  The goal in quantum state transfer is to avoid the need to physically
transport carriers of quantum information. This is achieved by using a suitably
engineered Hamiltonian that induces the transfer of the state of one subsystem
to another. A less known generalization of state transfer considers multiple
systems such that any pair can exchange quantum information and transfers can
take place at any time, starting and stopping independently. This is sometimes
called routing of quantum states. State transfer in particular has received a
great deal of attention, however the vast majority of results in both state
transfer and routing concern qubits transferred in a network of restricted
structure. Here we consider routing of single-mode Gaussian states and
entanglement through complex networks of quantum harmonic oscillators. We
compare a protocol where the transfer is completed in a single step but the
effective Hamiltonian only approximately transfers the state with one where the
transfer can in principle be perfect but the transfer is done in two steps, and
also illustrate the state-dependency of the transfer fidelity. We find that
even in a random and homogeneous network, the transfer fidelity still depends
on the degree of the nodes for any link density, and that in both random and
complex networks it is the community structure that controls the appearance of
higher frequency normal modes useful for transfer. Finally, we find that
networks of sufficient complexity may have superior routing performance over
superficially similar random networks. Our results pave the way for further
exploration of the role of community structure in state transfer and related
tasks.
