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Unified Approach to Construct Quantum Error-correcting Code

Jianfa Qian, Wenping Ma

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Quantum error-correcting codes play an important role in not only quantum communication but also quantum computation.All kinds of cyclic codes,for example,Hamming codes,BCH codes and Reed-Solomon codes et al.,constacyclic codes and quasi-cyclic codes have been used to construct quantum error-correcting codes.An unified approach to construct quantum error-correcting codes was presented by using quasi-twisted codes.A sufficient and necessary condition for quasi-twisted contained its dual codes,and a new method for constructing quasi-twisted codes was given.Moreover,new quantum quasi-twisted codes were obtained by using quasi-twisted codes.

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Quantum error-correcting codes play an important role in not only quantum communication but also quantum computation.All kinds of cyclic codes,for example,Hamming codes,BCH codes and Reed-Solomon codes et al.,constacyclic codes and quasi-cyclic codes have been used to construct quantum error-correcting codes.An unified approach to construct quantum error-correcting codes was presented by using quasi-twisted codes.A sufficient and necessary condition for quasi-twisted contained its dual codes,and a new method for constructing quasi-twisted codes was given.Moreover,new quantum quasi-twisted codes were obtained by using quasi-twisted codes.
Quantum error-correcting codes play an important role in not only quantum communication but also quantum computation.Many good quantum error-correcting codes have been constructed by using classical linear codes,for example,Hamming codes,BCH codes,RS codes and Reed-Muller codes.A new method to obtain quantum codes and asymmetric quantum codes is presented.Based on line graph of the n-cube,infinite families of new quantum codes and asymmetric quantum codes are presented.Moreover,if the length of asymmetric quantum codes is large,the asymmetric quantum codes are able to correct quantum errors with great asymmetry.
Building large-scale quantum computers is a very active area of research. A critical part of this effort is the ability to control quantum errors. Recently, many good quantum error correcting codes (QECC) over various finite fields Fq have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from classical codes, including cyclic, constacyclic, quasicyclic (QC), quasi-twisted (QT), and polycyclic codes using the CSS construction, Hermitian construction, and Lisonek-Singh methods. We believe direct and more elementary methods should be preferred.
This thesis investigates the algebraic structure, construction, encoding, and decoding of quasi-cyclic (QC) and quasi-twisted (QT) codes and their applications to quantum error correction. For selected repeated-root and simple-root QC codes, generator-matrix-based construction methods are developed using the Generalized Discrete Fourier Transform (GDFT) and Discrete Fourier Transform (DFT), respectively. These methods enable efficient encoding by matrix-vector multiplication and provide computational advantages over codeword-based constructions. Spectral decoding procedures are also studied, including a GDFT-based folding framework that relates decoding of repeated-root QC codes to that of associated simple-root codes. Furthermore, a correspondence between certain repeated-root QC codes over finite fields and simple-root QT codes over finite chain rings is established. Building on these classical structures, one-generator QT codes of index 3 are investigated for quantum-code construction. Their Hermitian duals are characterized, and sufficient conditions for Hermitian self-orthogonality are derived, yielding stabilizer quantum error-correcting codes with good and, in several cases, optimal parameters. Structured syndrome-based decoding is developed by relating stabilizer measurements to classical decoding of the underlying QT codes. An extension technique preserving Hermitian self-orthogonality is introduced, producing additional stabilizer codes and corresponding decoding procedures. Finally, entanglement-assisted quantum error-correcting codes are constructed from 1-generator QT codes through Hermitian rank conditions, yielding maximal-entanglement codes with competitive parameters. Overall, the thesis establishes QC and QT codes as a unified algebraic framework for efficient classical coding, stabilizer quantum error correction, and entanglement-assisted quantum coding.
Abstract In this paper, we systematically investigate the construction of two important classes of quantum error-correcting codes, namely entanglement-assisted quantum error-correcting codes and asymmetric quantum error-correcting codes, from a generalized class of index-2 quasi-cyclic codes. For entanglement-assisted quantum error-correcting codes, we establish a systematic construction framework and derive sufficient conditions for obtaining maximal-entanglement entanglement-assisted quantum error-correcting codes, from which two new classes of such codes are obtained. For asymmetric quantum error-correcting codes, we derive necessary and sufficient conditions for the inclusion relation between any two index-2 QC codes and use them to establish a systematic construction framework. Several new entanglement-assisted quantum error-correcting codes and asymmetric quantum error-correcting codes with parameters improving upon previously reported results are obtained as applications of the proposed constructions.