Abstract
We consider the problem of 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, nondegenerate ground state. We determine for any graph state the minimal $d$ such that it is the nondegenerate ground state of a $d$-body interaction Hamiltonian, while we show for ${d}^{\ensuremath{'}}$-body Hamiltonians $H$ with ${d}^{\ensuremath{'}}ld$ 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 nondegenerate (quasi)ground states of a two-body Hamiltonian acting on ${n}^{\ensuremath{'}}gn$ spins.
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