A computação quântica adiabática tem sua pedra angular no teorema adiabático, cuja eficiência está relacionada tradicionalmente à proporção da variação temporal do Hamiltoniano que descreve o sistema e o gap mínimo entre o estado fundamental e o primeiro excitado. Normalmente, esse gap tende a diminuir quando aumenta o número de recursos (bit quântico: qubit) de um processador quântico, exigindo dessa maneira variações lentas do Hamiltoniano para assim garantir uma dinâmica adiabática. Entre os candidatos para a sua implementação física, estão os qubits baseados em circuitos supercondutores os quais têm um grande potencial, por causa de seu alto controle e escalabilidade promissora. No entanto, quando esses qubits são implementados, eles têm uma fonte intrínseca de ruído devido a erros de fabricação, que não podem ser desprezados. Por isso, nesta tese nós estudamos como os efeitos causados pelas flutuações dos parâmetros físicos do qubit afetam o comportamento da fidelidade da computação, realizando com esse propósito a simulação da dinâmica de cadeias de spin pequenas desordenadas. A partir do análise exaustivo desse estúdio foi possível propor uma estratégia que permite aumentar a fidelidade considerando um sistema ruidoso. Por outro lado, motivados pelo interesse de obter critérios suficientes e necessários para satisfazer uma computação quântica adiabática e pelo fato que ainda não existe uma condição de adiabaticidade geral apesar de existir inúmeras propostas, nós apresentamos um novo critério que manifesta suficiência para sistemas mais gerais e finalmente apresentamos evidências de que tal condição seria um quantificador consistente. / Adiabatic quantum computation has its cornerstone in the adiabatic theorem, whose efficiency is traditionally related to the ratio of the Hamiltonian temporal variation that describes the system and the minimum gap between the ground state and the first excited state. Usually, this gap tends to decrease when the number of quantum resources (quantum bit: qubit) of a quantum processor increases, thus it requires slow variations of the Hamiltonian to ensure an adiabatic dynamic. Among the candidates for its physical implementation are the qubits superconducting circuit-based which have great potential because of their high control and promising scalability. However, when these qubits are implemented, they have an intrinsic source of noise due to manufacturing errors that can not be despised. Therefore, in this thesis we study how the effects caused by the fluctuations of the physical parameters of the qubit affect the behavior of the fidelity of the computation, accomplishing with this purpose the simulation of the dynamics of small disordered spin chains. From the exhaustive analysis of this studio, it was possible to propose a strategy that allows to increase the fidelity considering a noisy system. On the other hand, motivated by the interest of obtaining sufficient and necessary criteria to satisfy an adiabatic quantum computation and the fact that there is still no general adiabaticity condition despite there being numerous proposals, we present a new criterion that manifests sufficiency for more general systems and we finally presented evidence that such a condition would be a consistent quantifier.

Experimental and theoretical progress has been made toward the production and manipulation of novel atomic and molecular states. The design, construction and characterization of a driver for an acousto-optic modulator is presented which achieves a maximum diffraction efficiency of 54 % at 200 MHz, using a commercial modulator. A novel design is presented for a highly sensitive optical spectrum analyzer for displaying laser mode structure in real time. Utilizing programmable microcontrollers to read data from a CMOS image sensor illuminated by the diffraction pattern from a Fabry-Perot interferometer, this device can operate with beam powers as low as 3.3 micro-watts, at a fraction of the cost of equivalent products. Computational results are presented analyzing the behaviour of a model quantum system in the vicinity of an avoided crossing. The results are compared with calculations based on the Landau-Zener formula, with discussion of its limitations. Further computational work is focused on simulating expected conditions in the implementation of the STIRAP technique for coherent control of atoms and molecules in the beam experiment. The work presented provides tools to further the aim of producing large, mono-energetic populations of heavy Rydberg systems. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2007-10-03 17:17:56.841

A computação quântica adiabática tem sua pedra angular no teorema adiabático, cuja eficiência está relacionada tradicionalmente à proporção da variação temporal do Hamiltoniano que descreve o sistema e o gap mínimo entre o estado fundamental e o primeiro excitado. Normalmente, esse gap tende a diminuir quando aumenta o número de recursos (bit quântico: qubit) de um processador quântico, exigindo dessa maneira variações lentas do Hamiltoniano para assim garantir uma dinâmica adiabática. Entre os candidatos para a sua implementação física, estão os qubits baseados em circuitos supercondutores os quais têm um grande potencial, por causa de seu alto controle e escalabilidade promissora. No entanto, quando esses qubits são implementados, eles têm uma fonte intrínseca de ruído devido a erros de fabricação, que não podem ser desprezados. Por isso, nesta tese nós estudamos como os efeitos causados pelas flutuações dos parâmetros físicos do qubit afetam o comportamento da fidelidade da computação, realizando com esse propósito a simulação da dinâmica de cadeias de spin pequenas desordenadas. A partir do análise exaustivo desse estúdio foi possível propor uma estratégia que permite aumentar a fidelidade considerando um sistema ruidoso. Por outro lado, motivados pelo interesse de obter critérios suficientes e necessários para satisfazer uma computação quântica adiabática e pelo fato que ainda não existe uma condição de adiabaticidade geral apesar de existir inúmeras propostas, nós apresentamos um novo critério que manifesta suficiência para sistemas mais gerais e finalmente apresentamos evidências de que tal condição seria um quantificador consistente. / Adiabatic quantum computation has its cornerstone in the adiabatic theorem, whose efficiency is traditionally related to the ratio of the Hamiltonian temporal variation that describes the system and the minimum gap between the ground state and the first excited state. Usually, this gap tends to decrease when the number of quantum resources (quantum bit: qubit) of a quantum processor increases, thus it requires slow variations of the Hamiltonian to ensure an adiabatic dynamic. Among the candidates for its physical implementation are the qubits superconducting circuit-based which have great potential because of their high control and promising scalability. However, when these qubits are implemented, they have an intrinsic source of noise due to manufacturing errors that can not be despised. Therefore, in this thesis we study how the effects caused by the fluctuations of the physical parameters of the qubit affect the behavior of the fidelity of the computation, accomplishing with this purpose the simulation of the dynamics of small disordered spin chains. From the exhaustive analysis of this studio, it was possible to propose a strategy that allows to increase the fidelity considering a noisy system. On the other hand, motivated by the interest of obtaining sufficient and necessary criteria to satisfy an adiabatic quantum computation and the fact that there is still no general adiabaticity condition despite there being numerous proposals, we present a new criterion that manifests sufficiency for more general systems and we finally presented evidence that such a condition would be a consistent quantifier.

Made available in DSpace on 2017-07-21T19:25:55Z (GMT). No. of bitstreams: 1 Gustavo Uhdre.pdf: 2107286 bytes, checksum: 7ad35f5b79eaca9ffa73277e4eda912d (MD5) Previous issue date: 2017-03-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work aims to compare the eficiency between two alternative ways of performing quantum computing protocols. The one is called adiabatic quantum computation, which is to realize through the concepts of the adiabatic theorem. The second is called nonadiabatic quantum computation, which is performed through ideas of dynamic invariants. These protocols will be presented in a theoretical context of Nuclear Magnetic Resonance, without the experimental realization. / Este trabalho tem como objetivo comparar a eficiência entre duas maneiras alternativas de realizar protocolos de computação quântica. A primeira, é chamada de computação quântica adiabática, que é realizada através dos conceitos do teorema adiabático. A segunda, é chamada de computação quântica não adiabática, que é realizada através das ideias de invariantes dinâmicos. Esses protocolos serão apresentados em um contexto teórico de Ressonância Magnética Nuclear, sem a realização experimental.

abstract: The theory of geometric quantum mechanics describes a quantum system as a Hamiltonian dynamical system, with a projective Hilbert space regarded as the phase space. This thesis extends the theory by including some aspects of the symplectic topology of the quantum phase space. It is shown that the quantum mechanical uncertainty principle is a special case of an inequality from J-holomorphic map theory, that is, J-holomorphic curves minimize the difference between the quantum covariance matrix determinant and a symplectic area. An immediate consequence is that a minimal determinant is a topological invariant, within a fixed homology class of the curve. Various choices of quantum operators are studied with reference to the implications of the J-holomorphic condition. The mean curvature vector field and Maslov class are calculated for a lagrangian torus of an integrable quantum system. The mean curvature one-form is simply related to the canonical connection which determines the geometric phases and polarization linear response. Adiabatic deformations of a quantum system are analyzed in terms of vector bundle classifying maps and related to the mean curvature flow of quantum states. The dielectric response function for a periodic solid is calculated to be the curvature of a connection on a vector bundle. / Dissertation/Thesis / Ph.D. Mathematics 2011

L'approximation adiabatique en mécanique quantique stipule que si un système quantique évolue assez lentement, alors il demeurera dans le même état propre. Récemment, une faille dans l'application de l'approximation adiabatique a été découverte. Les limites du théorème seront expliquées lors de sa dérivation. Ce mémoire à pour but d'optimiser la probabilité de se maintenir dans le même état propre connaissant le système initial, final et le temps d'évolution total. Cette contrainte sur le temps empêche le système d'être assez lent pour être adiabatique. Pour solutionner ce problème, une méthode variationnelle est utilisée. Cette méthode suppose connaître l'évolution optimale et y ajoute une petite variation. Par après, nous insérons cette variation dans l'équation de la probabilité d'être adiabatique et développons en série. Puisque la série est développée autour d'un optimum, le terme d'ordre un doit nécessairement être nul. Ceci devrait nous donner un critère sur l'évolution la plus adiabatique possible et permettre de la déterminer. Les systèmes quantiques dépendants du temps sont très complexes. Ainsi, nous commencerons par les systèmes ayant des énergies propres indépendantes du temps. Puis, les systèmes sans contrainte et avec des fonctions d'onde initiale et finale libres seront étudiés. / The adiabatic approximation in quantum mechanics states that if the Hamiltonian of a physical system evolves slowly enough, then it will remain in the instantaneous eigenstate related to the initial eigenstate. Recently, two researchers found an inconsistency in the application of the approximation. A discussion about the limit of this idea will be presented. Our goal is to optimize the probability to be in the instantaneous eigenstate related to the initial eigenstate knowing the initial and final system, with the total time of the experiment fixed to $T$. This last condition prevents us from being slow enough to use the adiabatic approximation. To solve this problem, we turn to the calculus of variation. We suppose the ideal evolution is known and we add a small variation to it. We take the result, put it in the probability to be adiabatic and expand in powers of the variation. The first order term must be zero. This enables us to derive a criterion which will give us conditions on the ideal Hamiltonian. Those conditions should define the ideal Hamiltonian. Time dependent quantum systems are very complicated. To simplify the problem, we will start by considering systems with time independent energies. Afterward, the general case will be treated.

L'approximation adiabatique en mécanique quantique stipule que si un système quantique évolue assez lentement, alors il demeurera dans le même état propre. Récemment, une faille dans l'application de l'approximation adiabatique a été découverte. Les limites du théorème seront expliquées lors de sa dérivation. Ce mémoire à pour but d'optimiser la probabilité de se maintenir dans le même état propre connaissant le système initial, final et le temps d'évolution total. Cette contrainte sur le temps empêche le système d'être assez lent pour être adiabatique. Pour solutionner ce problème, une méthode variationnelle est utilisée. Cette méthode suppose connaître l'évolution optimale et y ajoute une petite variation. Par après, nous insérons cette variation dans l'équation de la probabilité d'être adiabatique et développons en série. Puisque la série est développée autour d'un optimum, le terme d'ordre un doit nécessairement être nul. Ceci devrait nous donner un critère sur l'évolution la plus adiabatique possible et permettre de la déterminer. Les systèmes quantiques dépendants du temps sont très complexes. Ainsi, nous commencerons par les systèmes ayant des énergies propres indépendantes du temps. Puis, les systèmes sans contrainte et avec des fonctions d'onde initiale et finale libres seront étudiés. / The adiabatic approximation in quantum mechanics states that if the Hamiltonian of a physical system evolves slowly enough, then it will remain in the instantaneous eigenstate related to the initial eigenstate. Recently, two researchers found an inconsistency in the application of the approximation. A discussion about the limit of this idea will be presented. Our goal is to optimize the probability to be in the instantaneous eigenstate related to the initial eigenstate knowing the initial and final system, with the total time of the experiment fixed to $T$. This last condition prevents us from being slow enough to use the adiabatic approximation. To solve this problem, we turn to the calculus of variation. We suppose the ideal evolution is known and we add a small variation to it. We take the result, put it in the probability to be adiabatic and expand in powers of the variation. The first order term must be zero. This enables us to derive a criterion which will give us conditions on the ideal Hamiltonian. Those conditions should define the ideal Hamiltonian. Time dependent quantum systems are very complicated. To simplify the problem, we will start by considering systems with time independent energies. Afterward, the general case will be treated.

The Landau-Zener-Stückelberg-Majorana (LZSM) problem models diabatic transitions between energy levels in quantum two-level systems with an avoided level-crossing. The diabatic transition is a consequence of quantum tunneling in energy space when the system's Hamiltonian is perturbed with a fast-acting bias. The probability of transition between the energy states for a linear bias is known as the LZSM transition probability. The objective of this work is to investigate the LZSM problem through analytical and numerical lenses. The LZSM transition probability is derived in two ways. The first approach is based on Majorana's solution using contour integrals. The second derivation follows Landau's quasi-classical treatment. The derivations demonstrate methods for transitions in the presence of time-dependent perturbations. The ubiquity of the two-level system is discussed and an application on qubits concerning LSZM interferometry is presented, with the latter arising after considering periodic biases. Lastly, a simulation of the two-level system is conducted using Trotter-decomposed time-evolution operators, perturbation theory, and vectorization. The simulated transition probabilities for linear and periodic biases are obtained for varied parameters. The results show that the simulation achieves an accurate and efficient emulation of the LZSM problem.