In this paper, we successfully obtain an explicit expression of the limit distribution [Formula: see text] of three-state quantum walks on cycles, the total variation distance between [Formula: see text] and the average probability [Formula: see text], and lower bound on the difference between two eigenvalues, among others. Based on the above conclusions, we finally get the mixing time [Formula: see text] of the quantum walk of Grover coin on the N-cycle. [Formula: see text] is the time required to characterize [Formula: see text] approaching [Formula: see text]. Our results show that the average probability of a three-state quantum walk on a cycle can approach its limit distribution faster than that of a two-state quantum walk, which might be of significance to quantum computation.

In this paper, on the basis of constructing a new shift operator [Formula: see text] and choosing the Grover coin [Formula: see text] as coin operator [Formula: see text], we get the standard evolution operator [Formula: see text] on cycles. Using [Formula: see text], we not only got the analytical expression of wavefunction [Formula: see text], but also obtained the conclusion that the limit distribution [Formula: see text] of [Formula: see text], which is not uniform distribution, regardless of [Formula: see text] is odd or even.

In this paper, continuous-time quantum walk on hypercube is discussed in view of Cartesian product structure. We find that the [Formula: see text]-fold Cartesian power of the complete graph [Formula: see text] is the [Formula: see text]-dimensional hypercube, which give us new ideas for the study of quantum walk on hypercube. Combining the product structure, the spectral distribution of the graph and the quantum decomposition of the adjacency matrix, the probability amplitudes of the continuous-time quantum walker’s position at time [Formula: see text] are given, and it is discussed that the probability distribution for the continuous-time case is uniform when [Formula: see text]. The application of this product structure greatly improves the study of quantum walk on complex graphs, which has far-reaching influence and great significance.

In this paper, continuous-time quantum walk is discussed based on the view of quantum probability, i.e. the quantum decomposition of the adjacency matrix A of graph. Regard adjacency matrix A as Hamiltonian which is a real symmetric matrix with elements 0 or 1, so we regard [Formula: see text] as an unbiased evolution operator, which is related to the calculation of probability amplitude. Combining the quantum decomposition and spectral distribution [Formula: see text] of adjacency matrix A, we calculate the probability amplitude reaching each stratum in continuous-time quantum walk on complete bipartite graphs, finite two-dimensional lattices, binary tree, [Formula: see text]-ary tree and [Formula: see text]-fold star power [Formula: see text]. Of course, this method is also suitable for studying some other graphs, such as growing graphs, hypercube graphs and so on, in addition, the applicability of this method is also explained.