Quantum error control codes

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Title: Quantum error control codes
Author: Abdelhamid Awad Aly Ahmed, Sala
Abstract: It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic computer . For instance , Shor's algorithm is able to factor large integers in polynomial time on a quantum computer . A quantum computer exploits the rules of quantum mechanics to speed up computations . However , it is a formidable task to build a quantum computer , since the quantum mechanical systems storing the information unavoidably interact with their environment . Therefore , one has to mitigate the resulting noise and decoherence effects to avoid computational errors . In this dissertation , I study various aspects of quantum error control codes - the key component of fault -tolerant quantum information processing . I present the fundamental theory and necessary background of quantum codes and construct many families of quantum block and convolutional codes over finite fields , in addition to families of subsystem codes . This dissertation is organized into three parts : Quantum Block Codes . After introducing the theory of quantum block codes , I establish conditions when BCH codes are self -orthogonal (or dual -containing ) with respect to Euclidean and Hermitian inner products . In particular , I derive two families of nonbinary quantum BCH codes using the stabilizer formalism . I study duadic codes and establish the existence of families of degenerate quantum codes , as well as families of quantum codes derived from projective geometries . Subsystem Codes . Subsystem codes form a new class of quantum codes in which the underlying classical codes do not need to be self -orthogonal . I give an introduction to subsystem codes and present several methods for subsystem code constructions . I derive families of subsystem codes from classical BCH and RS codes and establish a family of optimal MDS subsystem codes . I establish propagation rules of subsystem codes and construct tables of upper and lower bounds on subsystem code parameters . Quantum Convolutional Codes . Quantum convolutional codes are particularly well -suited for communication applications . I develop the theory of quantum convolutional codes and give families of quantum convolutional codes based on RS codes . Furthermore , I establish a bound on the code parameters of quantum convolutional codes - the generalized Singleton bound . I develop a general framework for deriving convolutional codes from block codes and use it to derive families of non -catastrophic quantum convolutional codes from BCH codes . The dissertation concludes with a discussion of some open problems .
URI: http : / /hdl .handle .net /1969 .1 /85910
Date: 2008-10-10

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Quantum error control codes. Available electronically from http : / /hdl .handle .net /1969 .1 /85910 .

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