  In this paper, we consider discrete time quantum walks on graphs with coin
focusing on the decentralized model, where the coin operation is allowed to
change with the vertex of the graph. When the coin operations can be modified
at every time step, these systems can be looked at as control systems and
techniques of geometric control theory can be applied. In particular, the set
of states that one can achieve can be described by studying controllability.
Extending previous results, we give a characterization of the set of reachable
states in terms of an appropriate Lie algebra. We then prove general results
and criteria relating controllability to the algebraic and topological
properties of the walk. As a consequence of these results, we prove that if the
degree of the underlying graph is larger than $\frac{N}{2}$, where $N$ is the
number of nodes, the quantum walk is always completely controllable, i.e., it
is possible to having it to evolve according to an arbitrary unitary evolution.
Another result is that controllability for decentralized models only depends on
the graph and not on the particular quantum walk defined on it. We also provide
explicit algorithms for control and quantify the number of steps needed for an
arbitrary state transfer. The results of the paper are of interest in quantum
information theory where quantum walks are used and analyzed in the development
of quantum algorithms.
