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arXiv:quant-ph/0610153

Subsystem Codes

Salah A. Aly, Andreas Klappenecker, Pradeep Kiran Sarvepalli

Published Oct 18, 2006• 25 citations
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Abstract

We investigate various aspects of operator quantum error-correcting codes or, as we prefer to call them, subsystem codes. We give various methods to derive subsystem codes from classical codes. We give a proof for the existence of subsystem codes using a counting argument similar to the quantum Gilbert-Varshamov bound. We derive linear programming bounds and other upper bounds. We answer the question whether or not there exist [[n,n-2d+2,r>0,d]]q subsystem codes. Finally, we compare stabilizer and subsystem codes with respect to the required number of syndrome qudits.

Related papers

On subsystem codes beating the quantum Hamming or Singleton bound

Andreas Klappenecker, Pradeep Kiran Sarvepalli

Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to q -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal q -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.

arXiv:quant-ph/0703213•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Aug 21, 2007•26 citations

Structures and Constructions of Subsystem Codes over Finite Fields

Salah A. Aly

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.

•Mar 3, 2012•0 citations

Quantum error control codes

Abdelhamid Awad Aly Ahmed, Sala

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.

•May 1, 2008•1 citation

Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes

Dave Bacon, Andrea Casaccino

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.

arXiv:quant-ph/0610088•Oct 11, 2006•30 citations

Subsystem code constructions

Salah A. Aly, Andreas Klappenecker

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.

arXiv:0712.4321•IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2008•28 citations

Quantum stabilizer codes and beyond

Sarvepalli, Pradeep Kiran

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.

arXiv:0810.2574•Oct 14, 2008•8 citations
We investigate various aspects of operator quantum error-correcting codes or, as we prefer to call them, subsystem codes. We give various methods to derive subsystem codes from classical codes. We give a proof for the existence of subsystem codes using a counting argument similar to the quantum Gilbert-Varshamov bound. We derive linear programming bounds and other upper bounds. We answer the question whether or not there exist [[n,n-2d+2,r>0,d]]q subsystem codes. Finally, we compare stabilizer and subsystem codes with respect to the required number of syndrome qudits.
Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to q -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal q -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.
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.
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.
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.
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.