摘要
量子纠错码因其在量子计算,量子通信和量子密码协议的设计和安全性证明中有广泛的应用成为研究的热点。本文主要研究了对称和非对称量子纠错码的构造,取得了以下研究成果:
1)利用逻辑函数构造对称量子纠错码。讨论了以逻辑函数对应的态为基态,构造对称量子纠错码的方法。逻辑函数的APC距离在这种构造方法中起着重要的作用,我们具体的分析了利用此方法构造得到的对称量子纠错码的极小距离和逻辑函数APC距离的关系,并且研究了构造的量子码的维数与函数APC距离的关系。进一步地,给出了利用此方法构造的对称量子纠错码的一组基态以及能够利用此方法得到对称量子MDS码的充分条件。特别地,举例说明了利用此方法能构造对称量子MDS码,从而证明了此方法是构造对称量子纠错码的好方法。
2)利用矩阵构造对称量子MDS码。通过寻找满足特殊性质的矩阵,构造性的证明了三个对称量子MDS码的存在,即对任意的素数p > 3,存在量子MDS码[[9,5,3]]p和[[8,4,3]]p ,对任意的素数p > 7,存在量子MDS码[[9,3,4]]p。
3)研究了非对称量子纠错码的构造。我们首先讨论了p态( p是奇素数)非对称图论量子纠错码,将对称量子纠错码的矩阵(或称图论)构造方法推广到非对称量子码,给出了p态非对称图论量子码[[n , k , d z / d x ]]p存在的一个充分条件,并给出了详细的数学证明。由此出发得到非对称图论量子MDS码存在的一个充分条件。这种构造非对称量子码的实质为寻找满足特殊性质的矩阵,我们通过构造满足这种性质的矩阵得到非对称量子MDS码,证明了这是一种有效的构造方法。
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.
引文
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