  We present a general procedure for constructing lattices of qubits with a
Hamiltonian composed of nearest-neighbour two-body interactions such that the
ground state encodes a cluster state. We give specific details for lattices in
one-, two-, and three-dimensions, investigating both periodic and fixed
boundary conditions, as well as present a proof for the applicability of this
procedure to any graph. We determine the energy gap of these systems, which is
shown to be independent of the size of the lattice but dependent on the type of
lattice (in particular, the coordination number), and investigate the scaling
of this gap in terms of the coupling constants of the Hamiltonian. We provide a
comparative analysis of the different lattice types with respect to their
usefulness for measurement-based quantum computation.
