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arXiv2026

Construction of Partial Join Graphs with Perfect State Transfer in Shunt Decomposition-Based Quantum Walks

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.

quant-ph↗

arXiv2026

Perfect State Transfer on Quotient Graphs in Shunt Decomposition-Based Quantum Walks

This paper investigates perfect state transfer (PST) in discrete-time quantum walks constructed via the shunt decomposition method. The walks are defined on a graph $G$ and its associated quotient graph $G/π$, induced by an equitable partition $π$. Through the shunt decomposition of $G$, we derive an explicit relation between the shift operator of the parent graph $G$ and that of its quotient graph $G/π$. We construct a reflection operator based on the characteristic matrix, which establishes a connection between the transition operator of the parent graph and that of its lower-dimensional quotient graph. We then prove that PST occurs on $G$ if and only if it occurs on $G/π$. Furthermore, we express the unitary evolution operator of the quotient graph in terms of Chebyshev polynomials of the first kind, from which we derive explicit criteria for PST. As an application, we establish PST on the cycle graph $C_{n}$ at time $k = n/2$, and lift the result to the parent graph $C_{2n}$ via the equitable partition $π$. We further show that if an equitable partition $π$ of $G$ induces a quotient isomorphic to $K_n^{\circlearrowleft}$, the complete digraph on $n$ vertices with a loop at every vertex, then PST occurs at step $k = n$, and the walk is periodic at $k = 2n$. This framework is applied to two families of graphs, which are the complete bipartite digraph $K_{n,n}^{\rightleftharpoons}$ and the circulant graph $\operatorname{Circ}(2n, S)$, where $S$ consists of all odd residues modulo $2n$ and $n = 2^s$ for some $s \geq 1$, establishing PST in their respective line digraphs. Collectively, these results also answer the question posed by Godsil and Zhan concerning which shunt decompositions or embeddings of a graph admit PST.

math.CO↗