Exploring topological signatures of quantum Fisher information scaling in graph states

arXiv:2611.08593v1 Announce Type: new Abstract: Graph states are a versatile resource for quantum metrology, and their quantum Fisher information (QFI) has a closed-form expression in terms of the underlying graph: it counts pairs of vertices with identical neighborhoods. It remains unclear, however, which global structural properties of a graph determine whether its QFI exceeds the standard quantum limit (SQL). Here we ask whether topological invariants, namely the Betti numbers of the graph's clique complex, capture this behavior, and we address the question numerically with two complementary approaches. First, we randomly add or delete edges in balanced complete multipartite graphs, which combine a QFI beyond the SQL with a single nonvanishing higher Betti number.