Download or read book Good Families of Quantum Low density Parity check Codes and a Geometric Framework for the Amplitude damping Channel written by Reina Riemann and published by . This book was released on 2011 with total page 70 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classical low-density parity-check (LDPC) codes were first introduced by Robert Gallager in the 1960's and have reemerged as one of the most influential coding schemes. We present new families of quantum low-density parity-check error-correcting codes derived from regular tessellations of Platonic 2-manifolds and from embeddings of the Lubotzky-Phillips-Sarnak Ramanujan graphs. These families of quantum error-correcting codes answer a conjecture proposed by MacKay about the existence of good families of quantum low-density parity-check codes with nonzero rate, increasing minimum distance and a practical decoder. For both families of codes, we present a logarithmic lower bound on the shortest noncontractible cycle of the tessellations and therefore on their distance. Note that a logarithmic lower bound is the best known in the theory of regular tessellations of 2-manifolds. We show their asymptotic sparsity and non-zero rate. In addition, we show their decoding performance with simulations using belief propagation. Furthermore, we present a general geometrical model to design non-additive quantum error-correcting codes for the amplitude-damping channel. Non-additive quantum error-correcting codes are more general than stabilizer or additive quantum errorcorrecting codes, and in some cases non-additive quantum codes are more optimal. As an example, we provide an 8-qubit amplitude-damping code, which can encode 1 qubit and correct for 2 errors. This violates the quantum Hamming bound which requires that its length start at 9.
Download or read book Topological Quantum Error Correcting Codes Beyond Dimension 2 written by Vivien Londe and published by . This book was released on 2019 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Error correction is the set of techniques used in order to store, process and transmit information reliably in a noisy context. The classical theory of error correction is based on encoding classical information redundantly. A major endeavor of the theory is to find optimal trade-offs between redundancy, which we try to minimize, and noise tolerance, which we try to maximize. The quantum theory of error correction cannot directly imitate the redundant schemes of the classical theory because it has to cope with the no-cloning theorem: quantum information cannot be copied. Quantum error correction is nonetheless possible by spreading the information on more quantum memory elements than would be necessary. In quantum information theory, dilution of the information replaces redundancy since copying is forbidden by the laws of quantum mechanics. Besides this conceptual difference, quantum error correction inherits a lot from its classical counterpart. In this PhD thesis, we are concerned with a class of quantum error correcting codes whose classical counterpart was defined in 1961 by Gallager [Gal62]. At that time, quantum information was not even a research domain yet. This class is the family of low density parity check (LDPC) codes. Informally, a code is said to be LDPC if the constraints imposed to ensure redundancy in the classical setting or dilution in the quantum setting are local. More precisely, this PhD thesis focuses on a subset of the LDPC quantum error correcting codes: the homological quantum error correcting codes. These codes take their name from the mathematical field of homology, whose objects of study are sequences of linear maps such that the kernel of a map contains the image of its left neighbour. Originally introduced to study the topology of geometric shapes, homology theory now encompasses more algebraic branches as well, where the focus is more abstract and combinatorial. The same is true of homological codes: they were introduced in 1997 by Kitaev [Kit03] with a quantum code that has the shape of a torus. They now form a vast family of quantum LDPC codes, some more inspired from geometry than others. Homological quantum codes were designed from spherical, Euclidean and hyperbolic geometries, from 2-dimensional, 3-dimensional and 4- dimensional objects, from objects with increasing and unbounded dimension and from hypergraph or homological products. After introducing some general quantum information concepts in the first chapter of this manuscript, we focus in the two following ones on families of quantum codes based on 4-dimensional hyperbolic objects. We highlight the interplay between their geometric side, given by manifolds, and their combinatorial side, given by abstract polytopes. We use both sides to analyze the corresponding quantum codes. In the fourth and last chapter we analyze a family of quantum codes based on spherical objects of arbitrary dimension. To have more flexibility in the design of quantum codes, we use combinatorial objects that realize this spherical geometry: hypercube complexes. This setting allows us to introduce a new link between classical and quantum error correction where classical codes are used to introduce homology in hypercube complexes.
Download or read book Degenerate Codes and Capacities of Quantum Channels written by Tyler Benjamin Jackson and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: No quantum system can be perfectly isolated from the environment and, as a result, no physical implementation of quantum information processing tasks can be completely free from noise. The best tool for combating such noise is the use of quantum error correcting codes (QECCs). The general requirements for QECCs have been documented for a while and yet construction of good codes and understanding their effect remains a difficult and active area of study. In this thesis I outline the work I have completed looking into both of these problems. My general technique throughout this work is to first reduce the problem space as much as possible through use of group theory. Then to use numerical methods and bring computational power to bear on the problem. In chapter 2, I investigate the construction of good codes for the amplitude damping channel, using the codeword stabilized quantum code(CWS) framework. Through an exhaustive search method many new codes with better parameters than previously known are found. In chapter 3, I continue constructing good codes for the amplitude damping channel, this time using code concatenation techniques to find results that would be unfeasible to find via an exhaustive search. Finally, chapter 4 broaches the difficult problem of determining if a quantum channel has capacity; the ultimate use of QECCs in a sense. Expanding on the key works done on the problem, I develop the theory surrounding effective noise channels obtained from applying a QECC on multiple uses of a channel in order to determine if capacity exists. Using this framework and making use of computational power available today, these techniques allow us to find many very noisy non-Pauli channels that have positive capacity which previously had not been shown to have capacity.
Download or read book Study of Quantum Low Density Parity Check and Quantum Degenerate Codes written by Ki-Hiu Ho and published by Open Dissertation Press. This book was released on 2017-01-27 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation, "Study of Quantum Low Density Parity Check and Quantum Degenerate Codes" by Ki-hiu, Ho, 何其曉, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. DOI: 10.5353/th_b4189710 Subjects: Error-correcting codes (Information theory) Quantum theory
Download or read book Low Density Parity Check Codes written by Robert G. Gallager and published by MIT Press (MA). This book was released on 1963-09-15 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Flexible Low Density Parity Check Codes Rate Length and Complexity written by Moritz Beermann and published by . This book was released on 2015-11-20 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Study of Quantum Low Density Parity Check and Quantum Degenerate Codes written by Ki-hiu Ho and published by . This book was released on 2009 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Searching for Good Low density Parity check Codes written by Mihaela Irina Enachescu and published by . This book was released on 2002 with total page 66 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Side Information in Bandit Problems Low density Parity check Codes for Non Symmetric Channels written by Chih-Chun Wang and published by . This book was released on 2005 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Research on Low density Parity Check Codes written by Tao Tian and published by . This book was released on 2003 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book On the Construction of Some Deterministic Low density Parity check Codes written by Rich Echard and published by . This book was released on 2002 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Low density Parity check Codes for Data Storage and Memory Systems written by Seungjune Jeon and published by . This book was released on 2011 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Topics in Low Density Parity Check Codes for the Binary Erasure Channel written by Ohad Barak and published by . This book was released on 2002 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Study of Quantum Low Density Parity Check and Quantum Degeneratecodes written by and published by . This book was released on 2009 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Quantum Stabilizer Codes and Beyond written by Pradeep Kiran Sarvepalli and published by . This book was released on 2008 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. Despite the large body of literature in quantum coding theory, many important questions, especially those centering on the issue of "good codes" are unresolved. In this dissertation the dominant underlying theme is that of constructing good quantum codes. It approaches this problem from three rather different but not exclusive strategies. Broadly, its contribution to the theory of quantum error correction is threefold. 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. In particular it provides many explicit constructions of stabilizer codes, most notably it simplifies the criteria by which quantum BCH codes can be constructed from classical 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. Prior to our work however, systematic methods to construct these codes were few and it was not clear how to fairly compare them with other classes of quantum codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work established a close link between subsystem codes and classical codes and it became clear 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. This approach is based on a Calderbank- Shor-Steane construction that combines BCH and finite geometry LDPC codes.
Download or read book Grassmannian Low Density Parity Check Codes written by Val B. Villegas and published by . This book was released on 2007 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Construction Applications and Implementations of Low density Parity Check Codes written by Christopher R. Jones and published by . This book was released on 2003 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt:
