Quantum decomposition of random walk on Cayley graph of finite group

•I first introduce the decomposition of quantum random walk operator by the help of quantum probability approach.•We then propose the state decomposition of quantum random walk with the help of eigenvalue and eigenvector.•As an application we then discuss the probability distribution for the quantum random walk on Cayley graph.

In the paper, A quantum decomposition (QD, for short) of random walk on Cayley graph of finite group is introduced, which contains two cases. One is QD of quantum random walk operator (QRWO, for short), another is QD of Quantum random walk state (QRWS, for short). Using these findings, I finally obtain some applications for quantum random walk (QRW, for short), which are of interest in the study of QRW, highlighting the role played by QRWO and QRWS.