We systematically investigated perfect state transfer between antipodal nodes of discrete time quantum walks on variants of the cycles $C_4$, $C_6$ and $C_8$ for three choices of coin operator. Perfect state transfer was found, in general, to be very rare, only being preserved for a very small number of ways of modifying the cycles. We observed that some of our useful modifications of $C_4$ could be generalised to an arbitrary number of nodes, and present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state transfer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state, interpolating between structures and by adding decoherence.

We introduce a variation of the discrete time quantum walk, the nonreversal quantum walk, which does not step back onto a position which it has just occupied. This allows us to simulate a dimer and we achieve it by introducing a new type of coin operator. The nonrepeating walk, which never moves in the same direction in consecutive time steps, arises by a permutation of this coin operator. We describe the basic properties of both walks and prove that the even-order joint moments of the nonrepeating walker are independent of the initial condition, being determined by five parameters derived from the coin instead. Numerical evidence suggests that the same is the case for the nonreversal walk. This contrasts strongly with previously studied coins, such as the Grover operator, where the initial condition can be used to control the standard deviation of the walker.

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