Academic literature on the topic 'Quantum Computation,Quantum Circuit,Boolean Circuit'

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2008.<br>Includes bibliographical references (p. 133-144).<br>The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of quantum computers. However, several other models of quantum computation exist which provide useful alternative frameworks for both discovering new quantum algorithms and devising new physical implementations of quantum computers. In this thesis, I first present necessary background material for a general physics audience and discuss existing models of quantum computation. Then, I present three new results relating to various models of quantum computation: a scheme for improving the intrinsic fault tolerance of adiabatic quantum computers using quantum error detecting codes, a proof that a certain problem of estimating Jones polynomials is complete for the one clean qubit complexity class, and a generalization of perturbative gadgets which allows k-body interactions to be directly simulated using 2-body interactions. Lastly, I discuss general principles regarding quantum computation that I learned in the course of my research, and using these principles I propose directions for future research.<br>by Stephen Paul Jordan.<br>Ph.D.

This thesis considers various paradigms of quantum computation in an attempt to understand the nature of the underlying physics. A standard approach is to consider unitary computation on pure input states, such that the outcome of the computation is determined by single computational basis measurement on the output state. It has been shown that there exists equivalent models of computation, such as measurement based quantum computing (MBQC), which provide insight into the role of entanglement and measurement. Furthermore, constraining or relaxing available resources can directly impacts the power of the computation, allowing one to gauge their role in the process. Here, we first extend known constructions such as Matrix Product States, MBQC and the oneclean qubit model to a mixed state formalism, in an attempt to develop computational models where noise acting on the physical resources, as might be experienced in laboratory settings, may be mapped to logical noise on the computation. Next, we introduce Measurement-Based Classical Computing, an essentially classical model of computation, wherein the complexity hard wired into probability distributions generated via quantum means yields surprising non classical results. Finally, we consider postselection the ability to discard displeasing measurement outcomes and argue that it may be used in a tame way, which does not provide a dramatic increase in computational power. From here, we develop a new Hamiltonian, based on a circuit to Hamiltonian construction, presenting evidence of QMA-hardness.

In this thesis, classical computation is first introduced. Finite quantum systems are considered with D-dimensional Hilbert space, and position x and momentum p taking values in Z(D) (the integers modulo D). An analytic rep resentation of finite quantum systems that use Theta function is presented and considered. The first novel part of this thesis is contribution to study reversible classical CNOT gates and their binary inputs and outputs with reversible cir cuits. Furthermore, a reversible classical Toffoli gates are considered, as well as implementation of a Boolean expression with classical CNOT and Toffoli gates. Reversible circuits with classical CNOT and Toffoli gates are also considered. The second novel part of this thesis the study of quantum computation in terms of CNOT and Toffoli gates. Analytic representations and their zeros are considered, while zeros of the inputs and outputs for quantum CNOT and Toffoli gates are studied. Also, approximate computation of their zeros on the output are calculated. Finally, some quantum circuits are discussed. i

Submitted by Biblioteca do Instituto de Física (bif@ndc.uff.br) on 2017-03-16T19:23:13Z No. of bitstreams: 1 Tese_rdsilva_final.pdf: 10087516 bytes, checksum: d388e8ea89e24003b63d88b1aa7ea4a4 (MD5)<br>Made available in DSpace on 2017-03-16T19:23:13Z (GMT). No. of bitstreams: 1 Tese_rdsilva_final.pdf: 10087516 bytes, checksum: d388e8ea89e24003b63d88b1aa7ea4a4 (MD5)<br>Conselho Nacional de Desenvolvimento Científico e Tecnológico<br>In this thesis I study the one-way quantum computation (1WQC) model and some applications of the different ways of translating 1WQC algorithms into the circuit model. In a series of recent results, different sets of conditions for implementing a computation deterministically in the one-way model have been proposed, each of them with their own properties. Some of those sets of conditions - generically known as flow conditions - try to explore the distinct parallel power of the 1WQC model, by increasing the number of operations that can be performed simultaneously. Here I contribute to this line of research by defining a new type of flow, which I call the signal-shifted flow (SSF), which has an interesting parallel structure that equals that of a depth-optimal flow.I also introduce a new framework for translating 1WQC algorithms into the circuit model. This translation preserves not only the computation performed but also some features of the 1WQC algorithm design. Within this framework I give two algorithms, each implementing a different translation procedure: the first gives compact (in space use) circuits for Regular Flow one-way computations, and the second does the same for SSF one-way computations. As an application of the SSF translation procedure, I combine it with other translation and optimization techniques to give an automated quantum circuit optimization procedure. This procedure is based on back-and-forth translation between the 1WQC and the circuit model, using 1WQC techniques to time-optimize computations in the circuit model. In the second part of this thesis, I use 1WQC tools to analyze quantum circuits interacting with closed timelike curves (CTCs). I do so by translating to the 1WQC model CTC-assisted circuits, and then showing that in some cases they can be shown to be equivalent to time-respecting circuits. The predictions obtained in those cases are exactly those of the quantum CTC model based on post-selected teleportation, proposed by Bennett, Schumacher and Svetlichny (BSS). This enabled us to show that the BSS model for quantum CTCs makes predictions which disagree with those of the highly influential CTC model proposed by David Deutsch.<br>Nesta tese eu estudo o modelo de computação quântica baseada em medições (CQBM) e algumas aplicações das diferentes maneiras de traduzir algoritmos de CQBM para o modelo de circuitos. Em uma série de resultados recentes, vários conjuntos de condições para implementar uma computação deterministicamente no modelo de CQBM têm sido propostas, cada um deles com diferentes propriedades. Alguns desses conjuntos de condições - genericamente conhecidos como condições de fluxo (flow) - tentam explorar o poder de paralelização do modelo de CQBM, aumentando o número de operações que podem ser realizadas simultaneamente. Aqui eu contribuo para essa linha de pesquisa definindo um novo tipo de fluxo, chamado fluxo de sinal deslocado (FSD), que tem uma estrutura paralela interessante que se iguala ao de um fluxo ótimo, do ponto de vista temporal. Eu também introduzo um novo sistema para traduzir algoritmos de CQBM para o modelo de circuitos. Esta tradução preserva não só a computação, mas também outras características de algoritmos em CQBM. Usando esse sistema eu desenvolvo dois algoritmos, cada um capaz de executar um procedimento de tradução diferente: o primeiro obtém circuitos compactos a partir de computações com fluxo regular, e o segundo faz o mesmo para computações com FSD. Como uma aplicação do procedimento de tradução de computações com FSD, eu combino esse procedimento com outras técnicas de tradução e otimização para desenvolver um procedimento automático de otimização de circuitos quânticos. Esse procedimento é baseado em traduções nos dois sentidos entre os modelos de CQBM e de circuitos, usando técnicas de CQBM para otimizar circuitos quânticos Na segunda parte desta tese, eu uso ferramentas do modelo de CQBM para analisar circuitos quânticos interagindo com curvas temporais fechadas (CTFs). Essa análise é feita traduzindo circuitos interagindo com CTFs para o modelo de CQBM e em seguida mostrando que, em alguns casos, esses circuitos podem ser transcritos como circuitos sem CTFs que realizam a mesma computação. As predições obtidas nesses casos são exatamente as mesmas daquelas obtidas usando o modelo para estudar CTFs proposto por Bennett, Schumacher e Svetlichny (BSS). Isso nos permitiu mostrar que o modelo BSS para CTFs faz predições que não concordam com aquelas dadas pelo influente modelo de CFTs proposto por David Deutsch.

Dans cette thèse, nous développons plusieurs outils pour la Correction d’Erreur Quantique (CEQ) autonome avec les qubits supraconducteurs.Nous proposons un schéma de CEQ autonome qui repose sur la technique du « reservoir engineering », dans lequel trois qubits de type transmon sont couplés à un ou plusieurs modes dissipatifs. Grâce à la mise au point d’une interaction effective entre les systèmes, l’entropie créée par les éventuelles erreurs est évacuée à travers les modes dissipatifs.La deuxième partie de ce travail porte sur un type de code récemment développé, le code des chats, à travers lequel l’information logique est encodée dans le vaste espace de Hilbert d’un oscillateur harmonique. Nous proposons un protocole pour réaliser des mesures continues et non-perturbatrices de la parité du nombre de photons dans une cavité micro-onde, ce qui correspond au syndrome d’erreur pour le code des chats. Enfin, en utilisant les résultats précédents, nous présentons plusieurs protocoles de CEQ continus et/ou autonomes basés sur le code des chats. Ces protocoles offrent une protection robuste contre les canaux d’erreur dominants en présence de dissipation stimulée à plusieurs photons<br>In this thesis, we develop several tools in the direction of autonomous Quantum Error Correction (QEC) with superconducting qubits. We design an autonomous QEC scheme based on quantum reservoir engineering, in which transmon qubits are coupled to lossy modes. Through an engineered interaction between these systems, the entropy created by eventual errors is evacuated via the dissipative modes.The second part of this work focus on the recently developed cat codes, through which the logical information is encoded in the large Hilbert space of a harmonic oscillator. We propose a scheme to perform continuous and quantum non-demolition measurements of photon-number parity in a microwave cavity, which corresponds to the error syndrome in the cat code. In our design, we exploit the strongly nonlinear Hamiltonian of a highimpedance Josephson circuit, coupling ahigh-Q cavity storage cavity mode to a low-Q readout one. Last, as a follow up of the above results, we present several continuous and/or autonomous QEC schemes using the cat code. These schemes provide a robust protection against dominant error channels in the presence of multi-photon driven dissipation

Cette thèse étudie deux modèles de calcul: les marches quantiques (QW) et les automates cellulaires quantiques (QCA), en vue de les appliquer en simulation quantique. Ces modèles ont deux avantages stratégiques pour aborder ce problème: d'une part, ils constituent un cadre mathématique privilégié pour coder la description du système physique à simuler; d'autre part, ils correspondent à des architectures expérimentalement réalisables.Nous effectuons d'abord une analyse des QWs en tant que schéma numérique pour l'équation de Dirac, en établissant leur borne d'erreur globale et leur taux de convergence. Puis nous proposons une notion de transformée de Lorentz discrète pour les deux modèles, QW et QCA, qui admet une représentation diagrammatique s'exprimant par des règles locales et d'équivalence de circuits. Par ailleurs, nous avons caractérisé la limite continue d'une grande classe de QWs, et démontré qu'elle correspond à une classe d'équations aux dérivées partielles incluant l'équation de Dirac massive en espace-temps courbe de $(1+1)$-dimensions.Finalement, nous étudions le secteur à deux particules des automates cellulaires quantiques. Nous avons trouvé les conditions d'existence du spectre discret (interprétable comme une liaison moléculaire) pour des interactions à courte et longue portée, à travers des techniques perturbatives et d'analyse spectrale des opérateurs unitaires<br>This thesis is devoted to the development of two well-known models of computation for their application in quantum computer simulations. These models are the quantum walk (QW) and quantum cellular automata (QCA) models, and they constitute doubly strategic topics in this respect. First, they are privileged mathematical settings in which to encode the description of the actual physical system to be simulated. Second, they offer an experimentally viable architecture for actual physical devices performing the simulation.For QWs, we prove precise error bounds and convergence rates of the discrete scheme towards the Dirac equation, thus validating the QW as a quantum simulation scheme. Furthermore, for both models we formulate a notion of discrete Lorentz covariance, which admits a diagrammatic representation in terms of local, circuit equivalence rules. We also study the continuum limit of a wide class of QWs, and show that it leads to a class of PDEs which includes the Hamiltonian form of the massive Dirac equation in (1+1)-dimensional curved spacetime.Finally, we study the two particle sector of a QCA. We find the conditions for the existence of discrete spectrum (interpretable as molecular binding) for short-range and for long-range interactions. This is achieved using perturbation techniques of trace class operators and spectral analysis of unitary operators

Submitted by Mario BC (mario@bc.ufrpe.br) on 2017-02-08T13:00:48Z No. of bitstreams: 1 Mouglas Eugenio Nasario Gomes.pdf: 2441879 bytes, checksum: 71064821936a79cf37326006ed006c46 (MD5)<br>Made available in DSpace on 2017-02-08T13:00:48Z (GMT). No. of bitstreams: 1 Mouglas Eugenio Nasario Gomes.pdf: 2441879 bytes, checksum: 71064821936a79cf37326006ed006c46 (MD5) Previous issue date: 2015-07-27<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>This paper presents the LinDCQ tool — a description language and programming quantum circuits — which enables the creation of quantum circuits with calculus of operations performed in parallel on the GPU, using JOCL. The tool also allows the generation of graphically circuit. Used as a mechanism to generate grammars of languages and automata as language recognizer and the regular expression engine. In this context a discussion of the phases of compilers and on quantum computation is presented as well as an explanation of the main technologies used for the development of quantum circuits. LinDCQ The tool consists of: grammar in BNF form (Backus-Naur-Form), the compiler verifies that the incidence of errors in the code to be executed, a graphical interface to facilitate the programming features that allow the construction of the circuit graphically and parallel algorithms JOCL to perform operations that require greater computational cost in the GPU. At the end of an experiment is performed in order to assess the usability of the tool, to thereby ensure a higher level of user acceptance, facilitating interaction thereof with the tool developed in this work.<br>Este trabalho apresenta a ferramenta LinDCQ - uma linguagem de descrição e programação de circuitos quânticos — a qual possibilita a criação de circuitos quânticos com cálculo das operações realizados de forma paralela na GPU, utilizando JOCL. A ferramenta também permite a geração do circuito de forma gráfica. Utiliza gramáticas como mecanismo na geração de linguagens e autômatos como mecanismo reconhecedor de linguagens e de expressões regulares. Nesse contexto é apresentada uma discussão sobre as fases dos compiladores e sobre a computação quântica, assim como uma explanação sobre as principais tecnologias utilizadas para o desenvolvimento de circuitos quânticos. A ferramenta LinDCQ é composta de: gramática no formato BNF (Backus-Naur-Form), compilador que verifica a incidência de erros no código a ser executado, de uma interface gráfica com características facilitadoras à programação que permite a construção do circuito de forma gráfica e de algoritmos paralelos em JOCL para executar as operações que requerem maior custo computacional na GPU. Ao final é realizado um experimento com o intuito de aferir a usabilidade da ferramenta, para, deste modo, garantir um maior um nível de aceitação do usuário, facilitando a interação do mesmo com a ferramenta desenvolvida nesta dissertação.

碩士<br>國立高雄第一科技大學<br>電腦與通訊工程所<br>94<br>Abstract This thesis is theoretically based on the Shor’s quantum algorithm of breaking prime factorization in polynomial time. By following Shor’s algorithm, the NMR quantum computer of 7 qubits was made by IBM laboratory in 2001 that is the one of the most significant applications. The NMR quantum computer is mainly composed of two quantum circuits including quantum Fourier transform circuit and quantum modular exponentiation circuit. In this thesis, there are two design methods proposed for quantum modular exponentiation, that is, Shannon expansion algorithm and quantum swapping gate with assisting gate algorithm. In the Shannon expansion design algorithm, the large circuit is decomposed systematically many small circuits based on recursion relation of algorithm, and then design them respectively. The other proposed design method is based on the elementary quantum swapping gate and using Toffoli gate as assisting gate to design modular exponential circuit systematically. The number of quantum elementary gates such as single qubit gate and control not gate is chosen as the criterion to define the complexity of the circuit for comparing the performance between the two design methods. Then, we’ll have a discussion on comparison of the complexity of modular exponentiation circuits including the Shannon expansion algorithm, quantum swapping gate with assisting gate algorithm, and IBM design method in 2001 respectively. First, because the Shannon expansion is an algorithm based on recursion relation, its designed modular exponentiation circuit is much better than the circuit designed by IBM in term of design complexity. Second, since quantum swapping gate with assisting gate is an algorithm that can design circuit systematically, it can design any kind of modular exponentiation circuit. Finally, we summarize the above discussion as following. The modular exponentiation circuits designed systematically by Shannon expansion algorithm, and quantum swapping gate with assisting gate algorithm are both better than IBM design method in 2001 from the view of design complexity and systematization. This is the main contribution of this thesis.

碩士<br>國立高雄第一科技大學<br>電腦與通訊工程所<br>95<br>The Shor''s quantum algorithm for breaking prime factorization in polynomial time was developed in 1994.This algorithm is mainly composed of two quantum circuits including quantum Fourier transform circuit and quantum modular exponentiation circuit .In this thesis, two methods are proposed for the design quantum modular Exponentiation. One is revision item by item method ,the other is tree structure search. These two methods use Toffoli gate to design modular exponential circuit .The number of quantum elementary gates such as single qubit gate and control NOT gate and chosen as the criterion to study the complexity of the circuit. The complexity of the designed modular exponentiation circuits is compared with the conventional IBM design method and Shannon expansion method .Finally, the advantages and disadvantages of proposed methods are described in details.