Lively quantum walks on cycles

We introduce a family of quantum walks on cycles parametrized by their liveliness, defined by the ability to execute a long-range move. We investigate the behaviour of the probability distribution and time-averaged probability distribution. We show that the liveliness parameter, controlling the magnitude of the additional long-range move, has a direct impact on the periodicity of the limiting distribution. We also show that the introduced model provides a method for network exploration which is robust against trapping.