Unitary coin discrete-time quantum walk on complete graph K4 based on adjacency matrix decomposition

The Unitary Coin Discrete-Time Quantum Walk (UCDTQW) serves as a universal model for quantum computers. In this paper, we conduct matrix analysis based on the unitary operator of UCDTQW, constructing the system’s shift operator as the unitary form of the adjacency matrix corresponding to the complete graph Formula: see text. Through a set of transformation equations, we realize the reciprocal mapping between the adjacency matrix and the shift operator. Specifically, by splitting the undirected edges of Formula: see text into multiple directed arcs, we construct a directed multigraph structure. Our study reveals that the unitary evolution operator of this multigraph exhibits properties analogous to those of the shift operator, particularly uncovers the periodic characteristics of the wave function. Furthermore, under another edge connection scheme, we achieve the construction of the quantum circuit representation for the shift operator.