Abstract In this paper, within the framework of continuous-time quantum walks (CTQWs), we decompose graphs into disjoint strata by adopting the idea of equitable partitions. Based on the thought, we investigate the success probability of finding marked vertices on the N -fold star power graph G ⋆ N (where G is a cycle graph with three vertices), as well as the time required for perfect state transfer. Additionally, we explore the time for perfect state transfer on G ⋆ N under partition conditions from the perspective of quantum probability. Specifically, we surprisingly discover an interesting result: the times for perfect state transfer under quantum probability-based partitions and equitable partitions are completely consistent when the coupling strength J = 1 and the parameter λ = 1.Abstract In this paper, within the framework of continuous-time quantum walks (CTQWs), we decompose graphs into disjoint strata by adopting the idea of equitable partitions. Based on the thought, we investigate the success probability of finding marked vertices on the N -fold star power graph G ⋆ N (where G is a cycle graph with three vertices), as well as the time required for perfect state transfer. Additionally, we explore the time for perfect state transfer on G ⋆ N under partition conditions from the perspective of quantum probability. Specifically, we surprisingly discover an interesting result: the times for perfect state transfer under quantum probability-based partitions and equitable partitions are completely consistent when the coupling strength J = 1 and the parameter λ = 1.

Physica ScriptaDec 1, 20250 citations

Perfect state transfer via quantum probability theory

S. Salimi, S. Ghoraishipour, A. Sorouri

arXiv:1007.3002Quantum Information ProcessingMar 26, 20127 citations

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

Banita Katuwal, Y Lakshmi Naidu, Srinath M S +1 more

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.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-pharXiv:2609.30490Sep 28, 20260 citations

Perfect state transfer on Cayley graphs over dihedral groups

Xiwang Cao, Keqin Feng

Recently, there are extensive studies on perfect state transfer on graphs due to their significant applications in quantum information processing and quantum computations. However, most of the graphs previously investigated are abelian Cayley graphs. In this paper, we study perfect state transfer on Cayley graphs over dihedral groups. Using the representations of the dihedral group , we present some necessary and sufficient conditions for the Cayley graph to have a perfect state transfer between two distinct vertices for some connection set S. Based on these conditions, we show that cannot have PST if n is odd and S is conjugation-closed. For some even integers n, it is possible for to have PST, some concrete constructions are provided.Recently, there are extensive studies on perfect state transfer on graphs due to their significant applications in quantum information processing and quantum computations. However, most of the graphs previously investigated are abelian Cayley graphs. In this paper, we study perfect state transfer on Cayley graphs over dihedral groups. Using the representations of the dihedral group , we present some necessary and sufficient conditions for the Cayley graph to have a perfect state transfer between two distinct vertices for some connection set S. Based on these conditions, we show that cannot have PST if n is odd and S is conjugation-closed. For some even integers n, it is possible for to have PST, some concrete constructions are provided.

Linear and Multilinear AlgebraApr 3, 201928 citations

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.