Minimising the number of edges in LC-equivalent graph states

Graph states are a powerful class of entangled states with numerous applications in quantum communication and quantum computation. Local Clifford (LC) operations that map one graph state to another can alter the structure of the corresponding graphs, including changing the number of edges. Here, we tackle the associated edge-minimisation problem: finding graphs with the minimum number of edges in the LC-equivalence class of a given graph. Such graphs are called minimum edge representatives (MER) and are crucial for minimising the resources required to create a graph state. We leverage Bouchet's algebraic formulation of LC-equivalence to encode the edge-minimisation problem as an integer linear program (EDM-ILP). We further propose a simulated annealing (EDM-SA) approach guided by the local clustering coefficient for edge minimisation. We identify new MERs for graph states with up to 16 qubits by combining EDM-SA and EDM-ILP. We extend the ILP to weighted-edge minimisation, where each edge has an associated weight, and prove that this problem is NP-complete. Finally, we employ our tools to minimise the resources required to create all-photonic generalised repeater graph states using fusion operations.