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arXiv:0803.0764

Asymmetric and Symmetric Subsystem BCH Codes and Beyond

Salah A. Aly

Published Mar 6, 2008• 10 citations
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Abstract

Recently, the theory of quantum error control codes has been extended to subsystem codes over symmetric and asymmetric quantum channels -- qubit-flip and phase-shift errors may have equal or different probabilities. Previous work in constructing quantum error control codes has focused on code constructions for symmetric quantum channels. In this paper, we develop a theory and establish the connection between asymmetric quantum codes and subsystem codes. We present families of subsystem and asymmetric quantum codes derived, once again, from classical BCH and RS codes over finite fields. Particularly, we derive an interesting asymmetric and symmetric subsystem codes based on classical BCH codes with parameters $[[n,k,r,d]]_q$, $[[n,k,r,d_z/d_x]]_q$ and $[[n,k',0,d_z/d_x]]_q$ for arbitrary values of code lengths and dimensions. We establish asymmetric Singleton and Hamming bounds on asymmetric quantum and subsystem code parameters; and derive optimal asymmetric MDS subsystem codes. Finally, our constructions are well explained by an illustrative example. This paper is written on the occasion of the 50th anniversary of the discovery of classical BCH codes and their quantum counterparts were derived nearly 10 years ago.

DOI
10.48550/arxiv.0803.0764

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Recently, the theory of quantum error control codes has been extended to include quantum codes over asymmetric quantum channels - qubit-flip and phase-shift errors may have equal or different probabilities. Previous work in constructing quantum error control codes has focused on code constructions for symmetric quantum channels. In this paper we establish a method to construct asymmetric quantum codes based on classical codes. We derive, once again, families of asymmetric quantum codes from classical BCH and RS codes over finite fields. Particularly, we present interesting asymmetric quantum codes based on nonprimitive narrow-sense BCH codes with parameters [[n, k, dz/dx]]qfor certain values of code lengths, dimensions, and various minimum distance. Finally, our constructions are well explained by an illustrative example.
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.
Abstract. It is recently conjectured in quantum information processing that phase-shift errors occur with high probability than qubit-flip errors, hence the former is more disturbing to quantum information than the later one. This leads us to construct asymmetric quantum error controlling codes to protect quantum information over asymmetric channels, Pr Z ≥ Pr X. In this paper we present two generic methods to derive asymmetric quantum cyclic codes using the generator polynomials and defining sets of classical cyclic codes. Consequently, the methods allow us to construct several families of asymmetric quantum BCH, RS, and RM codes. Finally, the methods are used to construct families of asymmetric subsystem codes. 1
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.
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