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.

Quantum mechanics is the physics of the very small. Quantum computers are devices that utilize the power of quantum mechanics for their computational primitives. Associated to each quantum system is an abstract space known as the Hilbert space. A subspace of the Hilbert space is known as a quantum code. Quantum codes allow to protect the computational state of a quantum computer against decoherence errors. The well-known classes of quantum codes are stabilizer or additive codes, non-additive codes and Clifford codes. This thesis aims at demonstrating a general approach to the construction of the various classes of quantum codes. The framework utilized is the Fourier transform over finite groups. The thesis is divided into four chapters. The first chapter is an introduction to basic quantum mechanics, quantum computation and quantum noise. It lays the foundation for an understanding of quantum error correction theory in the next chapter. The second chapter introduces the basic theory behind quantum error correction. Also, the various classes and constructions of active quantum error-control codes are introduced. The third chapter introduces the Fourier transform over finite groups, and shows how it may be used to construct all the known classes of quantum codes, as well as a class of quantum codes as yet unpublished in the literature. The transform domain approach was originally introduced in (Arvind et al., 2002). In that paper, not all the classes of quantum codes were introduced. We elaborate on this work to introduce the other classes of quantum codes, along with a new class of codes, codes from idempotents in the transform domain. The fourth chapter details the computer programs that were used to generate and test for the various code classes. Code was written in the GAP (Groups, Algorithms, Programming) computer algebra package. The fifth and final chapter concludes, with possible directions for future work. References cited in the thesis are attached at the end of the thesis.

<p>Holographic duality as a rigorous approach to quantum gravity claims that a quantum gravitational system is exactly equal to a quantum theory without gravity in lower spacetime dimensions living on the boundary of the quantum gravitational system. The duality maps key questions about the emergence of spacetime to questions on the non-gravitational boundary system that are accessible to us theoretically and experimentally. Recently, various aspects of quantum information theory on the boundary theory have been found to be dual to the geometric aspects of the bulk theory. In this thesis, we study the exact and approximate quantum error corrections (QEC) in a general quantum system (von Neumann algebras) focused on QFT and gravity. Moreover, we study entanglement theory in the presence of conserved charges in QFT and the multiparameter multistate generalization of quantum relative entropy.</p>

[pt] Os protocolos de comunicação quântica são essenciais para a transmissão segura de informações, utilizando os princípios da mecânica quântica para alcançar uma segurança inatingível por sistemas criptográficos clássicos. Diferentemente dos métodos tradicionais que dependem de chaves criptográficas convencionais, os protocolos quânticos exploram propriedades únicas dos sistemas quânticos para garantir a segurança da comunicação. No entanto, a implementação prática da distribuição quântica de chaves (QKD) é desafiada por erros introduzidos durante a geração e transmissão de estados quânticos e pela possível presença de espiões. Esta dissertação explora algumas das estratégias mais usadas para estimativa de erros, correção de erros e amplificação de privacidade em sistemas de QKD. Por meio de uma revisão bibliográfica e simulações abrangentes, o estudo avalia as técnicas mais eficazes em cada área. O objetivo final desta análise é desenvolver um método a ser implementado na Rede Rio Quântica, uma rede de comunicação quântica metropolitana que interliga as instituições PUC-Rio, CBPF e UFRJ por meio de fibras ópticas e a UFF através de um canal de espaço livre. Os resultados destacam a importância de otimizar as medidas de correção de erros e privacidade para melhorar a confiabilidade e a segurança das redes de comunicação quântica. / [en] Quantum communication protocols are essential for secure transmission of information, utilizing the principles of quantum mechanics to achieve security unattainable by classical cryptographic systems. Unlike traditional methods that rely on conventional cryptographic keys, quantum protocols exploit unique properties of quantum systems to ensure communication security. However, the practical implementation of quantum key distribution (QKD) is challenged by errors introduced during the generation and transmission of quantum states and the potential presence of eavesdroppers. This thesis explores some of the most commonly used strategies for error estimation, error reconciliation, and privacy amplification within QKD systems. Through a literature review and comprehensive simulations, the study evaluates the most effective techniques in each area. The ultimate goal of this analysis is to develop a method to be implemented on Rede Rio Quântica, a metropolitan quantum communication network interlinking the institutions PUC-Rio, CBPF and UFRJ via optical fibers and UFF through a free-space channel. The findings underscore the importance of optimizing error correction and privacy measures to enhance the reliability and security of quantum communication networks.

Dans ce mémoire, nous nous pencherons tout particulièrement sur une primitive cryptographique connue sous le nom de partage de secret. Nous explorerons autant le domaine classique que le domaine quantique de ces primitives, couronnant notre étude par la présentation d’un nouveau protocole de partage de secret quantique nécessitant un nombre minimal de parts quantiques c.-à-d. une seule part quantique par participant. L’ouverture de notre étude se fera par la présentation dans le chapitre préliminaire d’un survol des notions mathématiques sous-jacentes à la théorie de l’information quantique ayant pour but primaire d’établir la notation utilisée dans ce manuscrit, ainsi que la présentation d’un précis des propriétés mathématique de l’état de Greenberger-Horne-Zeilinger (GHZ) fréquemment utilisé dans les domaines quantiques de la cryptographie et des jeux de la communication. Mais, comme nous l’avons mentionné plus haut, c’est le domaine cryptographique qui restera le point focal de cette étude. Dans le second chapitre, nous nous intéresserons à la théorie des codes correcteurs d’erreurs classiques et quantiques qui seront à leur tour d’extrême importances lors de l’introduction de la théorie quantique du partage de secret dans le chapitre suivant. Dans la première partie du troisième chapitre, nous nous concentrerons sur le domaine classique du partage de secret en présentant un cadre théorique général portant sur la construction de ces primitives illustrant tout au long les concepts introduits par des exemples présentés pour leurs intérêts autant historiques que pédagogiques. Ceci préparera le chemin pour notre exposé sur la théorie quantique du partage de secret qui sera le focus de la seconde partie de ce même chapitre. Nous présenterons alors les théorèmes et définitions les plus généraux connus à date portant sur la construction de ces primitives en portant un intérêt particulier au partage quantique à seuil. Nous montrerons le lien étroit entre la théorie quantique des codes correcteurs d’erreurs et celle du partage de secret. Ce lien est si étroit que l’on considère les codes correcteurs d’erreurs quantiques étaient de plus proches analogues aux partages de secrets quantiques que ne leur étaient les codes de partage de secrets classiques. Finalement, nous présenterons un de nos trois résultats parus dans A. Broadbent, P.-R. Chouha, A. Tapp (2009); un protocole sécuritaire et minimal de partage de secret quantique a seuil (les deux autres résultats dont nous traiterons pas ici portent sur la complexité de la communication et sur la simulation classique de l’état de GHZ). / In this thesis, we will focus on a cryptographic primitive known as secret sharing. We will explore both the classical and quantum domains of such schemes culminating our study by presenting a new protocol for sharing a quantum secret using the minimal number of possible quantum shares i.e. one single quantum share per participant. We will start our study by presenting in the preliminary chapter, a brief mathematical survey of quantum information theory (QIT) which has for goal primarily to establish the notation used throughout the manuscript as well as presenting a précis of the mathematical properties of the Greenberger-Horne-Zeilinger (GHZ)-state, which is used thoroughly in cryptography and in communication games. But as we mentioned above, our main focus will be on cryptography. In chapter two, we will pay a close attention to classical and quantum error corrections codes (QECC) since they will become of extreme importance when we introduce quantum secret sharing schemes in the following chapter. In the first part of chapter three, we will focus on classical secret shearing, presenting a general framework for such a primitive all the while illustrating the abstract concepts with examples presented both for their historical and analytical relevance. This first part (chapters one and two) will pave the way for our exposition of the theory of Quantum Secret Sharing (QSS), which will be the focus of the second part of chapter three. We will present then the most general theorems and definitions known to date for the construction of such primitives putting emphasis on the special case of quantum threshold schemes. We will show how quantum error correction codes are related to QSS schemes and show how this relation leads to a very solid correspondence to the point that QECC’s are closer analogues to QSS schemes than are the classical secret sharing primitives. Finally, we will present one of the three results we have in A. Broadbent, P.-R. Chouha, A. Tapp (2009) in particular, a secure minimal quantum threshold protocol (the other two results deal with communication complexity and the classical simulation of the GHZ-state).

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. / 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.