  Here we consider the speed at which quantum information can be transferred
between the nodes of a linear network. Because such nodes are linear
oscillators, this speed is also important in the cooling and state preparation
of mechanical oscillators, as well as frequency conversion. We show that if
there is no restriction on the size of the linear coupling between two
oscillators, then there exist control protocols that will swap their respective
states with high fidelity within a time much less than a single oscillation
period. Standard gradient search methods fail to find these fast protocols. We
were able to do so by augmenting standard search methods with a path-tracing
technique, demonstrating that this technique has remarkable power to solve
time-optimal control problems, as well as confirming the highly challenging
nature of these problems. As a further demonstration of the power of
path-tracing, first introduced by Moore-Tibbets et al. [Phys. Rev. A 86, 062309
(2012)], we apply it to the generation of entanglement in a linear network.
