On Quantum and Classical Error Control Codes: Constructions and Applications

Abstract: It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic 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 work, 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 over symmetric and asymmetric channels. Particularly, many families of quantum BCH, RS, duadic, and convolutional codes are constructed over finite fields. Families of subsystem codes and a class of optimal MDS subsystem codes are derived over asymmetric and symmetric quantum channels. In addition, propagation rules and tables of upper bounds on subsystem code parameters are established. Classes of quantum and classical LDPC codes based on finite geometries and Latin squares are constructed.