Entanglement of weighted graphs uncovering transitions in variable-range interacting models

The cluster state acquired by evolving the nearest-neighbor (NN) Ising model from a completely separable state is the resource for measurement-based quantum computation. Instead of an NN system, a variable-range power-law interacting Ising model can generate a weighted graph state (WGS) whose genuine multipartite entanglement may reveal intrinsic characteristics of the evolving Hamiltonian. We establish that the generalized geometric measure (GGM) in the time evolved state with an arbitrary number of qubits is sensitive to falloff rates and the range of interactions of the evolving Hamiltonian. We report that the time-derivative and time-averaged GGM at a particular time can detect the transition points present in the falloff rates of the interaction strength, separating different regions, namely, long-range, quasilocal, and local ones in one- and two-dimensional lattices with deformation. Moreover, we illustrate that in the quasilocal and local regimes there exists a minimum coordination number in the evolving Ising model for a fixed total number of qubits which can mimic the GGM of the long-range model. In order to achieve a finite-size subsystem from the entire system, we design a local measurement strategy that allows a WGS of an arbitrary number of qubits to be reduced to a local unitarily equivalent WGS having fewer qubits with modified weights.