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双语推荐：量子纠错码

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量子纠错码在量子通信和量子计算中起着非常重要的作用,之前的量子纠错码的构造大部分都是利用经典的纠错码来构造得到,如Hamming码,BCH码,RS码,Reed-Muller码等各种经典纠错码。目前,很少有人利用图生成的线性码方法来构造量子纠错码,提出了一个新的构造量子纠错码和非对称量子纠错码的方法,即利用n立方图的线图生成的二元线性码来构造量子纠错码和非对称量子纠错码,得到了一类新的量子纠错码和非对称量子纠错码,并且,当码字的长度较大时,对所构造的非对称量子纠错码,在非对称信道上有更大的纠错能力。
@@@@Quantum error-correcting codes play an important role in not only quantum communication but also quantum compu-tation. Many good quantum error-correcting codes have been constructed by using classical linear codes, for example, Hamming codes, BCH codes, RS codes and Reed-Muller codes. A new method to obtain quantum codes and asymmetric quantum codes is presented. Based on line graph of the n-cube, infinite families of new quantum codes and asymmetric quantum codes are presented. Moreover, if the length of asymmetric quantum codes is large, the asymmetric quantum codes are able to correct quantum errors with great asymmetry.

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文章提出了一个新的构造量子纠错码和非对称量子纠错码的方法,即利用图的邻接矩阵生成的二元线性码来构造量子纠错码和非对称量子纠错码,得到了一类新的量子纠错码和非对称量子纠错码,而且当码字的长度较大时,所构造的非对称量子纠错码,在非对称信道上相位翻转对量子系统的影响比量子比特翻转的影响要大,因此,所构造的非对称量子纠错码在非对称信道上有更大的纠错能力。
A new method to constrcut quantum error-correcting codes and asymmetric quantum error-correcting codes is presented .Using the binary linear codes generated by the adjacency matrix of graphs ,the infinite families of new quantum error-correcting codes and asymmetric quantum error-correcting codes are presented . Moreover ,if the length of asymmetric quantum error-correcting codes is large ,the quantum phase flip in asymmetric channel exerts a greater influence on quantum system than the quantum bit flip does .So the asymmetric quantum error-correcting codes are able to correct quantum errors with great asymmetry .

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本文找到了一种研究优质差错基和量子纠错码的新方法，即群代数方法，它为差错基和量子码提供了一种代数表示。利用这种代数表示，建立了一系列关于最一般量子纠错码的线性规划限。
We find a new approach to study nice error bases and quantum error-correcting codes, namely the group algebra which gives us an algebraic notation for nice error bases and quantum codes. From this algebraic notation we establish a series of linear programming bounds on the most general quantum error-correcting codes.

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依据经典四元线性码理论和纠缠辅助量子纠错码理论,由四元线性码的生成矩阵给出四元线性码稳定极大纠缠的纠缠辅助量子码的几何特征。在给定几何特征基础上,由射影空间的Cap理论,设法用组合数学方法和搜索算法构造出给定几何特征的Cap,确定Cap码的参数。利用所得到的参数优良的Cap码,结合纠缠理论,构造出一些参数优良的极大纠缠的纠缠辅助量子码。其中,所构造的极大纠缠的纠缠辅助量子码有许多是最优码,还有一些纠缠辅助量子码改进了前人所得到的纠缠辅助量子码的参数,这些纠缠辅助量子纠错码是无法用已有方法得到的。这也证明了结合组合与搜索的方法来构造极大纠缠的纠缠辅助量子纠错码是有效的。
According to the entanglement-assisted quantum error-correcting code (EAQECC)theory,a ge-ometry character of a quaternary linear code stabilized a maximal entanglement EAQECC is presented.On the basis of this character and the theory of Cap in proj ective space,methods for constructing such Caps by combinational mathematics and search algorithm are designed,and good codes from such Caps are deter-mined.Many maximal entanglement EAQECCs are constructed from these obtained Cap codes.Almost all the maximal entanglement EAQECCs constructed here are optimal or near optimal EAQECCs,and some of them are the improved parameters of EAQECCs previously known;and some of them are new ones,but are very difficult to be obtained by using the known methods.

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量子纠错编码技术在量子信息理论中一直以来有着重要的地位。在量子纠错编码方案中，Schingemann和Werner两人提出了通过构造具有某些性质的图(矩阵)来构造非二元量子码的方法，他们利用这种图论方法构造出了很多好的量子码，并给出了量子码[[5,1,3]]p (p为大于2的素数)存在性的一个新证明。本文利用此法，通过构造Fp上满足特殊性质的8阶对称矩阵，证明对任意大于3的素数p，码长n与维数k之和等于8的所有MDS码(达到量子Singleton界)都存在。
Quantum error correction plays a crucial role in quantum information theory. Schlingemann and Werner presented a new way to construct quantum stabilizer codes by find-ing certain graphs (or matrices) with specific properties, and they constructed several new non-binary quantum codes, in particular, they gave a new proof on the existence of quantum codes [[5, 1, 3]] for all odd primes. In this paper, using the same method, we prove the existence of MDS quantum codes with the sum n and k being 8 for all primes exceeding three.

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在小型量子网络中采用量子隐形传态通信从物理机制上保证通信信息的绝对安全，但是由于量子信道存在噪声，干扰信息的正确性从而产生误码。为保证通信信息的可靠性，本文提出基于量子纠错码的小型量子网络路由通信协议。根据小型量子网络的路由特点构建路由表；依据路由表实现源量子节点到一跳、两跳目的量子节点的量子隐形传态；利用量子纠错码纠正因噪声产生的误码信息；对该协议的安全性进行理论证明。
In small quantum network ,the security of communication information can be guaranteed physically by the applica-tion of quantum teleportation ,the noise of quantum channel will be bound to interfere with the communication information ,leading to the error of code .In order to guarantee the reliability of information communication ,a routing protocol of small quantum network is proposed based on quantum error correcting code .According to the routing characteristics of small quantum network ,correspond-ing routing tables are made .On the basis of these routing tables ,quantum teleportation can be realized from source quantum node to the nodes within one-hop or two-hop .Correct the errors of code resulting from noise of quantum channel based on error correcting code theory ,and handle it accordingly ,to ensure the reliability of communication information at data link layer .The security of the protocol is proved in theory .

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构造一般二元自正交码是经典纠错码和量子纠错码研究的难点。研究基于并置二元循环矩阵的1-生成子拟循环码结构。以向量移位等价、线性码等价以及二元自正交码码字偶重量特点等为基础，设计特殊二元拟循环码结构，构造了28个最优或已知最优二元拟循环自正交码。提出自正交码截短-删除方法，构造出所获得自正交码的62个衍生码。文中的90个二元自正交码与文献[13]中最优或已知最优线性码比较，分别有67和23个二元自正交码是最优和已知最优。构造结果验证2个方法对一般二元自正交码构造的有效性，同时能较好解决量子纠错码构造中具有尽可能大对偶重量自正交码的设计问题。
Designing general binary self-orthogonal codes is a difficult problem in both classical coding theo-ry and quantum coding theory.The structure of one-generator quasi-cyclic codes constructed by concatena-ting binary circulant matrices is investigated.Twenty-eight optimal or best known binary self-orthogonal codes are built by designing the structure of a special subclass of quasi-cyclic codes,which takes advantage of some restrictions such as the shifting equivalence relation on vector,the equivalence relation on linear codes and even weight property of binary self-orthogonal codes.A puncturing-expurgating construction method for binary self-orthogonal codes is proposed,and sixty-two derived codes from these obtained self-orthogonal codes are constructed.In comparison with Literature (13),67 and 23 among our ninety self-orthogonal codes are separately optimal and best known.The construction results indicate that these two methods are effective to design general self-orthogona

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Non-Clifford操作不能在量子纠错码上自然横向实现,但可通过辅助量子态和在量子纠错码上能横向实现的Clifford操作来容错实现,从而取得容错量子计算的通用性.非平庸的单量子比特操作是Non-Clifford操作,可以分解为绕z轴和绕x轴非平庸旋转操作的组合.本文首先介绍了利用非稳定子态容错实现绕z轴和绕x轴旋转的操作,进而设计线路利用魔幻态容错制备非稳定子态集,最后讨论了运用制备的非稳定子态集模拟任意非平庸单量子比特操作的问题.与之前工作相比,制备非稳定子态的线路得到简化,成功概率提高,且在高精度模拟任意单量子比特操作时所消耗的非稳定子态数目减少了50%.
Based on the quantum error-correction codes and concatenation, quantum logical gates can be implemented transversally, which is called the fault-tolerant quantum computation. Clifford gates can be directly and fault-tolerantly performed, but they cannot reach universal quantum computation. How to implement the non-Clifford gate fault-tolerantly is a vital technique in fault-tolerant universal quantum computation. Here the magic state is selected to help the implementing of the non-Clifford gate transversally. Based on the non-stabilizer state cosθi|0?+sinθi|1?, circuits which can execute 2θi rotation around X-axis and Z-axis fault-tolerantly are proposed. Then new non-stabilizer states in this form are developed and produced from the distilled magic state. By using these states, a number of non-Clifford gates can be performed transversally, which makes profound implication in fault-tolerant quantum computation. We calculate the number of the non-stabilizer states needed fo

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将多变量公钥密码MI体制与基于QC-LDPC码的纠错码密码串联起来,提出了一种新的具有80比特安全性的抗量子计算的多变量公钥密码体制.新体制基于有限域上多变量非线性方程组求解的困难性问题及任意线性码的译码困难性问题,分析表明新体制是安全有效的,具有私钥较小的优点.
By combining multivariate public‐key cryptosystems (MI) and QC‐LDPC code‐based public‐key encryption schemes ,a novel encryption scheme which has 80 bits security is presented .The new scheme is based on NP‐hard problem of multivariate nonlinear polynomial equations over a finite field and NP‐com‐plete problem of decoding a large linear block code .Analysis shows that the scheme is secure and efficient and has the advantages of smaller private key size .

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图态是可以与数学上的图对应起来的多组分纠缠态，图的顶点在此扮演多进制量子位而连线则表示两个多进制量子位之间的相互作用．图态在量子纠错码、多体量子计算和单向量子计算中起重要作用．本文系统研究多进制图态纠缠，使用迭代计算等方法计算了局域幺正变换和图同构下不等价的所有9点图以下的三进制图态的纠缠及一部分四进制和五进制图态的纠缠，纠缠测度可以是几何纠缠、相对熵纠缠和鲁棒性纠缠．我们对计算结果进行了分类，并分析了所得到的最近分离态．
Graph states are multipartite entangled states that correspond to mathematical graphs, where the vertices of the graph now play the role of quantum multilevel systems and edges represent interactions of the systems. Graph states are the basis of quantum error correction and one-way quantum computer. We systematically study the entanglement of non-binary graph states. Using iterative algorithm and entanglement bounds, we calculate the entanglement of all the ternary graph states up to nine vertices and parts of quaternary and quinary graph states modulo local unitary transformations and graph isomorphisms. The entanglement measure can be the geometric measure, the measure of relative entropy of entanglement or the measure of logarithmic robustness. We classify the graph states according to the entanglement values obtained. The closest product states obtained in the calculations are studied.

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