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International Journal of Modern Physics B

GRAPHICAL QUANTUM LOW-DENSITY PARITY-CHECK CODES

Yuanli Cai, Mantao Xu, Yinkuo Meng, Ying Guo

Published Jul 18, 2012• 0 citations
Publisher

Abstract

Graphical approach provides a direct way to construct error correction codes. Motivated by its good properties, associating low-density parity-check (LDPC) codes, in this paper we present families of graphical quantum LDPC codes which contain no girth of four. Because of the fast algorithm of constructing for graphical codes, the proposed quantum codes have lower encoding complexity.

DOI
10.1142/s0217979212501184

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Graphical approach provides a direct way to construct error correction codes. Motivated by its good properties, associating low-density parity-check (LDPC) codes, in this paper we present families of graphical quantum LDPC codes which contain no girth of four. Because of the fast algorithm of constructing for graphical codes, the proposed quantum codes have lower encoding complexity.
In this paper we construct three types of low-density parity-check codes with column weight j = 3 based on geometries in graphical models. Low-density parity-check codes with j > 2 are desired because their minimum distance improves linearly with the code block length n. The codes we present here have girth 8 and girth 10. All codes are regular and well-structured. These codes have flexible block lengths and code rates, and may be used in the area of communications and data storage. Our simulation results show that they have better bit-error-rate decoding performance and lower error floors in additive white Gaussian noise channels than randomly constructed low-density parity-check codes.
Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. As a concrete demonstration, we construct a girth-8, (3,12)-regular [[9216,4612,≤48]][[9216,4612, \leq 48]][[9216,4612,≤48]] quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as 10−810^{-8}10−8 on the depolarizing channel with error probability 4%4 \%4%.
This study proposes an explicit construction method for quantum quasi-cyclic low-density parity-check (QC-LDPC) codes with a girth of 12. The proposed method designs parity-check matrices that maximize the girth while maintaining an orthogonal structure suitable for quantum error correction. By utilizing algebraic techniques, short cycles are eliminated, which improves error correction performance. Additionally, this method is extended to non-binary LDPC codes and spatially-coupled LDPC codes, demonstrating that both the girth and orthogonality can be preserved. The results of this study enable the design of high-performance quantum error-correcting codes without the need for random search.
This article develops a general method for constructing entanglement-assisted quantum low-density parity-check (LDPC) codes, which is based on combinatorial design theory. Explicit constructions are given for entanglement-assisted quantum error-correcting codes with many desirable properties. These properties include the requirement of only one initial entanglement bit, high error-correction performance, high rates, and low decoding complexity. The proposed method produces several infinite families of codes with a wide variety of parameters and entanglement requirements. Our framework encompasses the previously known entanglement-assisted quantum LDPC codes having the best error-correction performance and many other codes with better block error rates in simulations over the depolarizing channel. We also determine important parameters of several well-known classes of quantum and classical LDPC codes for previously unsettled cases.
There are many algorithms to construct good low-density parity-check (LDPC) code, the most typical algorithms are bit-filling algorithm, randomly construction algorithm, and PEG (progressive edge growth) algorithm. As shown in [1], the error floor of LDPC Code is decreased by using PEG algorithm, but the error correction performance in waterfall region is compromised since a stopping set with small size will form the codeword with small Hamming weight over AWGN [2]. In this paper, we propose a novel algorithm to construct LDPC Codes. In our algorithm, we construct LDPC Code to avoid small girth and small stopping set by detecting the complete associated matrix of check node (defined in this paper) that converted from bipartite graph of LDPC Code based on the graph theory. Simulation shows that the LDPC code constructed by our algorithm has lower error floor than randomly constructed LDPC Code. The performance improvement of our algorithm is 0.1dB at BER of 10-3compared with PEG algorithm.