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Metodos de construção de codigos quanticos CSS e conexões entre codigos quanticos e matroides

Giuliano G. La Guardia

Published Jul 12, 2008• 0 citations
Publisher

Abstract

This thesis proposes, as the main contributions, constructions method of new families of quantum CSS codes.These constructions are based on classical cyclic codes of the types BCH, Reed-Solomon, Reed-Muller, Quadratic Residue and also are based on product codes of classical Reed-Solomon codes.The main family of quantum codes constructed in this work, i. e., quantum codes having better parameters, are the ones derived from classical BCH codes.Moreover, we present some new conditions in which quantum CSS cyclic codes are quantum MDS codes.In addition, we provide the elements to connect matroid theory and quantum coding theory.More specifically, we show that the weight enumerator of a CSS quantum code is equivalent to evaluating the Tutte polynomial of the direct sum of the matroid associated to the classical codes used in the CSS construction.

DOI
10.47749/t/unicamp.2008.433145

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Abstract CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes (C1,C2)(C_1, C_2)(C1​,C2​) ( C 1 , C 2 ) such that C1C_1C1​ C 1 contains C2C_2C2​ C 2 , C2C_2C2​ C 2 is even, and the shortening of the dual of C1C_1C1​ C 1 with respect to the support of each codeword of C2C_2C2​ C 2 is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes (C1,C2)(C_1, C_2)(C1​,C2​) ( C 1 , C 2 ) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.
After the pioneering work of Shor and Steane, we are able to establish links between quantum codes and classical codes with certain self-orthogonality. Therefore, constructing classical self-orthogonal codes with small dimension and large dual distance becomes our centre point due to the interesting application to quantum codes. \n \n \n \nOne of the most interesting and useful families of classical codes are MDS codes. We will show a systematic construction for classical Hermitian self-orthogonal MDS codes through generalized Reed-Solomn codes. Afterwards, new families of quantum MDS codes can be produced. \n \n \n \nClassical BCH codes would be another good choice for self-orthogonal codes. We show the dual containment through polynomial evaluations, simply by choosing suitable cyclotomic cosets. As a result, quantum codes with good parameters can be derived. \n \n \n \nAnother good candidate of classical codes to investigate for dual containment comes from the class of \n \nalgebraic geometry codes. Instead of testing for dual containment directly, this thesis explores an equivalent condition for the existence of self-orthogonal AG codes. \n \n \n \nAfter various constructions for quantum codes, this thesis ends with a study on bounds for estimating the parameters of codes.
In this paper, two classes of quantum MDS codes are constructed. The main tools are multiplicative structures on finite fields. Carefully choosing different cosets can make the corresponding generalized Reed-Solomon codes Hermitian self-orthogonal, which yields quantum MDS codes by utilizing Hermitian type CSS construction. In some cases, these codes have minimal distances larger than q/2q/2q/2. Also, the parameters of these codes have never been reported before.
Quantum error-correcting codes are essential for protecting quantum information against decoherence. This work investigates the construction of quantum CSS codes from classical self-orthogonal binary linear codes derived from extended quadratic residue codes. By exploiting the large automorphism groups of these codes, the minimum-weight codewords are partitioned according to the action of maximal subgroups of the automorphism group. Codes generated from these partitions are shown to be self-orthogonal and to retain large automorphism groups, allowing the construction of associated quantum CSS codes with good parameters. The resulting codes include new quantum codes with lengths ranging from 16 to 48 and minimum distances up to 11. Their automorphism groups and permutation-decoding properties are also analyzed, including the number of permutations required for complete decoding. The results provide new families of quantum CSS codes combining good error-correcting parameters, large automorphism groups, and efficient permutation decoding.