We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular, the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m²,2, m] code as compared to the toric code which is a [2m²,2, m]code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

L'objet de cette thèse est l'étude des codes LDPC quantiques. Dans un premier temps, nous travaillons sur des constructions topologiques de codes LDPC quantiques. Nous proposons de construire une famille de codes couleur basée sur des pavages hyperboliques. Nous étudions ensuite les paramètres d'une famille de codes basée sur des graphes de Cayley.Dans une seconde partie, nous examinons les performances de ces codes. Nous obtenons une borne supérieure sur les performances des codes LDPC quantiques réguliers sur le canal à effacement quantique. Ceci prouve que ces codes n'atteignent pas la capacité du canal à effacement quantique. Dans le cas du canal de dépolarisation, nous proposons un nouvel algorithme de décodage des codes couleur basé sur trois décodages de codes de surface. Nos simulations numériques montrent de bonnes performances dans le cas des codes couleur toriques.Pour finir, nous nous intéressons au phénomène de percolation. La question centrale de la théorie de la percolation est la détermination du seuil critique. Le calcul exacte de ce seuil est généralement difficile. Nous relions la probabilité de percolation dans certains pavages réguliers du plan hyperbolique à la probabilité d'erreur de décodage pour une famille de codes hyperboliques. Nous en déduisons une borne sur le seuil critique de ces pavages hyperboliques basée sur des résultats de théorie de l'information quantique. Il s'agit d'une application de la théorie de l'information quantique à un problème purement combinatoire. / This thesis is devoted to the study of quantum LDPC codes. The first part presents some topological constructions of quantum LDPC codes. We introduce a family of color codes based on tilings of the hyperbolic plane. We study the parameters of a family of codes based on Cayley graphs.In a second part, we analyze the performance of these codes. We obtain an upper bound on the performance of regular quantum LDPC codes over the quantum erasure channel. This implies that these codes don't achieve the capacity of the quantum erasure channel. In the case of the depolarizing channel, we propose a new decoding algorithm of color codes based on three surface codes decoding. Our numerical results show good performance for toric color codes.Finally, we focus on percolation theory. The central question in percolation theory is the determination of the critical probability. Computing the critical probability exactly is usually quite difficult. We relate the probability of percolation in some regular tilings of the hyperbolic plane to the probability of a decoding error for hyperbolic codes on the quantum erasure channel. This leads to an upper bound on the critical probability of these hyperbolic tilings based on quantum information. It is an application of quantum information to a purely combinatorial problem.

Le fameux Théorème d’Extension de MacWilliams affirme que, pour les codes classiques, toute isométrie deHamming linéaire d'un code linéaire se prolonge en une application monomiale. Cependant, pour les codeslinéaires sur les alphabets de module, l'existence d'un analogue du théorème d'extension n'est pas garantie.Autrement dit, il existe des codes linéaires sur certains alphabets de module dont les isométries de Hammingne sont pas toujours extensibles. Il en est de même pour un contexte plus général d'un alphabet de module munid'une fonction de poids arbitraire. Dans la présente thèse, nous prouvons des analogues du théorèmed'extension pour des codes construits sur des alphabets et fonctions de poids arbitraires. La propriétéd'extension est analysée notamment pour les codes de petite longueur sur un alphabet de module de matrices,les codes MDS généraux, ou encore les codes sur un alphabet de module muni de la composition de poidssymétrisée. Indépendamment de ce sujet, une classification des deux groupes des isométries des codescombinatoires est donnée. Les techniques développées dans la thèse sont prolongées aux cas des codesstabilisateurs quantiques et aux codes de Gabidulin dans le cadre de la métrique rang. / The famous MacWilliams Extension Theorem states that for classical codes each linear Hamming isometry ofa linear code extends to a monomial map. However, for linear codes over module alphabets an analogue of theextension theorem does not always exist. That is, there may exists a linear code over a module alphabet with anunextendable Hamming isometry. The same holds in a more general context of a module alphabet equippedwith a general weight function. Analogues of the extension theorem for different classes of codes, alphabetsand weights are proven in the present thesis. For instance, extension properties of the following codes arestudied: short codes over a matrix module alphabet, maximum distance separable codes, codes over a modulealphabet equipped with the symmetrized weight composition. As a separate result, a classification of twoisometry groups of combinatorial codes is given. The thesis also contains applications of the developedtechniques to quantum stabilizer codes and Gabidulin 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.