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Chinese Physics B

New entanglement-assistant and asymmetric quantum codes from quasi-cyclic codes of index two

Zhen-Hui Zhang, Zhuo Li, Lijuan Xing

Published Sep 23, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/aeab5b

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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.
Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.
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.
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.
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.
Abstract It is known that entanglement-assisted quantum error-correcting codes (EAQECCs), a type of quantum error correction codes, can be easily constructed using linear codes that satisfy the property called linear complementary duals (LCD). For quasi-cyclic (QC) codes, which are a class of linear codes, we have already published the methods for constructing the codes with properties such as self-orthogonality, self-duality and reversibility according to the prime-factor decomposition of −1+xm-1+x^m−1+xm - 1 + x m , which reduce the amount of calculation by assembling several small generator polynomial matrices into a large generator polynomial matrix. In this paper, we propose a method to construct LCD–QC codes according to the prime-factor decomposition. The main idea of this method is to decompose the generator polynomial matrix of the QC code into several small generator polynomial matrices corresponding to the prime factors and perform LCD determination, which leads to a reduction in the amount of calculation. As an application of our construction method, we create EAQECCs from the constructed LCD–QC codes, compare those minimum weights with the maximum values of the minimum weights of existing EAQECCs and find 15 EAQECCs with larger minimum weights than the existing ones.