二面体群上Cayley图的完美态转移：完整且实用的表征

中文摘要 AI 辅助

图上的完美态转移因在量子信息和量子计算中的应用受到广泛关注。目前，关于Cayley图中允许完美态转移的连接集的明确表征十分罕见，仅对少数阿贝尔Cayley图有相关结果。本文对二面体群上连通Cayley图的共轭闭连接集进行表征，这类连接集允许完美态转移。通过应用拉马努金和、莫比乌斯反演，以及基于有理数的p进指数赋值的论证，我们将完美态转移施加的特征值约束转化为连接集上的明确结构条件。这一成果给出了完整且实用的表征，为识别和构建此类Cayley图提供了有效准则，还确定了精确的最小完美态转移时间。

英文摘要

Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. Explicit characterizations of connection sets admitting perfect state transfer in Cayley graphs are rare and, so far, are known only for a few abelian Cayley graphs. In this paper, we characterize the conjugation-closed connection sets of connected Cayley graphs over dihedral groups that admit perfect state transfer. By applying Ramanujan sums, Möbius inversion, and arguments based on the $p$-adic exponential valuation of rational numbers, we convert the eigenvalue constraints imposed by perfect state transfer into explicit structural conditions on the connection set. This yields a complete and practical characterization, which gives an effective criterion for recognizing and constructing such Cayley graphs and also determines the exact minimum perfect state transfer time.

发表机构

• Beijing Normal University（北京师范大学）

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