| Since 1996, quantum coding has been one of the most popular topics in the field of quantum information. As a kind of quantum coding scheme, quantum error-correction coding is the foundation of realization of quantum communication and quantum computation. So far, the theory of quantum error-correction coding has become more and more perfect. Many corresponding methods of classical coding techniques have been found in quantum field.In the field of classical error-correcting codes, low-density parity-check codes (LDPC codes) are one class of the codings which attract the most attention currently. The parity check matrix of an LDPC code is sparse and the Shannon limit can be closely approached by LDPC codes. The decoding algorithm of LDPC codes are simple, and can be operated in parallel. Of all LDPC codes, quasi-cyclic LDPC (QC-LDPC) codes have linear encoding complexity and need less storage space, which are also the current research focus in the field of classical error-correcting codes. In the field of quantum error-correcting codes, stabilizer codes are a richly structured class of quantum error-correcting codes. As a class of stabilizer codes with special structures, CSS codes can be constructed by a dual-containing binary linear code or a pair of classical binary linear codes with twisted relation. Based on CSS codes, we construct quantum LDPC codes from classical LDPC codes without four-cycles, and those quantum LDPC codes will inherit the better performance of their classical counterparts. But CSS codes constructed by dual-containing codes must contain cycles of length 4, which has a negative impact on the decoding. So the key question of quantum LDPC codes research is to construct a parity check matrix of classical LDPC codes with twisted relation and without four-cycles.In this paper, based on a method of constructing QC-LDPC codes, we propose a method to construct a pair of QC-LDPC codes with twisted relation and without four-cycles, and thus obtain quantum LDPC codes via CSS codes.This paper consists of the following three chapters.The first chapter briefly reviews the development and research significance of classical error-correcting coding theory and quantum error-correcting coding theory.The second chapter provides some basic knowledge of LDPC codes and quantum codes, which contain the basic concepts and results of LDPC codes, Tanner graph, stabilizer codes and CSS codes.In the third chapter, based on perfect cyclic difference sets and cyclic difference sets in combinatorial mathematics, we propose a method to construct the parity check matrix of a pair of QC-LDPC codes with twisted relation and without four-cycles, and thus obtain quantum LDPC codes without four-cycles. CSS codes constructed by the above method are prior to those constructed by dual-containing codes. |
