  In the literature, there are numerous studies of one-dimensional
discrete-time quantum walks (DTQWs) using a moving shift operator. However,
there is no exact solution for the limiting probability distributions of DTQWs
on cycles using a general coin or swapping shift operator. In this paper, we
derive exact solutions for the limiting probability distribution of quantum
walks using a general coin and swapping shift operator on cycles for the first
time. Based on the exact solutions, we show how to generate symmetric quantum
walks and determine the condition under which a symmetric quantum walk appears.
Our results suggest that choosing various coin and initial state parameters can
achieve a symmetric quantum walk. By defining a quantity to measure the
variation of symmetry, deviation and mixing time of symmetric quantum walks are
also investigated.
