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Quantum Information Processing

Application of Euclidean sums of matrix-product codes to quantum codes

Jie Liu, Peng Hu, Xiusheng Liu

Published May 6, 2023• 5 citations
Publisher
DOI
10.1007/s11128-023-03954-x

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Advances in Mathematics of Communications•Aug 18, 2025•0 citations
A new class of quantum low-density parity-check (LDPC) codes whose parity-check matrices are dual-containing matrices constructed based on lines of Euclidean geometries (EGs) is presented. The parity-check matrices of our quantum codes contain one and only one 4-cycle in every two rows and have better distance properties. However, the classical parity-check matrix constructed from EGs does not satisfy the condition of dual-containing. In some parameter conditions, parts of the rows in the matrix maybe have not any nonzero element in common. Notably, we propose four families of fascinating structure according to changes in all the parameters, and the parity-check matrices are adopted to satisfy the requirement of dual-containing. Series of matrix properties are proved. Construction methods of the parity-check matrices with dual-containing property are given. The simulation results show that the quantum LDPC codes constructed by this method perform very well over the depolarizing channel when decoded with iterative decoding based on the sum-product algorithm. Also, the quantum codes constructed in this paper outperform other quantum codes based on EGs.
Quantum low-density parity-check (LDPC) codes based on the Calderbank-Shor-Steane construction have low encoding and decoding complexity. The sum-product algorithm (SPA) can be used to decode quantum LDPC codes; however, the decoding performance may be significantly decreased by the many four-cycles required by this type of quantum codes. All four-cycles can be eliminated using the entanglement-assisted formalism with maximally entangled states (ebits). The proposed entanglement-assisted quantum error-correcting code based on Euclidean geometry outperform differently structured quantum codes. However, the large number of ebits required to construct the entanglement-assisted formalism is a substantial obstacle to practical application. In this paper, we propose a novel class of entanglement-assisted quantum LDPC codes constructed using classical Euclidean geometry LDPC codes. Notably, the new codes require one copy of the ebit. Furthermore, we propose a construction scheme for a corresponding zigzag matrix and show that the algebraic structure of the codes could easily be expanded. A large class of quantum codes with various code lengths and code rates can be constructed. Our methods significantly improve the possibility of practical implementation of quantum error-correcting codes. Simulation results show that the entanglement-assisted quantum LDPC codes described in this study perform very well over a depolarizing channel with iterative decoding based on the SPA and that these codes outperform other quantum codes based on Euclidean geometries.
In this paper, we first give the definition of $ t_1 $-partitioned Euclidean orthogonal matrix-product (MP) codes over finite fields. We then provide a criterion for $ t_1 $-partitioned Euclidean orthogonal MP codes over finite fields to be Euclidean dual-containing. Further, we exhibit a new construction method for asymmetric quantum codes by using this criterion. This construction is applied to obtain six new infinite families of asymmetric quantum codes. In addition, we compare our asymmetric quantum codes with previously known asymmetric quantum codes or quantum codes available in the literature. As can be seen, our obtained asymmetric quantum codes have better parameters than the asymmetric quantum codes or quantum codes available in the literature.