Multi-partite entanglement in cluster Ising Hamiltonians in d- dimensions.*

Cluster Ising Hamiltonian is a frustration-free system consisting of qubits arranged on the  vertices of a square grid graph with nearest neighbour, "cluster-like" interactions. There has been extensive earlier study (by other papers/authors) of the one dimensional case, demonstrating the existence of symmetry protected topological order and the two dimensional case has been shown to be useful in measurement based quantum computation. We use methods from quantum error correction and graph theory to calculate the exact amount of multi-partite entanglement in the non degenerate ground state of the d-dimensional cluster Ising Hamiltonian.