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IEEE Communications Letters

A Rate Scalable Construction of Non-Homogeneous Quantum LDPC Codes of CSS Type Based on Balanced Incomplete Block Design

Yun-Jiang Wang, Zhuo-Yan Xiao, Xing-Yu Xiong, Junzhi Yan, Chao Wang, Sha Shi

Published Feb 25, 2021• 4 citations
Publisher

Abstract

Balanced incomplete block design (BIBD) offers an important way to construct non-homogeneous quantum low-density-parity-check (LDPC) codes of Calderbank-Shor-Steane (CSS) type. However, the rates of the quantum codes based on this method are limited to be 1-2/m approximately, where m, the number of base blocks chosen from the associated BIBD, is even. We overcome this limitation by providing a rate scalable way to construct non-homogeneous quantum LDPC codes of CSS type. With our method, the rates of the resultant quantum codes are 1-2/m', where m'=m/e and e ∈ {1, 2}. Note that here m' can be odd. Thus, we double the rate range of the codes yielded from the existing strategies. Moreover, one can switch codes between the two rates flexibly adapting to various application scenarios.

DOI
10.1109/lcomm.2021.3052228

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Balanced incomplete block design (BIBD) offers an important way to construct non-homogeneous quantum low-density-parity-check (LDPC) codes of Calderbank-Shor-Steane (CSS) type. However, the rates of the quantum codes based on this method are limited to be 1-2/m approximately, where m, the number of base blocks chosen from the associated BIBD, is even. We overcome this limitation by providing a rate scalable way to construct non-homogeneous quantum LDPC codes of CSS type. With our method, the rates of the resultant quantum codes are 1-2/m', where m'=m/e and e ∈ {1, 2}. Note that here m' can be odd. Thus, we double the rate range of the codes yielded from the existing strategies. Moreover, one can switch codes between the two rates flexibly adapting to various application scenarios.
Classical row-circulant quasi-cyclic (QC) low-density parity check (LDPC) matrices are known to generate efficient high-rate short and moderate-length QC-LDPC codes, while the comparable random structures exhibit numerous short cycles of length-4. Therefore, we conceive a general formalism for constructing nondual-containing Calderbank-Shor-Steane (CSS)-type quantum low-density parity check (QLDPC) codes from arbitrary classical row-circulant QC-LDPC matrices. We apply our proposed formalism to the classical balanced incomplete block design (BIBD)-based row-circulant QC-LDPC codes for demonstrating that our designed codes outperform their dual-containing CSS-type counterparts as well as the entanglement-assisted (EA)-QLDPC codes.
We present a series of structured LDPC codes suitable for use in quantum error correction. Those codes belong to the class of dual-containing Calderbank-Shor-Steane (CSS) codes. The component CSS code is designed using the combinatorial object known as balanced incomplete block design (BIBD) with an even index. The quantum LDPC codes have the rate around 0.9. Several examples of quantum LDPC codes from BIBDs from unity index, extended by addition of an all- ones column, are introduced as well. To improve the BER performance, we employed the method of removing the cycles
Two classes of well-structured binary low-density parity check codes (LDPC) are described. The first class is based on a branch of combinatorial mathematics, known as the balanced incomplete block design (BIBD). Construction of three types of codes derived from BIBD designs is illustrated in addition to a family of LDPC codes constructed by decomposition of incidence matrices of the proposed BIBD designs. The decomposition technique reduces the density of the parity check matrix and hence reduces the number of short cycles, which generally lead to better performing LDPC codes. The second class of well-structured LDPC codes, Vandermonde or array LDPC codes, are defined by a small number of parameters and cover a large set of code lengths and rates with different column weights. The presented LDPC codes are quasi-cyclic with no cycles of length four; hence simple encoding while maintaining good performance is achieved. Furthermore, the codes are compared with known random LDPC codes, in order to assess their relative achievable performance/complexity trade-offs. It is shown that well-structured LDPC codes perform very similar to the known random LDPC codes.
Taking advantages of the property of classical codes from balance incomplete block designs (BIBDs), an improved construction of quantum Low Density Parity Check (quantum LDPC) codes is proposed in this paper. In the construction, we modified the original "Construction B" procedures to assure the quantum parity check matrix commutativity. It is shown that this construction method is a systematic structured method, and it is easier to obtain quantum codes. By simulation, we find the quantum parity check matrix obtained by our method are sparser than those from the method proposed by Ivan, and the codes have more good performance with the belief propagation iterative decoding algorithms.
Quantum low-density generator matrix (QLDGM) codes based on Calderbank-Steane-Shor (CSS) constructions have shown unprecedented error correction capabilities, displaying much improved performance in comparison to other sparse-graph codes. However, the nature of CSS designs and the manner in which they must be decoded limit the performance that is attainable with codes that are based on this construction. This motivates the search for quantum code design strategies capable of avoiding the drawbacks associated with CSS codes. In this article, we introduce non-CSS quantum code constructions based on classical LDGM codes. The proposed codes are derived from CSS QLDGM designs by performing specific row operations on their quantum parity check matrices to modify the associated decoding graphs. The application of this method results in performance improvements in comparison to CSS QLDGM codes, while also allowing for greater flexibility in the design process. The proposed non-CSS QLDGM scheme outperforms the best quantum low-density parity check codes that have appeared in the literature.