  Perfect state transfer between qubits on a uniformly coupled network, with
interactions specified by a graph, has advantages over an engineered chain,
such as much faster transfer times (independent of the distance between the
input and output vertices). This is achieved by many couplings working in
parallel. The trade-offs seem to be the need for increasing connectivity
between qubits, and a large number of vertices in the graph. The size of
existing graph constructions scale exponentially in the transfer distance,
making these schemes impractical over anything but the shortest distances. This
prompts the question of "How low can you go?" for the size of the graph
achieving a particular transfer distance. In this paper, we present reductions
in the vertex count required, although the overall scaling with transfer
distance remains exponential. We also tighten existing bounds on the required
degree of the vertices of the graph.
