Subsystem codes are the most versatile class of quantum error-correcting codes known to date that combine the best features of all known passive and active error-control schemes. The subsystem code is a subspace of the quantum state space that is decomposed into a tensor product of two vector spaces: the subsystem and the co-subsystem. A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilizer codes. The existence of numerous families of MDS subsystem codes is established. Propagation rules are derived that allow one to obtain longer and shorter subsystem codes from given subsystem codes. Furthermore, propagation rules are derived that allow one to construct a new subsystem code by combining two given subsystem codes.
Subsystem codes are the most versatile class of quantum error-correcting codes known to date that combine the best features of all known passive and active error-control schemes. The subsystem code is a subspace of the quantum state space that is decomposed into a tensor product of two vector spaces: the subsystem and the co-subsystem. A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilizer codes. The existence of numerous families of MDS subsystem codes is established. Propagation rules are derived that allow one to obtain longer and shorter subsystem codes from given subsystem codes. Furthermore, propagation rules are derived that allow one to construct a new subsystem code by combining two given subsystem codes.
Subsystem codes protect quantum information by encoding it in a tensor factor of a subspace of the physical state space. They generalize all major quantum error protection schemes, and therefore are especially versatile. This paper introduces numerous constructions of subsystem codes. It is shown how one can derive subsystem codes from classical cyclic codes. Methods to trade the dimensions of subsystem and co-subsystem are introduced that maintain or improve the minimum distance. As a consequence, many optimal subsystem codes are obtained. Furthermore, it is shown how given subsystem codes can be extended, shortened, or combined to yield new subsystem codes. These subsystem code constructions are used to derive tables of upper and lower bounds on the subsystem code parameters.
Abstract—Quantum information processing is a rapidly mounting field that promises to accelerate the speed up of computations. The field utilizes the novel fundamental rules of quantum mechanics such as accelerations. Quantum states carrying quantum information are tempted to noise and decoherence, that’s why the field of quantum error control comes. In this paper, we investigate various aspects of the general theory of quantum error control- subsystem codes. Particularly, we first establish two generic methods to derive subsystem codes from classical codes that are defined over finite fields Fq and F q 2. Second, we derive cyclic subsystem codes and using our two methods, we derive all classes of subsystem codes. Consequently, we construct two famous families of cyclic subsystem BCH and RS codes. Cyclic subsystem RS codes are turned out to be Optimal and MDS codes saturating the singleton bound with equality. Third, we demonstrate several methods of subsystem code constructions by extending, shortening and combining given subsystem codes. Finally, we present tables of upper and lower bounds on subsystem codes parameters 1. I.
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.
Subsystem codes are a generalization of decoherence free subspaces, noiseless subsystems, and quantum error-correcting codes. The known constructions of subsystem codes from classical codes are limited to quantum systems that all have the same dimension, and this dimension must be a power of a prime. It is shown that one can remove these restrictions and obtain subsystem codes in quantum systems of arbitrary finite dimension from classical codes that are subgroups of an abelian group. The constructions are derived from Clifford codes over abstract error groups with abelian index groups.
The essential insight of quantum error correction was that quantum information can be protected by suitably encoding this quantum information across multiple independently erred quantum systems. Recently it was realized that, since the most general method for encoding quantum information is to encode it into a subsystem, there exists a novel form of quantum error correction beyond the traditional quantum error correcting subspace codes. These new quantum error correcting subsystem codes differ from subspace codes in that their quantum correcting routines can be considerably simpler than related subspace codes. Here we present a class of quantum error correcting subsystem codes constructed from two classical linear codes. These codes are the subsystem versions of the quantum error correcting subspace codes which are generalizations of Shor's original quantum error correcting subspace codes. For every Shor-type code, the codes we present give a considerable savings in the number of stabilizer measurements needed in their error recovery routines.