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arXiv:2506.07439

Perfect state transfer in Grover walks on association schemes and distance-regular graphs

Koushik Bhakta, Bikash Bhattacharjya

Published Jun 9, 2025• 0 citations
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Abstract

This paper investigates perfect state transfer in Grover walks, a model of discrete-time quantum walks. We establish a necessary and sufficient condition for the occurrence of perfect state transfer on graphs belonging to an association scheme. Our focus includes specific association schemes, namely the Hamming and Johnson schemes. We characterize all graphs on the classes of Hamming and Johnson schemes that exhibit perfect state transfer. Furthermore, we study perfect state transfer on distance-regular graphs. We provide complete characterizations for exhibiting perfect state transfer on distance-regular graphs of diameter $2$ and diameter $3$, as well as integral distance-regular graphs.

DOI
10.48550/arxiv.2506.07439

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Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.

•Aug 13, 2014•34 citations
This paper investigates perfect state transfer in Grover walks, a model of discrete-time quantum walks. We establish a necessary and sufficient condition for the occurrence of perfect state transfer on graphs belonging to an association scheme. Our focus includes specific association schemes, namely the Hamming and Johnson schemes. We characterize all graphs on the classes of Hamming and Johnson schemes that exhibit perfect state transfer. Furthermore, we study perfect state transfer on distance-regular graphs. We provide complete characterizations for exhibiting perfect state transfer on distance-regular graphs of diameter 222 and diameter 333, as well as integral distance-regular graphs.
A Cayley graph Cay⁡(Γ,S)\operatorname{Cay}(Γ,S)Cay(Γ,S) over a finite group ΓΓΓ is said to be normal if its connection set SSS is a union of some conjugacy classes of ΓΓΓ. This paper investigates perfect state transfer in Grover walks on normal Cayley graphs. The Grover walk is a widely studied discrete-time quantum walk. We establish a necessary and sufficient condition for the occurrence of perfect state transfer on normal Cayley graphs. As applications, we derive explicit spectral criteria for perfect state transfer on Cayley graphs over abelian groups, dicyclic groups, and dihedral groups. These results yield several infinite families of Cayley graphs exhibiting perfect state transfer. We further obtain simple combinatorial characterizations of the existence of perfect state transfer on Cayley graphs over dihedral and dicyclic groups. Our general characterization also recovers a number of previously known results as special cases. As a further consequence, we obtain a complete characterization of perfect state transfer on unitary Cayley graphs. In particular, we prove that exactly four graphs in the class of unitary Cayley graphs exhibit perfect state transfer.
The unitary Cayley graph has vertex set $\{0,1, \hdots ,n-1\}$, where two vertices uuu and vvv are adjacent if gcd⁡(u−v,n)=1\gcd(u - v, n) = 1gcd(u−v,n)=1. In this paper, we study periodicity and perfect state transfer of Grover walks on the unitary Cayley graphs. We characterize all periodic unitary Cayley graphs. We prove that periodicity is a necessary condition for occurrence of perfect state transfer on a vertex-transitive graph. Also, we provide a necessary and sufficient condition for the occurrence of perfect state transfer on circulant graphs. Using these, we prove that only four graphs in the class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walks on integral regular graphs.
The paper investigates perfect state transfer (PST) in Grover walks on Cayley graphs over the dihedral group DnD_nDn​. The Grover walk is a discrete-time quantum walk widely studied in quantum information processing. A Cayley graph Cay⁡(Γ,S)\operatorname{Cay}(Γ,S)Cay(Γ,S) is called normal if SSS is the union of some conjugacy classes of the group ΓΓΓ; otherwise, it is called non-normal. Most existing studies have been restricted to Cayley graphs over abelian groups. In contrast, we investigate both normal and non-normal cases for Cayley graphs over the non-abelian group DnD_nDn​. By examining the parity of nnn and the normality of the Cayley graph, we obtain a complete characterization of PST on Cay⁡(Dn,S)\operatorname{Cay}(D_n,S)Cay(Dn​,S). In particular, we establish necessary and sufficient conditions for the occurrence of PST in all possible cases, and prove that PST does not occur for normal Cayley graphs when nnn is odd. Furthermore, we construct several infinite families of normal and non-normal Cayley graphs Cay⁡(Dn,S)\operatorname{Cay}(D_n,S)Cay(Dn​,S) that exhibit PST, illustrating the application of the main result. Our approach is based on the representation theory of the dihedral group.
We study perfect state transfer in Grover walks on two important classes of graphs: strongly regular graphs and strongly walk-regular graphs. The latter class is a generalization of the former. We first give a complete classification of strongly regular graphs that admit perfect state transfer. The only such graphs are the complete bipartite graph K2,2K_{2,2}K2,2​ and the complete tripartite graph K2,2,2K_{2,2,2}K2,2,2​. We then show that, if a connected strongly walk-regular graph that is not a strongly regular graph admits perfect state transfer, then its spectrum must be of the form {[k]1,[k2]α,[0]β,[−k2]γ}\{[k]^1, [\frac{k}{2}]^α, [0]^β, [-\frac{k}{2}]^γ\}{[k]1,[2k​]α,[0]β,[−2k​]γ}, and we enumerate all feasible spectra of this form up to k=20k=20k=20 with the help of a computer. These results are obtained using techniques from algebraic number theory and spectral graph theory, particularly through the analysis of eigenvalues and eigenprojections of a normalized adjacency matrix. While the setting is in quantum walks, the core discussion is developed entirely within the framework of spectral graph theory.
We completely classify connected Cayley graphs over finite abelian groups that exhibit perfect state transfer in Grover walks. First, we show that a Cayley graph over any finite group exhibiting perfect state transfer has valency 111, 222, or 444. For valency 444, we prove that an abelian Cayley graph Cay⁡(Γ,S)\operatorname{Cay}(Γ,S)Cay(Γ,S) exhibits perfect state transfer if and only if S={a,−a,a+h,−a+h}S=\{a,-a,a+h,-a+h\}S={a,−a,a+h,−a+h}, where hhh has order 222, and 4∤∣Γ∣/24\nmid |Γ|/24∤∣Γ∣/2. Combining this characterization with the valency-one and valency-two cases, we explicitly determine all pairs (Γ,S)(Γ,S)(Γ,S) for which Cay⁡(Γ,S)\operatorname{Cay}(Γ,S)Cay(Γ,S) exhibits perfect state transfer. Our results extend the classification of circulant graphs of valency at most four obtained by Kubota and Yoshino [Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks, \emph{J. Comb. Theory Ser. A.}, 216 (2025) 106064] to Cayley graphs over arbitrary finite abelian groups and to all valencies.
Let X be a graph, A its adjacency matrix, and t a non-negative real number. The matrix exp(i t A) determines the evolution in time of a certain quantum system defined on the graph. It represents a continuous-time quantum walk in X. We say that X admits perfect state transfer from a vertex u to a vertex v if there is a time t such that \n| exp(i t A)_(u,v) | = 1. \n \nThe main problem we study in this thesis is that of determining which simple graphs admit perfect state transfer. For some classes of graphs the problem is solved. For example, a path on n vertices admits perfect state transfer if and only if n=2 or n=3. However, the general problem of determining all graphs that admit perfect state transfer is substantially hard. \n \nIn this thesis, we focus on some special cases. We provide necessary and sufficient conditions for a distance-regular graph to admit perfect state transfer. In particular, we provide a detailed account of which distance-regular graphs of diameter three do so. \n \nA graph is said to be spectrally extremal if the number of distinct eigenvalues is equal to the diameter plus one. Distance-regular graphs are examples of such graphs. We study perfect state transfer in spectrally extremal graphs and explore rich connections to the topic of orthogonal polynomials. We characterize perfect state transfer in such graphs. \n \nWe also provide a general framework in which perfect state transfer in graph products can be studied. We use this to determine when direct products and double covers of graphs admit perfect state transfer. As a consequence, we provide many new examples of simple graphs admitting perfect state transfer. We also provide some advances in the understanding of perfect state transfer in Cayley graphs for the groups (Z_2)^d and Z_n. \n \nFinally, we consider the problem of determining which trees admit perfect state transfer. We show more generally that, except for the path on two vertices, if a connected bipartite graph contains a unique perfect matching, then it cannot admit perfect state transfer. We also consider this problem in the context of another model of quantum walks determined by the matrix exp(i t L), where L is the Laplacian matrix of the graph. In particular, we show that no tree on an odd number of vertices admits perfect state transfer according to this model.