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arXiv:1004.4488

Apologizing Comment on `Quantum Quasi-Cyclic Low-Density Parity-Check codes$\,$"

Dazu Huang, Zhigang Chen, Li, Xin, Ying Guo

Published Apr 26, 2010• 0 citations
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Abstract

In our recent paper entitled "Quantum Quasi-Cyclic Low-Density Parity-Check codes" [ICIC 2009. LNCS 5754], it was claimed that some new quantum codes can be constructed via the CSS encoding/decoding approach with various lengths and rates. However, the further investigation shows that the proposed construction may steal some ideas from the paper entitled "Quantum Quasi-Cyclic LDPC codes" [quant-ph/0701020v2]. We feel that the apologizing point of the original protocol is that some results are almost similar to that of construction methods with algebraic combinatorics although we suggest the different approach for improving them. Also, there is a weak point of the original coding approach while considering the application of codes in imperfect channels.

DOI
10.48550/arxiv.1004.4488

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Apologizing Comment on `Quantum Quasi-Cyclic Low-Density Parity-Check codes \,"
In our recent paper entitled "Quantum Quasi-Cyclic Low-Density Parity-Check codes" [ICIC 2009. LNCS 5754], it was claimed that some new quantum codes can be constructed via the CSS encoding/decoding approach with various lengths and rates. However, the further investigation shows that the proposed construction may steal some ideas from the paper entitled "Quantum Quasi-Cyclic LDPC codes" [quant-ph/0701020v2]. We feel that the apologizing point of the original protocol is that some results are almost similar to that of construction methods with algebraic combinatorics although we suggest the different approach for improving them. Also, there is a weak point of the original coding approach while considering the application of codes in imperfect channels.
We investigate the construction of quantum low-density parity-check (LDPC) codes from classical quasi-cyclic (QC) LDPC codes with girth greater than or equal to 6. We have shown that the classical codes in the generalized Calderbank-Shor-Steane (CSS) construction do not need to satisfy the dual-containing property as long as pre-shared entanglement is available to both sender and receiver. We can use this to avoid the many 4-cycles which typically arise in dual-containing LDPC codes. The advantage of such quantum codes comes from the use of efficient decoding algorithms such as sum-product algorithm (SPA). It is well known that in the SPA, cycles of length 4 make successive decoding iterations highly correlated and hence limit the decoding performance. We show the principle of constructing quantum QC-LDPC codes which require only small amounts of initial shared entanglement.
Quantum quasi-cyclic (QC) and quantum low-density parity-check (LDPC) codes have received significant attention due to their algebraic regularity and performance under low-complexity decoding. In this work, we explore a class of structured quantum codes based on reproducible matrices, which generalize known families such as cyclic, QC, and dyadic codes. Focusing on sparse quasi-dyadic (QD) structures, we investigate how their symmetries influence Tanner graph cycles. Moreover, we construct quantum LDPC codes within the Calderbank-Shor-Steane (CSS) framework, analyzing dual-containing configurations, and assess their error rates.
Quantum error correction codes are needed to protect quantum systems from interference. An important family of quantum error correction codes are stabilizer codes. Many stabilizer codes in the literature belong to a subclass of stabilizer codes, referred to as CSS codes since they are based on the so-called CSS formalism. CSS codes have received significant attention because the dual-containing condition for the CSS formalism is relatively straightforward to satisfy and also because it is easy to produce CSS codes from classical error correction codes. However, some stabilizer codes in the literature show superior error correction performance than existing CSS codes, and are based on binary low-density parity-check (LDPC) codes. Since it has been shown that CSS codes based on nonbinary LDPC codes have good performance over quantum qubit channels, in this paper we investigate stabilizer codes based on nonbinary LDPC codes. In particular, we propose constructions of stabilizer codes based on nonbinary quasi-cyclic low-density parity-check (QC-LDPC) codes for qubit channels. The simulation results show that our QC-LDPC stabilizer codes decoded by a nonbinary sum-product algorithm have better performance than their binary counterparts.
CSS code is a simple,efficient quantum error correction code construction,and has been found applications in construction of varieties of quantum codes,so far.In this paper,we propose a new construction method of quantum parity check matrix codes based on CSS by taking advantage of the method used in classical quasi-cyclic LDPC codes.Using quick encoding algorithm,we can obtain the quantum code based on this proposed method.At last,a quantum code(3,8)(16,6) is selected as the example to illustrate the process of construction,and its encoding circuit by Controlled-not gate and Hadard gate is also presented.