On Quantum and Classical Error Control Codes: Constructions and Applications

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 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. This work is organized into these 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. Quantum and Classical LDPC Codes. LDPC codes are a class of modern error control codes that can be decoded using iterative decoding algorithms. In this part, I derive classes of quantum LDPC codes based on finite geometries, Latin squares and combinatorial objects. In addition, I construct families of LDPC codes derived from classical BCH codes and elements of cyclotomic cosets. Asymmetric Quantum Codes. Recently, the theory of quantum error control codes has been extended to include quantum codes over asymmetric quantum channels — qubit-flip and phase-shift errors may occur with different probabilities. I derive families of asymmetric quantum codes derived from classical BCH and RS codes over finite fields. In addition, I derive a generic method to derive asymmetric quantum cyclic codes.