés. / Traditionally, classification models (such as partitions and hierarchies) aim at separating without ambiguities and produce non-overlapping clusters (i.e two clusters are either disjoint or one is included in the other). However, this non ambiguity may lead to mask information such as in the case of hybrid plants in biology or of texts which belong to two (or more) different genres in textual analysis for instance. General models like hypergraphs or lattices allow to take into account overlapping clusters. More precisely, "totally balanced" models allows class infringement and presents some useful constraints for classification.In machine learning, decision trees are a widely used model as they are simple to use and understand. They are also based on the idea of partition of sets.We show in this work different links between traditional classification and supervised machine learning and show what each world can bring to the other.We propose two methods of classification which link the two universes. We then extend the notion of binarity, widely-used for trees, to hypergraphs and lattices. We show the equivalence between binarizable systems and totally balanced systems, which makes of totally balanced structures a great candidate for classification models with class infringement. We also propose some approximation methods of any system (lattice, hypergraph, dissimilarity) by a totally balanced one.

Neste trabalho, apresentamos estudos sobre o magnetotransporte em um sistema de bicamadas com uma rede de antipontos triangulares em campos magnéticos baixos sob a aplicação de campos elétricos externos, que são produzidos por voltagens de porta. A bicamada é feita em poços quânticos largos (wide quantum well) de alta densidade eletrônica, formado em heteroestruturas semicondutoras de AlxGa1xAs=GaAs. Oscila- ções magneto-inter-sub-banda (MIS) são observadas em poços quânticos largos de alta densidade eletrônica com duas sub-bandas ocupadas. Estas são originadas pelo espalhamento inter-sub-bandas e tem um máximo para campos magnéticos B que satisfazem a condição de alinhamento entre os leques dos níveis de Landau de cada sub-banda. Oscila- ções de comensurabilidade são observadas na magnetoresistência que é sensível ao arranjo do potencial dos antipontos. A aplicação de campos elétricos faz diminuir o número de oscilações na magnetoresistência para campos magnéticos compreendidos entre 0; 1T e 0; 4T, observando-se uma transição das oscilações MIS aos efeitos de comensurabilidade. Aplicando voltagens de porta podemos variar a amplitude do potencial dos antipontos. / In this work, we present studying about magnetotransport in a bilayer system with triangular antidot lattices in low magnetic elds under the application of external electric eld. The bilayer forms inside a wide quantum well of high electron density in semiconductor heterostructures formed by AlxGa1xAs=GaAs. Magneto-inter-subband (MIS) oscillations are observed in a wide quantum wells of high electron density with two subbands occupied, and they are caused by intersubband scattering and have a maximum for a magnetic eld B that satises the alignment condition between the staircase of Landau level. Commensurability oscillations are observed in magnetoresistance, which is sensitive to the potential of antidot arrangements. The application of electric elds decrease the number of oscillations in the magnetoresistance for magnetic elds between 0; 1T and 0:4T, showing a transition of MIS oscillations to commensurability oscillations. We varied the amplitude of the potential of the antidots applying dierent gate voltages.