Author: Reina Riemann Publisher: ISBN: Category : Languages : en Pages : 70
Book Description
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.
Author: Vivien Londe Publisher: ISBN: Category : Languages : en Pages : 0
Book Description
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.
Author: Tyler Benjamin Jackson Publisher: ISBN: Category : Languages : en Pages :
Book Description
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.
Author: Ki-Hiu Ho Publisher: Open Dissertation Press ISBN: 9781374683181 Category : Languages : en Pages :
Book Description
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
Author: Reina Riemann
Author: Vivien Londe
Author: Tyler Benjamin Jackson
Author: Ki-Hiu Ho
Author: Robert G. Gallager
Author: Moritz Beermann
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Author: Mihaela Irina Enachescu
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