arXiv 0712.36052007-12quant-ph

逻辑函数与量子纠错码

Logic Functions and Quantum Error Correcting Codes

Yajie Xu, Zhi Ma, Chunyuan Zhang, Xin Lü

备注：: 12 pages

摘要：

本文基于逻辑函数与量子纠错码的关系，将量子纠错码的构造统一到图、投影器和逻辑函数上。给出了素域GF（p）上QECC的一个构造，文[8]中的一个结果可以看作是本文一个定理的推论.借助布尔函数，我们从数学的角度给出了图QECC存在性的一个明确证明，并发现GF（p）上[[n，k，d]] QECC的存在性需要与文献[9]中描述的条件类似的条件.给出了在参考文献[17]定义的对应关系下，对应于简单无向图的GF（2）上的每个[[n，0，d]] QECC都有一个布尔基状态，该状态与图的邻接矩阵密切相关。在修改了算子的定义之后，我们发现通过参考文献[11]中描述的投影器构造的一些QECC可以具有布尔基态。给出了布尔函数用于投影作图的一个必要条件。我们也给出了一些例子来说明我们的结果。

英文摘要：

In this paper, based on the relationship between logic functions and quantum error correcting codes(QECCs), we unify the construction of QECCs via graphs, projectors and logic functions. A construction of QECCs over a prime field GF(p) is given, and one of the results given by Ref[8] can be viewed as a corollary of one theorem in this paper. With the help of Boolean functions, we give a clear proof of the existence of a graphical QECC in mathematical view, and find that the existence of an [[n,k,d]] QECC over GF(p) requires similar conditions with that depicted in Ref[9]. The result that under the correspondence defined in Ref[17], every [[n,0,d]] QECC over GF(2) corresponding to a simple undirected graph has a Boolean basis state, which is closely related to the adjacency matrix of the graph, is given. After a modification of the definition of operators, we find that some QECCs constructed via projectors depicted in Ref[11] can have Boolean basis states. A necessary condition for a Boolean function being used in the construction via projectors is given. We also give some examples to illustrate our results.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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