量子稳定子码的差错纠正与译码网络构建

Error correction and decoding for quantum stabilizer codes

寻找差错症状与差错算子之间映射关系是量子译码网络的核心内容,也是量子译码网络实现纠错功能的关键.给出了比特翻转差错症状矩阵和相位翻转差错症状矩阵的定义,将任意Pauli差错算子的差错症状表示为比特翻转差错症状矩阵和相位翻转差错症状矩阵的线性组合.研究发现,量子稳定子码的差错症状矩阵由其校验矩阵所决定,从而可将差错症状矩阵与差错算子之间的映射关系转化为校验矩阵与差错算子之间的映射关系,使得所有关于差错症状的分析都可以通过分析其校验矩阵来实现.这与经典线性码的差错症状与奇偶校验矩阵之间的关系类似,因此可以将经典线性码的差错检测和纠正相关成果扩展到量子码的译码过程.基于差错算子与差错症状之间的对应关系给出了构造量子差错纠正电路的方法,根据编码算子的酉性得到了基于编码算子逆算子的译码网络构建方法.

Mapping the error syndromes to error operators is the core of quantum decoding network and the key step to realize quantum error correction.The definitions of the bit flip error syndrome matrix and the phase flip error syndrome matrix are presented,and then the error syndromes of Pauli errors are expressed in terms of the columns of the bit flip error syndrome matrix and the phase flip error syndrome matrix.It is also shown that the error syndrome matrix of a stabilizer code is determined by its check matrix,which is similar to the relationship between the classical error and the parity check matrix of classical codes.So,the techniques of error detection and error correction for classical linear codes can be applied to quantum stabilizer codes after some modifications.The error correction circuits are constructed based on the relationship between the error operator and error syndrom.The decoding circuit is constructed by reversing the encoding circuit because the encoding operators are unitary.