Dukes (2014) and Konno, Shimizu, and Takei (2017) studied the periodicity for 2-state quantum walks whose coin operator is the Hadamard matrix on cycle graph CN with N vertices. The present paper treats the periodicity for 3-state quantum walks on CN . Our results follow from a new method based on cyclotomic field. This method shows a necessary condition for the coin operator of quantum walks to have the finite period. Moreover, we reveal the period TN of two kinds of typical quantum walks, the Grover and Fourier walks. We prove that both walks do not have any finite period except for N = 3, in which case T3 = 6 (Grover), = 12 (Fourier).

Dukes (2014) and Konno, Shimizu, and Takei (2017) studied the periodicity for 2-state quantum walks whose coin operator is the Hadamard matrix on cycle graph C_N with N vertices. The present paper treats the periodicity for 3-state quantum walks on C_N. Our results follow from a new method based on the cyclotomic field. This method gives a necessary condition for the coin operator for quantum walks to be periodic. Moreover, we reveal the period T_N of typical two kinds of quantum walks, the Grover and Fourier walks. We prove that both walks do not have any finite period except for N=3, in which case T_3=6 (Grover), =12 (Fourier).

We provide a detailed analysis of the survival probability of the Grover walk on the ladder graph with an absorbing sink. This model was discussed in Mareš et al., Phys. Rev. A 101, 032113 (2020), as an example of counter-intuitive behaviour in quantum transport where it was found that the survival probability decreases with the length of the ladder L, despite the fact that the number of dark states increases. An orthonormal basis in the dark subspace is constructed, which allows us to derive a closed formula for the survival probability. It is shown that the course of the survival probability as a function of L can change from increasing and converging exponentially quickly to decreasing and converging like L-1 simply by attaching a loop to one of the corners of the ladder. The interplay between the initial state and the graph configuration is investigated.