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arXiv:2009.02543

Extended quasi-cyclic constructions of quantum codes and entanglement-assisted quantum codes

Jingjie Lv, Ruihu Li, Yu Yao

Published Sep 5, 2020• 0 citations
arXivHTMLPDF

Abstract

Construction of quantum codes and entanglement-assisted quantum codes with good parameters via classical codes is an important task for quantum computing and quantum information. In this paper, by a family of one-generator quasi-cyclic codes, we provide quasi-cyclic extended constructions that preserve the self-orthogonality to obtain stabilizer quantum codes. As for the computational results, some binary and ternary stabilizer codes with good parameters are constructed. Moreover, we present methods to construct maximal-entanglement entanglement-assisted quantum codes by means of the class of quasi-cyclic codes and their extended codes. As an application, some good maximal-entanglement entanglement-assisted quantum codes are obtained and their parameters are compared.

DOI
10.48550/arxiv.2009.02543

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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.
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In this short letter, our main objective is to construct good quantum stabilizer codes over small fields, which have a more potential possibility to be employed in actual quantum processors. We firstly consider a suitable class of two-generators quasi-cyclic (QC) codes and determine their parameters. Then by virtue of their symplectic dual algebraic structure, we derive a sufficient condition to be symplectic dual-containing of these codes. Subsequently, an explicit construction of stabilizer codes from quasi-cyclic codes is presented. As for computational results, we provide many stabilizer codes with good parameters over binary, ternary and quaternary fields that all can not be deduced by the quantum Gilbert-Varshamov (GV) bound. In the binary case, our several codes strictly improve the lower bounds on the minimum distance in current records. In the ternary and quaternary occasions, numerous codes fill some gaps or have better parameters than the current results.