single module of trapped-ion qubits will likely contain no more than $10-100$ ions, noting that the speed at which gates are possible becomes slower as the module is expanded. On the network scale, we might imagine a network of nodes over longer distances connected by quantum repeaters [@Briegel1998]. In such a network, establishing direct links between every possible pair of $N$ nodes would require $\Theta(N^2)$ sets of quantum repeaters, a prohibitive cost as $N$ becomes large.

The state of the art in quantum technologies, such as ion traps and superconducting qubits, is the ability to control a small number ($\sim 10-100$) of physical qubits using certain fixed sets of one- and two-qubit operations. Instead of increasing the size of these modules, one could instead build a network out of many small modules that are connected at a higher level in a sparse way, perhaps by optical communication links [@Monroe2014].

Our first goal will be to describe modular architectures in the language of graph theory. This will then allow us to quantify and compare their connectivity properties against other network designs, notably the nearest-neighbor grid architecture.

An unweighted graph $G = (V,E)$ is conventionally specified by a set of vertices $V$, 