  The construction of large-scale quantum computers will require modular
architectures that allow physical resources to be localized in easy-to-manage
packages. In this work, we examine the impact of different graph structures on
the preparation of entangled states. We begin by explaining a formal framework,
the hierarchical product, in which modular graphs can be easily constructed.
This framework naturally leads us to suggest a class of graphs, which we dub
hierarchies. We argue that such graphs have favorable properties for quantum
information processing, such as a small diameter and small total edge weight,
and use the concept of Pareto efficiency to identify promising quantum graph
architectures. We present numerical and analytical results on the speed at
which large entangled states can be created on nearest-neighbor grids and
hierarchy graphs. We also present a scheme for performing circuit
placement--the translation from circuit diagrams to machine qubits--on quantum
systems whose connectivity is described by hierarchies.
