Exploring topological signatures of quantum Fisher information scaling in graph states

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. Edge addition eventually destroys both the metrological advantage and the nontrivial topology. Under edge deletion, by contrast, higher Betti numbers persist well after the QFI has dropped to the SQL. Second, we start from Erdős--Rényi random graphs, whose QFI does not typically exceed the SQL, and maximize a chosen Betti number with a Monte Carlo search. The search drives the graphs toward near-extremal multipartite structures whose QFI lies well beyond the SQL. Our results show that topological optimization offers a route to metrologically useful graph states, but also that Betti numbers alone do not determine the QFI.