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Since they were first envisioned, quantum computers have oft been portrayed as devices of limitless power, able to perform calculations in a mere instant that would take current computers years to determine. This is, of course, not the case. A huge amount of effort has been invested in trying to understand the limits of quantum computers---under which circumstances they outperform classical computers, how large a speed-up can be gained, and what draws the distinction between quantum and classical computing. In this Ph.D. thesis, I investigate a few intriguing properties of quantum computers involving quantum oracles and classically-simulatable quantum circuits. In Part I I study the notion of black-box unitary operations, and procedures for effecting the inverse operation. Part II looks at how quantum oracles can be used to test properties of probability distributions, and Part III considers classes of quantum circuits that can be simulated efficiently on a classical computer. In more detail, Part I studies procedures for inverting black-box unitary operations. Known techniques are generally limited in some way, often requiring ancilla systems, working only for restricted sets of operators, or simply being too inefficient. We develop a novel procedure without these limitations, and show how it can be applied to lift a requirement of the Solovay-Kitaev theorem, a landmark theorem of quantum compiling. Part II looks at property testing for probability distributions, and in particular considers a special type of access known as the \textit{conditional oracle}. The classical conditional oracle was developed by Canonne et al. in 2015 and subsequently greatly explored. We develop a quantum version of this oracle, and show that it has advantages over the classical process. We use this oracle to develop an algorithm that decides whether or not a mixed state is fully mixed. In Part III we study classically-simulatable quantum circuits in more depth. Two well-known classes are Clifford circuits and matchgate circuits, which we briefly review. Using these as inspiration, we use the Jordan-Wigner transform to develop new classes of non-trivial quantum circuits that are also classically simulatable.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2017-04-04T10:58:54Z No. of bitstreams: 2 Dissertação -Thais de Lima Silva - 2016.pdf: 2036779 bytes, checksum: d24a78c94accea059363389b0a888431 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-04-04T11:09:59Z (GMT) No. of bitstreams: 2 Dissertação -Thais de Lima Silva - 2016.pdf: 2036779 bytes, checksum: d24a78c94accea059363389b0a888431 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-04-04T11:09:59Z (GMT). No. of bitstreams: 2 Dissertação -Thais de Lima Silva - 2016.pdf: 2036779 bytes, checksum: d24a78c94accea059363389b0a888431 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-03-22 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Among the first attempts towards the unification of quantum and special relativity theories that which more highlighted was the Dirac’s electron theory. Amongst its achievements is the accurated calculation of the hydrogen spectrum and the antiparticles prediction. Even though it provides good results, some difficulties arise in this theory such as the existence of negative energy states for free particles. At the same time that it allows for the antiparticle discovery, it leads to questions like the trembling motion of a free electron (zitterbewegung). It is not a consensus whether this motion is real or just a theoretical failure in describing the reality. In this work we present a proposal to simulate the Dirac electron dynamics and, therefore, zitterbewegung, by means of a transformation of a paraxial light beam. The same transformation as is carried out on the vector state by the Dirac’s evolution operator is implemented in the transverse beam profile using wave plates and spatial light modulator. The proposal includes the cases of one and two spatial dimensions by simulating the spatial degrees of freedom into the transverse coordinates of the light beam, which offers a difficulty in simulating the three dimensional dynamics in general case. However, we show that it is possible for particular initial states. Our simulation presents some advantages over previously ones, namely: it does not require the construction of specificaly designed devices; the adjustable parameters are present on phases printed by the modulator and can be arbitrarily changed; it permits the measurement of the mean as well as the standard position operators; it permits the simulation in two and three spatial dimensions. / Dentre as primeiras tentativas de unificação da teoria quântica com a teoria de relatividade especial aquela que se destacou foi a teoria do elétron de Dirac. Entre seus sucessos está o cálculo com grande acurácia do espectro do átomo de hidrogênio e a predição da existência da antipartícula do elétron. Porém, apesar de seus bons resultados, alguns problemas surgem nessa teoria. A existência de estados com energia cinética negativa, ao mesmo tempo que permite a descoberta de antipartículas, leva a problemas como o movimento trêmulo zitterbewegung de um elétron livre. Não se sabe se esse movimento é real ou apenas uma inexatidão da teoria em descrever a realidade. Neste trabalho apresentamos uma proposta de emulação da dinâmica de um elétron de Dirac e, portanto, de zitterbewegung, por meio da transformação de um feixe luminoso paraxial. A mesma transformação efetuada pelo operador de evolução temporal de Dirac é implementada sobre o perfil transversal do feixe fazendo-o passar por placas de onda e imprimindo fases com um modulador espacial de luz. A proposta inclui os casos de uma e duas dimensões espaciais e, por emular a posição da partícula na posição sobre o plano transversal ao feixe, dispomos somente de dois graus de liberdade, dificultando a simulação no caso de o espaço ser tridimensional. Entretanto, mostramos que ainda assim é possível simular um caso particular nessa situação. Nossa simulação apresenta algumas vantagens sobre simulações implementadas anteriormente, são elas: não requer a construção de dispositivos para esse fim específico; os parâmetros ajustáveis estão presentes nas fases do modulador e podem ser variados arbitrariamente; permite a comparação entre o operador posição padrão e o operador posição média; permite a simulação nos casos de espaço bi e tridimensional.

Объект исследования – линейно-цепочечный углерод на подложке кремния. Цель работы – моделирование синтеза цепочечного углерода методом гибридной молекулярной динамики. Методы исследования: гибридная квантово-классическая молекулярная динамика. В ходе выполнения данной магистерской диссертации на языке Python были написаны программы, запускающие моделирование квантово-классической молекулярной динамики. В результате был разработан модуль для реализации электростатического подхода QM/MM вычислений. На основе данного модуля было произведено моделирование ионно-плазменного синтеза углеродной пленки. Анализа полученных структур показал, что содержание sp1, sp2, sp3 углерода в пленке максимально при энергиях аргона 40, 70, 90 эВ соответственно. / The object under study is linear-chained carbon on a silicon substrate. The aim of this work is linear-chained carbon synthesis simulation by the hybrid molecular dynamics. Research methods: hybrid quantum-classical molecular dynamics. In the course of this master's dissertation, Python programs were written. That programs run the simulation of quantum-classical molecular dynamics. As a result, a module was developed to implement the electrostatic approach of QM/MM calculations. On the basis of this module, modeling of the ion-beam plasma synthesis of a carbon film was carried out. The obtained structures analysis showed that the content of sp1, sp2, and sp3 carbon in the film is maximum at argon energies of 40, 70, and 90 eV, respectively.