introduce a binary operation on graphs known as the hierarchical product, describe how it can be used to produce families of graphs we call hierarchies, and discuss the properties of these hierarchies. In \[sec:comparison\], we will compare hierarchies to other families of graphs, examining how certain graph-theoretic quantities scale with the total number of included qubits. In \[sec:construction\], we will use analytic and numerical methods to examine how long is required to construct GHZ states spanning our graphs or to transfer states across them. Finally, in \[sec:placement\], we will show how the unique structure of hierarchies allows for simple heuristics to map qubits in an algorithm into physical locations in hardware.

Hierarchical Products of Graphs {#sec:hp}
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Background and Notation
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One of the defining features of modularity in a network is the presence of clusters of nodes that are well-connected. Qualitatively, a modular network can be partitioned into such node clusters, or *modules*, that have a sparse interconnectivity. In quantum networking, it is believed that fully connected architectures will suffer greatly decreasing performance or increasing costs as the number of nodes becomes larger, and this motivates the search for alternative network designs. For instance, Ref. [@Monroe2014] estimates that a 