Perfect state transfer in Grover walks on normal Cayley graphs

This paper investigates the phenomenon of perfect state transfer (PST) in Grover walks, a type of discrete-time quantum walk, specifically within the class of normal Cayley graphs. A Cayley graph is considered normal if its connection set is a union of conjugacy classes of the underlying group, a property that allows for the use of group representation theory to analyze the graph's adjacency matrix and spectral decomposition. The authors aim to provide a general characterization of PST in these graphs and apply this framework to specific group types, including abelian, dicyclic, and dihedral groups.

The authors prove that PST occurs between two vertices in a normal Cayley graph if and only if the relative group element connecting them is a central element of order 2, and the eigenvalues of the graph's discriminant matrix satisfy specific conditions related to Chebyshev polynomials. By leveraging the representation theory of finite groups, they derive explicit spectral criteria for PST. These criteria are then used to identify infinite families of graphs that exhibit PST. Additionally, the paper provides a complete characterization of PST in unitary Cayley graphs, demonstrating that only four such graphs exhibit this property.

Perfect state transfer is a critical requirement for quantum communication and information processing, as it allows for the reliable transmission of quantum states between nodes in a network. By extending the study of PST from simple circulant or abelian graphs to the broader class of normal Cayley graphs, this work provides a powerful algebraic toolset for designing quantum networks with high symmetry. The combinatorial and spectral criteria developed here simplify the search for graphs capable of perfect state transfer, offering a systematic approach to identifying structures that support efficient quantum information transport.