三类特殊的量子纠错码的构造研究

发布时间：2018-08-29 07:57

【摘要】：与环境不可避免的交互作用引起的量子比特的消相干是实现量子计算的一个主要障碍。量子纠错码提供了最有效的方法来克服消相干。Shor构造了第一个量子纠错码[[9,1,3]]。自此,量子纠错码理论发展迅速,许多构造量子码的方法被研究出来。本文以经典纠错码理论为基础,主要研究了三类特殊的量子纠错码:子系统码、非对称量子码和量子卷积码,得到一系列有新参数的量子纠错码。具体研究内容如下:1.子系统码的发现被认为是量子纠错理论的一个重要突破。在第三章中,利用三元图邻接矩阵生成的经典三元线性码构造新的三元子系统码。列出一些新的子系统码,并分析它们的性能,这些码可以纠正小于或等于3个量子错误,且码率随着码长的增加而增大。本文首次利用图上的经典线性码构造子系统码,而且,构造的子系统码在之前的文献中没有出现过。.2.在许多量子力学系统中,相对于比特翻转错误或组合的比特相位翻转错误,相位翻转错误发生更加频繁。这就需要在量子通道中设计具有非对称性优势的量子码。在第四章中,在两类经典常循环码基础上,构造两类非对称量子码。并证明它们达到Singleton界的上界,是最佳码。给出具体的例子,经过比较,发现文中构造的非对称量子码对相位翻转错误和量子比特翻转错误有更大的纠错能力。3.保护信息的量子特征是证明量子计算机可行性面临的一个重要挑战。量子卷积码的设计意图是在长距离通信中保护一连串的量子信息。在第五章中,运用经典常循环码构造了两类量子卷积码,给出利用常循环码构造卷积码的详细过程。并证明构造的量子卷积码是最佳码,且达到量子Singleton界的上界,与之前文献中的量子卷积码的参数不同。
[Abstract]:The decoherence of quantum bits caused by the inevitable interaction with the environment is a major obstacle to the realization of quantum computing. Quantum error-correcting codes provide the most effective method to overcome the declination. Shor constructs the first quantum error-correcting codes [9]. Since then, the theory of quantum error-correcting codes has developed rapidly, and many methods of constructing quantum codes have been developed. Based on the classical error-correcting code theory, three special types of quantum error-correcting codes, subsystem codes, asymmetric quantum codes and quantum convolutional codes, are studied in this paper, and a series of quantum error-correcting codes with new parameters are obtained. The specific contents of the study are as follows: 1. The discovery of subsystem codes is considered to be an important breakthrough in quantum error correction theory. In chapter 3, a new ternary subsystem code is constructed by using the classical ternary linear codes generated by the adjacency matrix of ternary graphs. Some new subsystem codes are listed, and their performance is analyzed. These codes can correct three quantum errors less than or equal to, and the bit rate increases with the increase of code length. In this paper, the classical linear codes on graphs are used to construct subsystem codes for the first time, and the constructed subsystem codes have not appeared in the previous literature. In many quantum mechanical systems, phase flip errors occur more frequently than bit flip errors or combination bit phase flip errors. This requires the design of quantum codes with asymmetric advantages in quantum channels. In chapter 4, two kinds of asymmetric quantum codes are constructed on the basis of two classical constant cyclic codes. It is proved that they reach the upper bound of Singleton bound and are the best codes. An example is given and it is found that the asymmetric quantum code constructed in this paper has a greater error correction capability of phase flip error and quantum bit inversion error. Protecting the quantum characteristics of information is an important challenge to prove the feasibility of quantum computer. Quantum convolution codes are designed to protect a series of quantum information in long distance communication. In chapter 5, two kinds of quantum convolutional codes are constructed by using classical constant cyclic codes, and the detailed process of constructing convolutional codes by using constant cyclic codes is given. It is proved that the constructed quantum convolutional codes are the best codes and reach the upper bound of the quantum Singleton bound, which is different from the parameters of the quantum convolutional codes in previous literatures.
【学位授予单位】：安徽理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O413;TN911.2

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