This dissertation focuses on one question: how should one drive an experimentally prepared state of a generic quantum system into a different target-state, simultaneously minimizing energy dissipation and maximizing the fidelity between the target and evolved-states? We develop optimal adiabatic driving protocols for general quantum systems, and show that these are geodesic paths. Geometric ideas have always played a fundamental role in the understanding and unification of physical phenomena, and the recent discovery of topological insulators has drawn great interest to topology from the field of condensed matter physics. Here, we discuss the quantum geometric tensor, a mathematical object that encodes geometrical and topological properties of a quantum system. It is related to the fidelity susceptibility (an important quantity regarding quantum phase transitions) and to the Berry curvature, which enables topological characterization through Berry phases. A refined understanding of the interplay between geometry and topology in quantum mechanics is of direct relevance to several emergent technologies, such as quantum computers, quantum cryptography, and quantum sensors. As a demonstration of how powerful geometric and topological ideas can become when combined, we present the results of an experiment that we recently proposed. This experimental work was done at the Google Quantum Lab, where researchers were able to visualize the topological nature of a two-qubit system in sharp detail, a startling contrast with earlier methods. To achieve this feat, the optimal protocols described in this dissertation were used, allowing for a great improvement on the experimental apparatus, without the need for technical engineering advances. Expanding the existing literature on the quantum geometric tensor using notions from differential geometry and topology, we build on the subject nowadays known as quantum geometry. We discuss how slowly changing a parameter of a quantum system produces a measurable output of its response, merely due to its geometric nature. Next, we topologically characterize different classes of Hamiltonians using the Berry monopole charges, and establish their topological protection. Finally, we explore how such knowledge allows one to access topologically forbidden regions by adiabatically breaking and reestablishing symmetries.

Materiais compósitos são conhecidos pela alta resistência mecânica e baixo peso, desempenho superior, resistência à corrosão e baixa densidade. A produção de um material composto possui vários processos com particularidades diferentes. O enrolamento filamentar (Filament Winding) é um processo no qual fibras de reforço contínuas são posicionadas de maneira precisa e de acordo com um padrão predeterminado para compor a forma do componente desejado. As fibras, submetidas à tração, são enroladas continuamente ao redor de um mandril que tem a forma do produto final, em geral utilizando equipamentos automáticos. No final do processo, após a cura da resina, o mandril é geralmente removido. Desta forma, é de fundamental importância que o projetista disponha de recursos computacionais adequados para o cálculo das trajetórias e sequenciamento de posicionamento das fibras. Esse trabalho tem como objetivo o desenvolvimento de procedimentos matemáticos para cálculo de trajetórias geodésicas no processo de \"Filament Winding\" e implementá-los em um ambiente computacional em linguagem de alto nível Java, considerando-se os casos de revestimento circunferencial, helicoidal e polar. São desenvolvidos dois estudos de caso: tubos cônicos e vasos de pressão, e os resultados apresentados e discutidos, validando os procedimentos e ambiente implementado. / Composite materials are well known by the high strength and low weight, superior performance, resistance to the corrosion and low density. The production of a composite material part involves some processes with different requirements. The filament winding process is an automated process in which continuous reinforcement fibers are lay down in prescribed paths on the surface of a mandrel, which is generally removed after the cure of the resin. In such a way, it is fundamental that the designer uses computational resources for the calculation of the paths and sequence of the fibers. In this work is developed the mathematical procedures for calculation of geodesic trajectories in the Filament Winding process and implements them in a computational environment in high level language Java, considering the circumferential, helical and polar strategies. Two case studies are developed successfully: conical pipes and pressure vessels, and the results presented and discussed, validating the procedures and implemented environment.

Materiais compósitos são conhecidos pela alta resistência mecânica e baixo peso, desempenho superior, resistência à corrosão e baixa densidade. A produção de um material composto possui vários processos com particularidades diferentes. O enrolamento filamentar (Filament Winding) é um processo no qual fibras de reforço contínuas são posicionadas de maneira precisa e de acordo com um padrão predeterminado para compor a forma do componente desejado. As fibras, submetidas à tração, são enroladas continuamente ao redor de um mandril que tem a forma do produto final, em geral utilizando equipamentos automáticos. No final do processo, após a cura da resina, o mandril é geralmente removido. Desta forma, é de fundamental importância que o projetista disponha de recursos computacionais adequados para o cálculo das trajetórias e sequenciamento de posicionamento das fibras. Esse trabalho tem como objetivo o desenvolvimento de procedimentos matemáticos para cálculo de trajetórias geodésicas no processo de \"Filament Winding\" e implementá-los em um ambiente computacional em linguagem de alto nível Java, considerando-se os casos de revestimento circunferencial, helicoidal e polar. São desenvolvidos dois estudos de caso: tubos cônicos e vasos de pressão, e os resultados apresentados e discutidos, validando os procedimentos e ambiente implementado. / Composite materials are well known by the high strength and low weight, superior performance, resistance to the corrosion and low density. The production of a composite material part involves some processes with different requirements. The filament winding process is an automated process in which continuous reinforcement fibers are lay down in prescribed paths on the surface of a mandrel, which is generally removed after the cure of the resin. In such a way, it is fundamental that the designer uses computational resources for the calculation of the paths and sequence of the fibers. In this work is developed the mathematical procedures for calculation of geodesic trajectories in the Filament Winding process and implements them in a computational environment in high level language Java, considering the circumferential, helical and polar strategies. Two case studies are developed successfully: conical pipes and pressure vessels, and the results presented and discussed, validating the procedures and implemented environment.