Usamos um modelo exatamente solúvel para calcular a dinâmica da fidelidade de uma computação baseada em medidas projetivas cujo sistema interage com um meio ambiente comum que insere erros de fase. Mostramos que a fidelidade do estado de Cluster canônico oscila como função do tempo e, como consequência, a computação quântica baseada em medidas projetivas pode apresentar melhores resultados computacionais mesmo para um conjunto sequencial de medidas lentas. Além disso, apresentamos uma condição necessária para que a dinâmica da fidelidade de um estado quântico geral apresente um comportamento não-monotônico. / We use an exact solvable model to calculate the gate fidelity dynamics of a measurement-based quantum computation that interacts with a common dephasing environment. We show that the fidelity of the canonical cluster state oscillates as a function of time and, as a consequence, the measurement-based quantum computer can give better computational results even for a set of slow measurement sequences. Furthermore, we present a necessary condition to the fidelity dynamics of a general quantum state presents a non-monotonical shape.

We investigate the geometric phase (GP) acquired by the states of a spin-1/2 nucleus which is subject to a static magnetic field. This nucleus as the carrier system of GP, is taken as coupled to a dissipative environment, so that it evolves non-unitarily. We study the effects of different characteristics of different environments on GP as nucleus evolves in time. We showed that magnetic field strength is the primary physical parameter that determines the stability of GP / its stability decreases as the magnetic field strength increases. (By decrease in stability what we mean is the increase in the time rate of change of GP.) We showed that this decrease can be very rapid, and so it could be impossible to make use of it as a quantum logic gate in quantum information theory (QIT). To see if these behaviors differ in different environments, we analyze the same system for a fixed temperature environment which is under the influence of an electromagnetic field in a squeezed state. We find that the general dependence of GP on magnetic field does not change, but this time the effects are smoother. Namely, increase in magnetic field decreases the stability of GP also for in this environment / but this decrease is slower in comparison with the former case, and furthermore it occurs gradually. As a second problem we examine the entanglement of two atoms, which can be used as a two-qubit system in QIT. The entanglement is induced by an external quantum system. Both two-level atoms are coupled to a third two-level system by dipole-dipole interaction. The two atoms are assumed to be in ordinary vacuum and the third system is taken as influenced by a certain environment. We examined different types of environments. We show that the steady-state bipartite entanglement can be achieved in case the environment is a strongly fluctuating, that is a squeezed-vacuum, while it is not possible for a thermalized environment.

The present work, briefly described below, consists of implementation of several quantum algorithms in an NMR Quantum Information Processor. Game theory gives us mathematical tools to analyze situations of conflict between two or more players who take decisions that influence their welfare. Classical game theory has been applied to various fields such as market strategy, communication theory, biological processes, foreign policies. It is interesting to study the behaviour of the games when the players share certain quantum correlations such as entanglement. Various games have been studied under the quantum regime with the hope of obtaining some insight into designing new quantum algorithms. Chapter 2 presents the NMR implementation of three such algorithms. Experimental NMR implementation given in this chapter are: (i) Three qubit ‘Dilemma’ game with corrupt sources’. The Dilemma game deals with the situation where three players have to choose between going/not going to a bar with a seating capacity of two. It is seen that in the players have a higher payoff if they share quantum correlations. However, the pay-off falls rapidly with increasing corruption in the source qubits. Here we report the experimental NMR implementation of the quantum version of the Dilemma game with and without corruption in the source qubits. (ii) Two qubit ‘Ulam’s game’. This is a two player game where one player has to find out the binary number thought by the other player. This problem can be solved with one query if quantum resources are used. This game has been implemented in a two qubit system in an NMR quantum information processor. (iii) Two qubit ‘Battle of Sexes’ game. This game deal with a situation where two players have conflicting choices but a deep desire to be together. This leads to a dilemma in the classical case. Quantum mechanically this dilemma is resolved and a unique solution emerges. The NMR implementation of the quantum version of this game is also given in this chapter. Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover’s search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, if one uses local adiabatic evolutions, their complexity is of the order √N (where N=2n) [37, 38]. In Chapter 3, the NMR implementation of (i) the Deutsch-Jozsa and the (ii) Grover’s search algorithm using local adiabatic evolution has been presented. In adiabatic algorithm, the system is first prepared in the equal superposition of all the possible states which is the ground state of the beginning Hamiltonian. The solution is encoded in the ground state of the final Hamiltonian. The system is evolved under a linear combination of the beginning and the final Hamiltonian. During each step of the evolution the interpolating Hamiltonian slowly changes from the beginning to the final Hamiltonian, thus evolving the ground state of the beginning Hamiltonian towards the ground state of the final Hamiltonian. At the end of the evolution the system is in the ground state of the final Hamiltonian which is the solution. The final Hamiltonian, for each of the two cases of adiabatic algorithm described in this chapter, are constructed depending on the problem definition. Adiabatic algorithms have been proved to be equivalent to standard quantum algorithms with respect to complexity [39]. NMR implementation of adiabatic algorithms in homonuclear spin systems face problems due to decoherence and complicated pulse sequences. The decoherence destroys the answer as it causes the final state to evolve to a mixed state and in homonuclear systems there is a substantial evolution under the internal Hamiltonian during the application of the soft pulses which prevents the initial state to converge to the solution state. The resolution of these issues are necessary before one can proceed for the implementation of an adiabatic algorithm in a large system. Chapter 4 demonstrates that by using ‘strongly modulated pulses’ for creation of interpolating Hamiltonian, one can circumvent both the problems and thus successfully implement the adiabatic SAT algorithm in a homonuclear three qubit system. The ‘strongly modulated pulses’ (SMP) are computer optimized pulses in which the evolution under the internal Hamiltonian of the system and RF inhomogeneities associated with the probe is incorporated while generating the SMPs. This results in precise implementation of unitary operators by these pulses. This work also demonstrates that the strongly modulated pulses tremendously reduce the time taken for the implementation of the algorithm, can overcome problems associated with decoherence and will be the modality in future implementation of quantum information processing by NMR. Quantum search algorithm, involving a large number of qubits, is highly sensitive to errors in the physical implementation of the unitary operators. This can put an upper limit to the size of the data base that can be practically searched. The lack of robustness of the quantum search algorithm for a large number of qubits, arises from the fact that stringent ‘phase-matching’ conditions are imposed on the algorithm. To overcome this problem, a modified operator for the search algorithm has been suggested by Tulsi [40]. He has theoretically shown that even when there are errors in implementation of the unitary operators, the search algorithm with his modified operator converges to the target state while the original Grover’s algorithm fails. Chapter 5, presents the experimental NMR implementation of the modified search algorithm with errors and its comparison with the original Grover’s search algorithm. We experimentally validate the theoretical predictions made by Tulsi that the introduction of compensatory Walsh-Hadamard and phase-flip operations refocuses the errors. Experimental Quantum Information Processing is in a nascent stage and it would be too early to predict its future. The excitement on this topic is still very prevalent and many options are being explored to enhance the hardware and software know-how. This thesis endeavors in this direction and probes the experimental feasibility of the quantum algorithms in an NMR quantum information processor.

This dissertation presents the implementation of a technique to generate atomic Fock states of Lithium 6 with ultra-high fidelity, called laser culling. Fock states, atomic states with a definite number of particles, are a mandatory step for studying few-body quantum phenomena such as quantum tunneling, quantum entanglement, and serve as building blocks for quantum simulators. The creation of ultra-high fidelity Fock states begins with a degenerate Fermi gas in an optical dipole trap. Being fermions, lithium-6 atoms fill the energy levels of the dipole trap with 2 atoms per energy level. Introducing a magnetic field gradient creates a linear potential that tilts the potential produced by the optical dipole trap. The initially bound energy levels become quasi-bound states, each with a different lifetime. By exploiting the difference between these lifetimes, one can generate a single pair of atoms in the ground state of the trap with fidelities that can exceed 99.9%. This dissertation first presents the details of the design and construction of an apparatus for laser culling, and then reports on the progress made towards the creation of atomic Fock states with ultra-high fidelity. / text

In this thesis we have focused on two topics: Discrete Quantum Walks and Quantum Image Processing. Our work is a contribution within the field of quantum computation from the perspective of a computer scientist. With the purpose of finding new techniques to develop quantum algorithms, there has been an increasing interest in studying Quantum Walks, the quantum counterparts of classical random walks. Our work in quantum walks begins with a critical and comprehensive assessment of those elements of classical random walks and discrete quantum walks on undirected graphs relevant to algorithm development. We propose a model of discrete quantum walks on an infinite line using pairs of quantum coins under different degrees of entanglement, as well as quantum walkers in different initial state configurations, including superpositions of corresponding basis states. We have found that the probability distributions of such quantum walks have particular forms which are different from the probability distributions of classical random walks. Also, our numerical results show that the symmetry properties of quantum walks with entangled coins have a non-trivial relationship with corresponding initial states and evolution operators. In addition, we have studied the properties of the entanglement generated between walkers, in a family of discrete Hadamard quantum walks on an infinite line with one coin and two walkers. We have found that there is indeed a relation between the amount of entanglement available in each step of the quantum walk and the symmetry of the initial coin state. However, as we show with our numerical simulations, such a relation is not straightforward and, in fact, it can be counterintuitive. Quantum Image Processing is a blend of two fields: quantum computation and image processing. Our aim has been to promote cross-fertilisation and to explore how ideas from quantum computation could be used to develop image processing algorithms. Firstly, we propose methods for storing and retrieving images using non-entangled and entangled qubits. Secondly, we study a case in which 4 different values are randomly stored in a single qubit, and show that quantum mechanical properties can, in certain cases, allow better reproduction of original stored values compared with classical methods. Finally, we briefly note that entanglement may be used as a computational resource to perform hardware-based pattern recognition of geometrical shapes that would otherwise require classical hardware and software.

This thesis deals with a series of quantum computer implementation issues from the Kane 31P in 28Si architecture to Shor’s integer factoring algorithm and beyond. The discussion begins with simulations of the adiabatic Kane CNOT and readout gates, followed by linear nearest neighbor implementations of 5-qubit quantum error correction with and without fast measurement. A linear nearest neighbor circuit implementing Shor’s algorithm is presented, then modified to remove the need for exponentially small rotation gates. Finally, a method of constructing optimal approximations of arbitrary single-qubit fault-tolerant gates is described and applied to the specific case of the remaining rotation gates required by Shor’s algorithm.

Nessa dissertação, abordamos o problema de dois qubits interagindo com campos externos e entre si controladamente, de acordo com um Hamiltoniano considerado realista para implementação da porta lógica quântica XOR. Introduzimos acoplamentos entre as observáveis do sistema de dois qubits e um banho de osciladores harmônicos a fim de tratarmos o problema da dissipação e da decoerência. Primeiramente nós consideramos o limite no qual a decoerência é mais rápida que qualquer processo gerado pelo Hamiltoniano do sistema. Prosseguimos então, através do método numérico conhecido como Integrador Unitário, com o estudo da matriz densidade do sistema durante a operação da porta lógica quântica sem incluir, inicialmente, o acoplamento com o banho de osciladores harmônicos. Finalmente, implementamos o método numérico conhecido como Propagador quase adiabático para estudar a decoerência e a dissipação durante a operação da porta lógica quântica XOR, a fim de analisarmos os aspectos perturbativos do sistema quântico de dois qubits. / In this dissertation, we approach the problem of two qubits interading with themselves and with externa1 fields in a controlled way, according to a Hamiltonian considered realistic to implement the XOR quantum gate. We introduce couplings between the observables of the two-qubits system and of a bath of harmonic oscillators, to treat the problems of dissipation and decoherence. Preliminarly, we consider the limit in which decoherence is faster than any process dictated by the Hamiltonian evolution of the system. Then, through a unitary-integrator numerical method, we proceed with the study of the evolution of the density matrix of the system during the operation of the logical quantum gate, initially, without the coupling with the bath of harmonic oscillators. Finally, we use the quasiadiabatic path integral method to study the dissipation and decoherence during the logical operation, through the inclusion of the bath.

Orientador: Carlile Campos Lavor / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-17T14:47:27Z (GMT). No. of bitstreams: 1 Melo_Nolmar_D.pdf: 2834013 bytes, checksum: 5639deabb953aa019e4e1c9c905e856d (MD5) Previous issue date: 2011 / Resumo: Este trabalho apresenta uma linguagem algébrica para dois elementos básicos da teoria da informação quântica (os bits quânticos e os operadores densidade), baseada nas propriedades de uma álgebra de Clifford de assinatura (n,3n). Demonstramos que a nova descrição desses elementos preserva as mesmas propriedades matemáticas obtidas com a descrição clássica. Com isso, estendemos alguns resultados apresentados na literatura que relaciona Álgebra de Clifford e Informação Quântica. / Abstract: This work presents an algebraic language for two basic elements of quantum information theory (the quantum bits and density operators), based in the properties of a Clifford algebra of signature (n,3n). We prove that the new description of these elements preserves the same mathematical properties obtained with the classical description. We also extend some results presented in the literature that relate Clifford algebra and quantum information. / Doutorado / Matematica Aplicada / Doutor em Matemática

Orientador: Walter Alexandre Carnielli / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias Humanas / Made available in DSpace on 2018-08-06T09:39:40Z (GMT). No. of bitstreams: 1 Agudelo_JuanCarlosAgudelo_M.pdf: 2515941 bytes, checksum: 58c117425f8731bb67cf3fc0e1181ad4 (MD5) Previous issue date: 2006 / Resumo: As diferentes interpretações da mecânica quântica levanta sérios problemas filosóficos a respeito da natureza do mundo físico e do estatuto das teorias físicas. Tais interpretações desempenham um papel importante na compreensão dos modelos de computação quântica, e por sua vez os modelos de computação quântica abrem a possibilidade de se confrontar as teses filosóficas que se atrevem a responder a tais problemas. Apesar das relevantes e surpreendentes promessas de uso pragmático e tecnológico da computação quântica, não é por essa vereda que caminha este trabalho: o que aqui se oferece é um novo paradigma de computação (um modelo de computação baseado no paradigma paraconsistente), e se propõe uma nova interpretação da computação quântica através desse novo paradigma, dessa forma colaborando simultaneamente na discussão filosófica a respeito da noção de computabilidade e da mecânica quântica. O presente trabalho introduz a definição do que será chamado de modelo de máquinas de Turing paraconsistentes (MTPs). Tal modelo de computação é uma generalização do modelo clássico de máquinas de Turing. No modelo de máquinas de Turing paraconsistentes, diferentemente do modelo clássico, permite-se a execução de múltiplas instruções de mancha simultânea, dando lugar a multiplicidade de símbolos em diferentes casas da fita, multiplicidade de estados e multiplicidade dc posições da máquina. Considerando que tal multiplicidade de configurações, embora essencial nas MTPs pode ser interpretado como incoerências com respeito às máquinas de Turing clássicas, permite-se acrescentar condições de consistência e inconsistência na execução das instruções nas MTPs. o que servirá então para controlar o estado dc incoerência do sistema. Depois de apresentar o modelo de MTPs, são descritos os modelo de máquinas de Turing quânticas (MTQs) e o modelo dc circuitos quânticos (CQs), ambos introduzidos inicialmente por David Dcutsch. os quais são, respectivamente, generalizações do modelo de máquinas dc Turing clássicas e de circuitos boolcanos clássicos usando as leis da mecânica quântica. Finalmente, estabelecem-se relações entre o modelo de MTPs e os modelos de computação quântica, simulando algoritmos quânticos simples (um CQ que soluciona o chamado problema de Dcutsch e um CQ que soluciona o chamado problema de Deutsch-Josza) e mostrando que o paralelismo quântico, uma característica essencial da computação quântica., pode em alguns casos ser simulado por meio de MTPs. Dessa forma, apesar de o particular modelo de MTPs aqui apresentado ter algumas restrições na simulação de certas características da computação quântica, abre-se a possibilidade de se definir outros modelos de MTPs de maneira a simular tais características. Em resumo, o presente trabalho, na medida cm que oferece um novo paradigma de computação (a saber, a computação paraconsistente) e uma nova interpretação dos modelos de computação quântica (a saber, a interpretação da computação quântica através da computação paraconsistente) contribui para a discussão filosófica a respeito da interpretação dos modelos de computação quântica, e possivelmente da interpretação da própria mecânica quântica. Contudo, não menos importante é o fato de que, apesar de o presente trabalho não pretender se dedicar a questões puramente técnicas da computabilidade, ele de fato abre um imenso campo de investigação a respeito da computação relativizada à lógica e suas implicações - no caso presente, relativizada à lógica paraconsistente / Abstract: The interpretations of quantum mechanics open serious philosophical questions about the nature of the physical world and about the status of physical theories. Such interpretations play an important role in the understanding of models of quantum computing, and models of quantum computing, by their turn, open possibilities to confront and test philosophical theses that dare to address such problems. Although the promising pragmatic and technological applications of quantum computing, this work goes in another way: what is here offered is a new paradigm of computation based upon the pa-raconsistency paradigm, and a new interpretation of quantum computing through this model, in this way simultaneously collaborating in the philosophical discussion about the concepts of computability and of quantum mechanics. This work introduces the definition of what will be called the model of paraconsistent Turing machines (PTMs). Such computational model is a generalization of the classical model of Turing machines. In the PTMs model, differently from the classical Turing machines model, simultaneous execution of multiple instructions is allowed, giving rise to a multiplicity of symbols on different cells of the tape, multiplicity of machine states and multiplicity of machine positions. Such multiplicity of configurations, though essential in the PTMs. can be seen as incoherencies with respect to classical Turing machines: to compensate this, the PTMs permit to operate with consistency and inconsistency conditions to control the global state of incoherence of the system. After introducing the PTMs, the model of quantum Turing machines (QTMs) and the model of quantum circuits (QCs) arc presented. This models arc due to David Dcutsch and arc. respectively, generalizations of the model of classical Turing machines and the model of classical boolean circuits, using the laws of quantum mechanics. Finally, relations between the model of PTMs and models of quantum computing arc established, which permits to simulate simple quantum algorithms (a QC to solve the so called Dcutsch problem and a QC to solve the so called Dcutsch-Jozsa. problem) by PTMs. It is also shown that quantum parallelism, an essential characteristic of quantum computation, may be simulated in some cases by PTMs. Although the particular model of PTMs here presented has some restrictions in the capacity to simulate certain quantum computing characteristics, our work opens the possibility to define other PTM models which could simulate such characteristics. To sum up, the present work, while offering a new paradigm of computation (namely, parconsistent computation) and a new interpretation of quantum computing (namely, interpretation of quantum computation by means of paraconsistent computation) contributes to the philosophical discussion about the interpretation of quantum computation and of the quantum mechanics itself. However, not less important is the fact that, besides its lack of explicit intention towards tccnical questions of computability theory, opens a new line of research about the possibilities of logic-relativized computation and its implications- in the present case, relativized to paraconsistent logics / Mestrado / Logica / Mestre em Filosofia

In 2004 Robert W. Spekkens introduced a toy theory designed to make a case for the epistemic view of quantum mechanics. But how does Spekkens’ toy model differ from quantum theory? While some differences are well-established, we attempt to approach this question from a tomographic point of view. More specifically, we provide experimentally viableprocedureswhichenablesustocompletelycharacterizethestatesandgatesthatare available in the toy model. We show that, in contrast to quantum theory, decompositions of transformations in the toy model must be done in a non-linear fashion.