Researches On The Constructions Of Three Kinds Of Special Quantum Error-Correcting Codes

A main obstacle to realize quantum computation is decoherence of quantum bits caused by inevitable interaction with environments.Quantum error-correcting codes provide the most efficient way to overcome decoherence.The first quantum error-correcting code[[9,1,3]]was discovered by Shor.After the work of Shor,the theory of quantum error-correcting codes has been progressed rapidly.Various constructions of good quantum codes have been done.In this paper,we study three classes of special quantum error-correcting codes based on classical error-correcting codes:subsystem codes,asymmetric quantum codes and quantum convolutional codes.After the study,we get a series of good quantum codes.The concrete research content is as follows:1.The discovery of subsystem codes was considered to be a major breakthrough in quantum error correction theory.In the third chapter,we construct new ternary subsystem codes by employing classical ternary linear codes generated by the adjacency matrix on the cubic graph.Giving some new subsystem codes,we analyze their performance.These subsystem codes are able to correct less than or equal to three quantum errors,and their rates increase with the increase of the lengths.It is the first time to utilize classical linear codes on the graph to construct subsystem codes.Moreover,these subsystem codes are different from the ones in the literature.2.In many quantum mechanical systems,the phase-flip errors happen more frequently than the bit-flip errors or the combined bit-phase flip errors.There is a need to design quantum codes that take advantage of this asymmetry in quantum channels.In the forth chapter,two classes of asymmetric quantum codes are constructed by using classical constacyclic codes.These new asymmetric codes are proved to be optimal in the sense that they achieve the Singleton bound.Through comparisons showed in the example,these asymmetric quantum codes can correct quantum errors with greater asymmetry.3.An important challenge to prove the feasibility of quantum computers is to protect the quantum nature of information.The design idea of quantum convolutional codes is to protect a series of quantum information in a long distance communication.In the fifth chapter,we construct two classes of quantum convolutional codes based on classical constacyclic codes.The detailed construction process of convolutional codes derived from constacyclic codes is given.These quantum convolutional codes which are different from the ones in the literature are proved to be optimal,in other words,they attain the quantum Singleton bound.