  We study the state transfer through quantum walks placed on a bounded
one-dimensional path. We first consider continuous-time quantum walks from a
Gaussian state. We find such a state when superposing centered on the starting
and antipodal positions preserves a high fidelity for a long time and when sent
on large circular graphs. Furthermore, it spreads with a null group velocity.
We also explore discrete-time quantum walks to evaluate the qubit fidelity
throughout the walk. In this case, the initial state is a product of states
between a qubit and a Gaussian superposition of position states. Then, we add
two $\sigma_x$ gates to confine this delocalized qubit. We also find that this
bounded system dynamically enables periodic recovery of the initial separable
state. We outline some applications of our results in dynamic graphs and
propose quantum circuits to implement them based on the available literature.
