In quantum information processing, the information is stored in the state of quantum mechanical systems. Since the interaction with the environment is unavoidable, there is a need for quantum error correction to protect the stored information. Until now, the methods for quantum error correction were mainly based on quantum codes that rely on the arithmetic in finite fields. In contrast, this thesis aims to develop a basic framework for quantum error correcting codes over a class of rings known as the Frobenius rings. This thesis focuses on developing the theory of stabilizer codes over the Frobenius rings and provides a systematic construction of codes over these rings. A special class of Frobenius rings called finite chain rings will be the emphasis of this thesis. The theory needed for comparing the minimum distance of stabilizer codes over the finite chain rings to that over the fields is studied in detail. This thesis finally derives that the minimum distance of stabilizer codes over finite chain rings cannot exceed the minimum distance over the fields.

It is believed that quantum computers would be able to solve complex problems more quickly than any other deterministic or probabilistic computer. Quantum computers basically exploit the rules of quantum mechanics for speeding up computations. However, building a quantum computer remains a daunting task. A quantum computer, as in any quantum mechanical system, is susceptible to decohorence of quantum bits resulting from interaction of the stored information with the environment. Error correction is then required to restore a quantum bit, which has changed due to interaction with external state, to a previous non-erroneous state in the coding subspace. Until now the methods for quantum error correction were mostly based on stabilizer codes over finite fields. The aim of this thesis is to construct quantum error correcting codes over finite Frobenius rings. We introduce stabilizer codes over quadratic algebra, which allows one to use the hamming distance rather than some less known notion of distance. We also develop propagation rules to build new codes from existing codes. Non binary codes have been realized as a gray image of linear Z4 code, hence the most natural class of ring that is suitable for coding theory is given by finite Frobenius rings as it allow to formulate the dual code similar to finite fields. At the end we show some examples of code construction along with various results of quantum codes over finite Frobenius rings, especially codes over Zm.

In classical and quantum information theory there are different types of error-correcting codes being used. We study the equivalence of codes via a classification of their isometries. The isometries of various codes over Frobenius alphabets endowed with various weights typically have a rich and predictable structure. On the other hand, when the alphabet is not Frobenius the isometry group behaves unpredictably. We use character theory to develop a duality theory of partitions over Frobenius bimodules, which is then used to study the equivalence of codes. We also consider instances of codes over non-Frobenius alphabets and establish their isometry groups. Secondly, we focus on quantum stabilizer codes over local Frobenius rings. We estimate their minimum distance and conjecture that they do not underperform quantum stabilizer codes over fields. We introduce symplectic isometries. Isometry groups of binary quantum stabilizer codes are established and then applied to the LU-LC conjecture.

Orientador : Reginaldo Palazzo Junior / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-04T00:27:05Z (GMT). No. of bitstreams: 1 Almeida_AntonioCarlosAidode_D.pdf: 2149041 bytes, checksum: 427f77a8e0ec2774c7b152dd209ba9fa (MD5) Previous issue date: 2004 / Resumo: A decoerencia é um dos maiores desafios obstrutivos da computação quantica. Os codigos corretores de erros quanticos tem sido desenvolvidos com o intuito de enfrentar este desafio. Uma estrutura de grupos e uma classe associada de codigos, a classe dos codigos estabilizadores, tem-se mostrado uteis na produção de codigos e no entendimento da estrutura de classes de codigos. Todos os codigos estabilizadores descobertos ate o momentos são codigos de bloco. Nesta tese, construiremos uma classe de codigos convolucional quanticos concatenados. Introduziremos o conceito de memoria convolucional quantica e algumas tecnicas simples para produzir bons codigos convolucionais quanticos a partir de classes de codigos concolucionais classicos / Abstract: Decoherence is one of the major challenges facing the field of quantum computation. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated class of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specified codes and class of codes. All stabilizer codes discovered so far are block codes. In this thesis we will construct a class of concatenated quantum convolutional codes. We will introduce the concept of quantum convolutional memory and some simple techniques to produce good quantum convolutional codes from classes of classical convolutional codes / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

[pt] Métodos computacionais se tornam essenciais diante de problemas complexos onde a intuição humana e métodos tradicionais falham. Trabalhos recentes apresentam redes neurais artificiais capazes de realizar eficientemente tarefas intratáveis por algoritmos convencionais com o emprego de aprendizado de máquina, tornando-se assim um dos métodos mais populares. Concomitantemente, algoritmos genéticos, inspirados pelos processos biológicos de seleção natural e mutação, têm sido utilizados como método metaheurístico para encontrar soluções de problemas de otimização. Levantamos então a questão se algoritmos genéticos possuem potencial para resolver problemas no contexto da computação quântica, onde a intuição humana decresce à medida que os sistemas físicos crescem. Especificamente, nos concentramos na evolução de códigos de correção de erros quânticos dentro do formalismo de códigos stabilizer. Ao especificar uma função de fitness apropriada, mostramos que somos capazes de evoluir códigos celebrados, como o código do Shor e o perfeito de 9 e 5 qubits respectivamente, além de novos exemplos não antecipados. Adicionalmente, comparamos com o método força bruta de busca aleatória e verificamos uma crescente superioridade do algoritmo genético conforme aumenta-se o número total de qubits. Diante dos resultados, imaginamos que algoritmos genéticos possam se tornar ferramentas valiosas para desempenhar aplicações complexas em sistemas quânticos e produzir circuitos sob medida que satisfaçam restrições impostas por hardware. / [en] Computational methods become essential in the face of complex problems where human intuition and traditional methods fail. Recent works present artificial neural networks capable of efficiently performing tasks intractable by conventional algorithms using machine learning, rendering it one of the most popular methods. Concomitantly, genetic algorithms, inspired by the biological processes of natural selection and mutation, have been used as a metaheuristic method to find solutions to optimization problems. We then raise the question of whether genetic algorithms have the potential to solve problems in the context of quantum computing, where human intuition decreases as physical systems grow. Specifically, we focus on the evolution of quantum error-correcting codes within the stabilizer code formalism. By specifying an appropriate fitness function, we show that we can evolve celebrated codes, such as the Perfect and Shor s code with respectively 5 and 9 qubits, in addition to new unanticipated examples. Additionally, we compared it with a brute force random search and verified an increasing superiority of the genetic algorithm as the total number of qubits increases. Given the results, we foresee that genetic algorithms can become valuable tools to perform complex applications in quantum systems and produce tailored circuits that satisfy restrictions imposed by hardware.

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. Despite the large body of literature in quantum coding theory, many important questions, especially those centering on the issue of "good codes" are unresolved. In this dissertation the dominant underlying theme is that of constructing good quantum codes. It approaches this problem from three rather different but not exclusive strategies. Broadly, its contribution to the theory of quantum error correction is threefold. Firstly, it extends the framework of an important class of quantum codes - nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. In particular it provides many explicit constructions of stabilizer codes, most notably it simplifies the criteria by which quantum BCH codes can be constructed from classical codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. Prior to our work however, systematic methods to construct these codes were few and it was not clear how to fairly compare them with other classes of quantum codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work established a close link between subsystem codes and classical codes and it became clear that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels. This approach is based on a Calderbank- Shor-Steane construction that combines BCH and finite geometry LDPC codes.