The Construction Of Quantum Error Correcting Codes

The theory of quantum error-correcting codes is the primary tool for fighting decoherencein quantum computers and quantum communication systems. Since the first quantumerror-correcting code was proposed by Shor, the theory of quantum error-correcting codesdevelops rapidly, has become a cross-frontier field in computer science, communications,physics and mathematics. We discuss the constructions of asymmetric quantum codes andentanglement-assisted quantum stabilizer codes in this thesis and obtain the following results:1. We investigate the constructions of non-binary asymmetric quantum BCH codes andsubsystem BCH codes. We analyze the properties of the cyclotomic cosets of cyclic code,compute the exact dimensions of these codes, and obtain new asymmetric quantum BCH codesand new subsystem BCH codes by using the CSS construction method of asymmetric quantumcodes. The asymmetric quantum BCH codes and subsystem BCH codes in this paper have betterparameters than the ones available in the literature.2. We investigate the constructions of asymmetric quantum codes using the method of logicfunctions. We construct asymmetric quantum codes with basic states corresponding to logicfunctions, and give the relationship between the minimum distances of asymmetric quantumcodes and the APC distance of the logic function. We also give some examples of asymmetricquantum MDS codes and illuminate that this is a valid way for constructing asymmetric quantumcodes.3. We investigate the constructions of p-asymmetric quantum codes via matrices. We giveexplicit mathematical proof to optimize the sufficient conditions for the existence of asymmetricgraphic quantum codes with parameters[[n, k, d z/d x]]p. As a result, we obtain the sufficientconditions for the existence of asymmetric quantum MDS codes. By finding matrices withspecial properties, we prove that there exist asymmetric quantum MDS codes [[6,2,4,2]]pand[[n,1, n1,2]]p.4. We investigate the constructions of non-binary entanglement-assisted quantum stabilizercodes. Through the appropriate unitary operationtrans we transform the nonabelian generatorsinto the abelian generators which form a valid quantum error-correcting code. We also give analgorithm to determine the circuit for non-binary entanglement-assisted quantum stabilizer codes.We use the pre-shared entanglement between the sender and receiver that the codes weconstructed do not require the dual-containing constraint of the CSS construction, and manynon-binary classical codes which do not satisfy the condition can be used to construct non-binaryentanglement-assisted quantum codes. So we can obtain quantum error-correcting codes withhigh performance.