  In this work, we study the controllability of the bilinear Schr\"odinger
equation on infinite graphs for periodic quantum states. We consider the
bilinear Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the
Hilbert space $L^2_p$ composed by functions defined on an infinite graph
$\mathscr{G}$ verifying periodic boundary conditions on the infinite edges. The
Laplacian $-\Delta$ is equipped with specific boundary conditions, $B$ is a
bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We
present the well-posedness of the system in suitable subspaces of
$D(|\Delta|^{3/2})$ . In such spaces, we study the global exact controllability
and we provide examples involving for instance tadpole graphs and star graphs
with infinite spokes.
