Pretty good state transfer in Grover walks on abelian Cayley graphs
中文速览 本文研究了在一种称为格罗弗行走（Grover walk）的量子行走模型中，实现“几乎完美状态转移”（Pretty Good State Transfer, PGST）的条件。论文的核心贡献在于，通过运用切比雪夫多项式、谱图论和数论方法，为任意图上的格罗弗行走建立了一个普适的、关于PGST的充要条件。随后，作者将此理论聚焦于阿贝尔群上的凯莱图，并最终完整地刻画了酉凯莱图上发生PGST的条件。研究发现，当且仅当酉凯莱图的阶数\(n\)为\(2m\)或\(4m\)（其中\(m\)为无平方因子奇数）时，才会出现PGST。这一结论揭示了大量仅能实现几乎完美状态转移、而无法实现完美状态转移（PST）的新图类，从而显著扩展了可用于高保真量子信息传输的图结构范围。 English Research Briefing Research Briefing: Pretty good state transfer in Grover walks on abelian Cayley graphs 1. The Core Contribution This paper establishes a comprehensive theoretical framework for characterizing Pretty Good State Transfer (PGST) in the context of Grover walks on graphs. Its central thesis is that the conditions for PGST can be precisely determined through a combination of spectral graph theory and number theory. The paper’s primary conclusion is a necessary and sufficient condition for the occurrence of PGST on abelian Cayley graphs, leading to the complete characterization of PGST on unitary Cayley graphs. This work successfully identifies infinite new families of graphs that exhibit PGST but do not admit the rarer Perfect State Transfer (PST), thereby significantly broadening the landscape of topologies considered viable for high-fidelity quantum communication. ...