a useful starting point is that, in nearest-neighbor connected systems, performing these tasks using unitary processes from an initial product state is limited by the Lieb-Robinson bound [@Bravyi2006a; @Bentsen2018]. It takes a time proportional to the distance between two points to establish maximal quantum correlation between them. By examining these tasks on a range of different graphs, we hope to understand how the graph structure can affect the limitations on quantum processes caused by locality considerations. Prior work has characterized the difficulty of creating graph states [@Cuquet2012], but preparation of such states is not limited by Lieb-Robinson considerations.

Our work in this paper should be contrasted with work on entanglement percolation [@Acin2007; @Perseguers2013]. Entanglement percolation describes the process of using low-quality entanglement between adjacent nodes on a graph to create one unit of long-range, high-quality entanglement (e.g., a Bell pair). The use of entanglement percolation to prepare large cluster states on a lattice was considered in Ref. [@Kieling2007]. The nature of entanglement growth in complex networks was considered in Refs. [@Cuquet2009; @Cuquet2011], showing that so-called “scale-free” networks are particularly easy to produce large entangled states in. We are interested in the overall capability of different graph structures to perform large 