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Author: Avanti Ulhas Ketkar Publisher: ISBN: Category : Languages : en Pages :
Book Description
Stabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Different ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.
Author: Avanti Ulhas Ketkar Publisher: ISBN: Category : Languages : en Pages :
Book Description
Stabilizer codes form a special class of quantum error correcting codes. Nonbinary quantum stabilizer codes are studied in this thesis. A lot of work on binary quantum stabilizer codes has been done. Nonbinary stabilizer codes have received much less attention. Various results on binary stabilizer codes such as various code families and general code constructions are generalized to the nonbinary case in this thesis. The lower bound on the minimum distance of a code is nothing but the minimum distance of the currently best known code. The focus of this research is to improve the lower bounds on this minimum distance. To achieve this goal, various existing quantum codes are studied that have good minimum distance. Some new families of nonbinary stabilizer codes such as quantum BCH codes are constructed. Different ways of constructing new codes from the existing ones are also found. All these constructions together help improve the lower bounds.
Author: Santosh Kumar Publisher: ISBN: Category : Languages : en Pages :
Book Description
The most popular class of quantum error correcting codes is stabilizer codes. Binary quantum stabilizer codes have been well studied, and Calderbank, Rains, Shor and Sloane (July 1998) have constructed a table of upper bounds on the minimum distance of these codes using linear programming methods. However, not much is known in the case of nonbinary stabilizer codes. In this thesis, we establish a bridge between self-orthogonal classical codes over the finite field containing q2 elements and quantum codes, extending and unifying previous work by Matsumoto and Uyematsu (2000), Ashikhmin and Knill (November 2001), Kim and Walker (2004). We construct a table of upper bounds on the minimum distance of the stabilizer codes using linear programming methods that are tighter than currently known bounds. Finally, we derive code construction techniques that will help us find new codes from existing ones. All these results help us to gain a better understanding of the theory of nonbinary stabilizer codes.
Author: Reza Dastbasteh Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
Quantum error-correcting codes (quantum codes) are applied to protect quantum information from errors caused by noise (decoherence) on the quantum channel in a way that is similar to that of classical error-correcting codes. The stabilizer construction is currently the most successful and widely used technique for constructing binary quantum codes. We explore new frontiers beyond the stabilizer construction. Our approach enables integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Our construction is particularly well-suited to certain families of classical codes, including duadic codes and additive twisted codes. For duadic codes, we provide various modifications of our construction and develop new computational strategies to bound the minimum distance. This enabled us to extend the tables of good duadic codes to much larger block lengths. The primary focus of this thesis is on additive twisted codes, which are highly structured but also technically much more difficult than the more common families of codes. They are widely referenced but have received relatively little development in previous studies. We discover new connections between twisted codes and linear cyclic codes and provide novel lower and upper bounds for the minimum distance of twisted codes. We show that classical tools such as the Hartmann-Tzeng minimum distance bound are applicable to twisted codes. This enabled us to discover five new infinite families and many other examples of record-breaking, and sometimes optimal, binary quantum codes. Another important contribution is the development of new criteria for code equivalence within the families of linear cyclic, constacyclic, and twisted codes. We introduce novel sufficient conditions for code equivalence and classify all equivalent codes of certain lengths. We prove a recent conjecture on a necessary condition for the formula describing affine equivalence. For twisted codes, we use algebraic methods, such as group actions, to determine many codes with the same parameters. These results have practical implications, as they are useful for pruning the search for new good codes, and they enabled us to discover many new record-breaking linear and binary quantum codes.
Author: Louis Kauffman Publisher: CRC Press ISBN: 1584889004 Category : Mathematics Languages : en Pages : 625
Book Description
Research and development in the pioneering field of quantum computing involve just about every facet of science and engineering, including the significant areas of mathematics and physics. Based on the firm understanding that mathematics and physics are equal partners in the continuing study of quantum science, Mathematics of Quantum Computation an
Author: Giuliano Gadioli La Guardia Publisher: Springer Nature ISBN: 303048551X Category : Computers Languages : en Pages : 234
Book Description
This text presents an algebraic approach to the construction of several important families of quantum codes derived from classical codes by applying the well-known Calderbank-Shor-Steane (CSS), Hermitian, and Steane enlargement constructions to certain classes of classical codes. In addition, the book presents families of asymmetric quantum codes with good parameters and provides a detailed description of the procedures adopted to construct families of asymmetric quantum convolutional codes. Featuring accessible language and clear explanations, the book is suitable for use in advanced undergraduate and graduate courses as well as for self-guided study and reference. It provides an expert introduction to algebraic techniques of code construction and, because all of the constructions are performed algebraically, it enables the reader to construct families of codes, rather than only codes with specific parameters. The text offers an abundance of worked examples, exercises, and open-ended problems to motivate the reader to further investigate this rich area of inquiry. End-of-chapter summaries and a glossary of key terms allow for easy review and reference.
Author: Sala Abdelhamid Awad Aly Ahmed Publisher: ISBN: Category : Languages : en Pages :
Book Description
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.
Author: Paul Bruillard Publisher: American Mathematical Soc. ISBN: 1470440741 Category : Education Languages : en Pages : 240
Book Description
This volume contains the proceedings of the AMS Special Session on Topological Phases of Matter and Quantum Computation, held from September 24–25, 2016, at Bowdoin College, Brunswick, Maine. Topological quantum computing has exploded in popularity in recent years. Sitting at the triple point between mathematics, physics, and computer science, it has the potential to revolutionize sub-disciplines in these fields. The academic importance of this field has been recognized in physics through the 2016 Nobel Prize. In mathematics, some of the 1990 Fields Medals were awarded for developments in topics that nowadays are fundamental tools for the study of topological quantum computation. Moreover, the practical importance of this discipline has been underscored by recent industry investments. The relative youth of this field combined with a high degree of interest in it makes now an excellent time to get involved. Furthermore, the cross-disciplinary nature of topological quantum computing provides an unprecedented number of opportunities for cross-pollination of mathematics, physics, and computer science. This can be seen in the variety of works contained in this volume. With articles coming from mathematics, physics, and computer science, this volume aims to provide a taste of different sub-disciplines for novices and a wealth of new perspectives for veteran researchers. Regardless of your point of entry into topological quantum computing or your experience level, this volume has something for you.
Author: Jurgen Bierbrauer Publisher: CRC Press ISBN: 135198960X Category : Mathematics Languages : en Pages : 318
Book Description
Although its roots lie in information theory, the applications of coding theory now extend to statistics, cryptography, and many areas of pure mathematics, as well as pervading large parts of theoretical computer science, from universal hashing to numerical integration. Introduction to Coding Theory introduces the theory of error-correcting codes in a thorough but gentle presentation. Part I begins with basic concepts, then builds from binary linear codes and Reed-Solomon codes to universal hashing, asymptotic results, and 3-dimensional codes. Part II emphasizes cyclic codes, applications, and the geometric desciption of codes. The author takes a unique, more natural approach to cyclic codes that is not couched in ring theory but by virtue of its simplicity, leads to far-reaching generalizations. Throughout the book, his discussions are packed with applications that include, but reach well beyond, data transmission, with each one introduced as soon as the codes are developed. Although designed as an undergraduate text with myriad exercises, lists of key topics, and chapter summaries, Introduction to Coding Theory explores enough advanced topics to hold equal value as a graduate text and professional reference. Mastering the contents of this book brings a complete understanding of the theory of cyclic codes, including their various applications and the Euclidean algorithm decoding of BCH-codes, and carries readers to the level of the most recent research.
Author: Publisher: ScholarlyEditions ISBN: 1490112502 Category : Computers Languages : en Pages : 1538
Book Description
Issues in Information Science Research / 2013 Edition is a ScholarlyEditions™ book that delivers timely, authoritative, and comprehensive information about Web and Grid Services. The editors have built Issues in Information Science Research: 2013 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about Web and Grid Services in this book to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in Information Science Research: 2013 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.
Author: Avanti Ulhas Ketkar
Author: Avanti Ulhas Ketkar
Author: Santosh Kumar
Author: Reza Dastbasteh
Author: Louis Kauffman
Author: Giuliano Gadioli La Guardia
Author: Sala Abdelhamid Awad Aly Ahmed
Author: Avaz Naghipour
Author: Paul Bruillard
Author: Jurgen Bierbrauer
Author: