  We study pretty good quantum state transfer (i.e., state transfer that
becomes arbitrarily close to perfect) between vertices of graphs with an
involution in the presence of an energy potential. In particular, we show that
if a graph has an involution that fixes at least one vertex or at least one
edge, then there exists a choice of potential on the vertex set of the graph
for which we get pretty good state transfer between symmetric vertices of the
graph. We show further that in many cases, the potential can be chosen so that
it is only non-zero at the vertices between which we want pretty good state
transfer. As a special case of this, we show that such a potential can be
chosen on the endpoints of a path to induce pretty good state transfer in paths
of any length. This is in contrast to the result of [6], in which the authors
show that there cannot be perfect state transfer in paths of length 4 or more,
no matter what potential is chosen.
