Graph states as ground states of many-body spin-1/2 Hamiltonians

We consider the problem whether graph states can be ground states of local interaction Hamiltonians. For Hamiltonians acting on n qubits that involve at most two-body interactions, we show that no n-qubit graph state can be the exact, non-degenerate ground state. We determine for any graph state the minimal d such that it is the non-degenerate ground state of a d-body interaction Hamiltonian, while we show for d'-body Hamiltonians H with d'<d that the resulting ground state can only be close to the graph state at the cost of H having a small energy gap relative to the total energy. When allowing for ancilla particles, we show how to utilize a gadget construction introduced in the context of the k-local Hamiltonian problem, to obtain n-qubit graph states as non-degenerate (quasi-)ground states of a two-body Hamiltonian acting on n'>n spins.