The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.
The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes. Firstly, it extends the framework of an important class of quantum codes -- nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work establishes a close link between subsystem codes and classical codes showing that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels.
Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposition of the theory of quantum stabilizer codes. We review the stabilizer formalism of quantum codes, establish the connection between classical codes and stabilizer codes and the main methods for constructing quantum codes from classical codes. In addition to the expository part, we include new results that cannot be found elsewhere. Specifically, after reviewing some important bounds for quantum codes, we prove the nonexistence of pure perfect quantum stabilizer codes with minimum distance greater than 3. Finally, we illustrate the general methods of constructing quantum codes from classical codes by explicitly constructing two new families of quantum codes and conclude by showing how to construct new quantum codes by shortening.
Abstract Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposition of the theory of quantum stabilizer codes. We review the stabilizer formalism of quantum codes, establish the connection between classical codes and stabilizer codes and the main methods for constructing quantum codes from classical codes. In addition to the expository part, we include new results that cannot be found elsewhere. Specifically, after reviewing some important bounds for quantum codes, we prove the nonexistence of pure perfect quantum stabilizer codes with minimum distance greater than 3. Finally, we illustrate the general methods of constructing quantum codes from classical codes by explicitly constructing two new families of quantum codes and conclude by showing how to construct new quantum codes by shortening.
It is conjectured that quantum computers are able to solve certain problems more \nquickly than any deterministic or probabilistic computer. For instance, Shor's algorithm \nis able to factor large integers in polynomial time on a quantum computer. \nA quantum computer exploits the rules of quantum mechanics to speed up computations. \nHowever, it is a formidable task to build a quantum computer, since the \nquantum mechanical systems storing the information unavoidably interact with their \nenvironment. Therefore, one has to mitigate the resulting noise and decoherence \neffects to avoid computational errors. \nIn this dissertation, I study various aspects of quantum error control codes - the key component of fault-tolerant quantum information processing. I present the \nfundamental theory and necessary background of quantum codes and construct many \nfamilies of quantum block and convolutional codes over finite fields, in addition to \nfamilies of subsystem codes. This dissertation is organized into three parts: \nQuantum Block Codes. After introducing the theory of quantum block codes, I \nestablish conditions when BCH codes are self-orthogonal (or dual-containing) \nwith respect to Euclidean and Hermitian inner products. In particular, I derive \ntwo families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum \ncodes, as well as families of quantum codes derived from projective geometries. \nSubsystem Codes. Subsystem codes form a new class of quantum codes in which \nthe underlying classical codes do not need to be self-orthogonal. I give an \nintroduction to subsystem codes and present several methods for subsystem \ncode constructions. I derive families of subsystem codes from classical BCH and \nRS codes and establish a family of optimal MDS subsystem codes. I establish \npropagation rules of subsystem codes and construct tables of upper and lower \nbounds on subsystem code parameters. \nQuantum Convolutional Codes. Quantum convolutional codes are particularly \nwell-suited for communication applications. I develop the theory of quantum \nconvolutional codes and give families of quantum convolutional codes based \non RS codes. Furthermore, I establish a bound on the code parameters of \nquantum convolutional codes - the generalized Singleton bound. I develop a \ngeneral framework for deriving convolutional codes from block codes and use it \nto derive families of non-catastrophic quantum convolutional codes from BCH \ncodes. \nThe dissertation concludes with a discussion of some open problems.
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.
Recently, the theory of quantum error control codes has been extended to subsystem codes over symmetric and asymmetric quantum channels -- qubit-flip and phase-shift errors may have equal or different probabilities. Previous work in constructing quantum error control codes has focused on code constructions for symmetric quantum channels. In this paper, we develop a theory and establish the connection between asymmetric quantum codes and subsystem codes. We present families of subsystem and asymmetric quantum codes derived, once again, from classical BCH and RS codes over finite fields. Particularly, we derive an interesting asymmetric and symmetric subsystem codes based on classical BCH codes with parameters [[n,k,r,d]]q, [[n,k,r,dz/dx]]q and [[n,k′,0,dz/dx]]q for arbitrary values of code lengths and dimensions. We establish asymmetric Singleton and Hamming bounds on asymmetric quantum and subsystem code parameters; and derive optimal asymmetric MDS subsystem codes. Finally, our constructions are well explained by an illustrative example. This paper is written on the occasion of the 50th anniversary of the discovery of classical BCH codes and their quantum counterparts were derived nearly 10 years ago.
This work introduces a symplectic framework for quantum error correcting codes in which local structure is analyzed through an anticode perspective. In this setting, a code is treated as a symplectic space, and anticodes arise as maximal symplectic subspaces whose elements vanish on a prescribed set of components, providing a natural quantum analogue of their classical counterparts. This framework encompasses several families of quantum codes, including stabilizer and subsystem codes, provides a natural extension of generalized distances in quantum codes, and yields new invariants that capture local algebraic and combinatorial features. The notion of anticodes also naturally leads to operations such as puncturing and shortening for symplectic codes, which in turn provide algebraic interpretations of key phenomena in quantum error correction, such as the cleaning lemma and complementary recovery and yield new descriptions of weight enumerators.