The Construction Of Quantum Error Correcting Codes

Because quantum error correcting codes have great application in quantum computation, quantum communication, designing quantum cryptographic protocols and proof of their security, quantum error correcting codes (QECCs) have being a hot topic.We discuss the construction of symmetric and asymmetric quantum codes in this thesis and obtain the following results:1) We construct symmetric quantum codes via logic functions. We construct symmetric quantum codes with basic states corresponding to logic functions. The APC distance plays a key role in this construction. It is proposed the relationship between the minimal distance of the constructed quantum codes and the APC distance of the logic functions. It is also discussed the relationship between the dimension of the constructed quantum codes and the APC distance of the logic functions. Further more, we present the basic states of the constructed quantum codes and the sufficient conditions of constructing symmetric quantum MDS codes in this way. Specially, we obtain some symmetric quantum MDS codes as examples constructed in this way and prove this is valid way of constructing quantum codes.2) We construct symmetric quantum codes via matrices. By finding matrices with special propertites, we prove that for all odd prime p > 3, symmetric quantum MDS codes [[9,5,3]]p and [[8,4,3]]p exist, for all odd prime p > 7, symmetric quantum MDS codes [[9,3,4]]p exist.3) We construct asymmetric quantum codes via matrices. We firstly discuss p -asymmetric quantum codes, where p is odd prime. We generalize the construction of symmetric quantum codes via matrices to asymmetric quantum codes. We propose the sufficient conditions and their proof for the existence of asymmetric graphic quantum codes with parameters [[n , k , d z / d x ]]p. As a result, we obtain the sufficient conditions for the existence of asymmetric quantum MDS codes. Constructing asymmetric quantum codes in this way is to find matrices with special propertites. By finding matrices with this kind of properties, we gain some asymmetric quantum MDS codes and prove this is valid method to construct asymmetric quantum codes.