基于分流分解的量子行走中具有完美态转移的部分连接图的构造
Construction of Partial Join Graphs with Perfect State Transfer in Shunt Decomposition-Based Quantum Walks
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- Sri Sathya Sai Institute of Higher Learning（斯里·萨蒂亚·赛高等学习研究所）
- National Institute of Technology Agartala（阿加塔拉国家技术学院）
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中文总结 AI 辅助
本文通过分流分解框架构造带符号耦合的有向部分连接图的量子行走转移算子，推导周期性与完美态转移条件，并利用双覆盖构造扩展了可实现完美态转移的图族类别。
中文摘要 AI 辅助
本文定义了带符号耦合的有向部分连接图，并利用分流分解框架为这些图构造了离散时间量子行走的转移算子。所得的转移算子适用于几个重要的图族，包括带环的完全图、循环部分连接图、完全二分图、完全二分图的张量幂，所有这些图都带有符号耦合。对于每个图族，我们确定了转移算子的相应结构，并推导了当两个有向正则图之一具有完美态转移（PST）时，周期性和完美态转移的充要条件。基于这些结果，我们识别了两种类型的态转移：内部PST，发生在同一图内的顶点之间；以及耦合PST，发生在连接图的两个分量之间。我们进一步为有向部分连接图开发了一种双覆盖构造，并推导了当相关转移算子不一定交换时周期性和PST的条件。利用这种构造，我们建立了带环完全图的双覆盖的PST结果。特别地，我们提供了一个例子，其中完全图\\(K_n\\)在\\(n\geq4\\)时不表现出PST，而\\(K_n\\)的适当部分连接图在\\(n=2^m\\)，\\(m\geq2\\)时表现出PST。因此，这些结果扩展了在基于分流分解的量子行走中允许PST的图族类别，并为研究图连接、乘积和覆盖中的量子态转移提供了一个统一框架。
英文摘要
In this paper, we define directed partial join graphs with signed couplings and construct discrete-time quantum-walk transition operators for these graphs using the shunt-decomposition framework. The resulting transition operators apply to several important graph families, including complete graphs with loops, circulant partial joins, complete bipartite graphs, tensor powers of complete bipartite graphs, all with signed couplings. For each family, we identify the corresponding structure of the transition operator and derive necessary and sufficient conditions for periodicity and perfect state transfer (PST), when one of the two directed regular graphs admits PST. Based on these results, we identify two types of state transfer; internal PST, which occurs between vertices within the same graph, and coupling PST, which occurs between two components of join graphs. We further develop a double-cover construction for directed partial join graphs and derive conditions for periodicity and PST when the associated transition operators do not necessarily commute. Using this construction, we establish PST results for double covers of complete graphs with loops. In particular, we provide an example in which the complete graph \(K_n\) does not exhibit PST for \(n\geq4\), whereas a suitable partial join of \(K_n\) exhibits PST when \(n=2^m\), \(m\geq2\). Hence, these results extend the class of graph families admitting PST in shunt-decomposition-based quantum walks and provide a unified framework for studying quantum state transfer in graph joins, products, and covers.
