  Doubly-parametric quantum transducers (DPTs), such as electro-opto-mechanical
devices, show promise as quantum interconnects between the optical and
microwave domains, thereby enabling long distance quantum networks between
superconducting qubit systems. However, any transducer will inevitably
introduce loss and noise that will degrade the performance of a quantum
network. We explore how DPTs can be used to construct a network capable of
distributing remote two-mode microwave entanglement over an optical link by
comparing fourteen different network topologies. The fourteen topologies we
analyze consist of combinations of different transducer operations, entangled
resources, and entanglement swapping measurements. For each topology, we derive
a necessary and sufficient analytic threshold on DPT parameters that must be
exceeded in order to distribute microwave-microwave entanglement. We find that
the thresholds are dependent on the given network topology, along with the
available entanglement resources and measurement capabilities. In the high
optical loss limit, which is relevant to realistic networks, we find that
down-conversion of each half of an optical two-mode squeezed vacuum state is
the most robust topology. Finally, we numerically evaluate the amount of
microwave--microwave entanglement generated for each topology using currently
achievable values for DPT parameters, entangled resources, and swapping
measurements, finding the encouraging result that several topologies are within
reach of current experimental capabilities.
