Theoretical Research On Quantum Error-Correcting Codes

In this paper, we mainly discussed quantum error-correction theory and some relative constructions of several kinds of quantum error-correcting codes.First, optimal quantum stabilizer codes of distance 3 are explicitly constructed for all lengths except for the following four families of lengths 8fm - {1,2} and fm+2 - {2,3} with fm=(4m-1)/3 and m≥2 being integer. For the lengths less than 128 three previously unknown codes [[36,29,3]], [[37,30.3]] and [[81.73,3]] have been found.Second, based on the graph state, we present a systematic way of constructing good quantum codes, both additive and nonadditive. We also generalize this method to the con-struction of entanglement-assisted quantum error-correcting codes(EAQECC). Through this new way, we found a lot of new quantum codes, including nonadditive codes ((9.12,3)) and ((10,24,3)), a family of EAQECCs that beat the quantum Hamming bound and even the entanglement-enhanced quantum error-correcting codes(EAQECC) [[9,5,3;1]].At last, we propose another family of QECCs, namely, the breeding QECCs, that can be built from any classical additive codes, either linear or nonlinear. Besides, since nonlinear codes are generally more efficient than linear codes, our breeding codes have better parameters than those codes built from linear codes. One example is the two-error-correcting breeding code [[13,13,5;11]], which is the first entanglement enhanced code with a larger coding distance than 3. Moreover, we presented the encoding and decoding operations of the whole family of the optimal one-error-correcting breeding codes in a direct and visualized way.