  We introduce a family of Hamiltonian systems for measurement-based quantum
computation with continuous variables. The Hamiltonians (i) are quadratic, and
therefore two body, (ii) are of short range, (iii) are frustration-free, and
(iv) possess a constant energy gap proportional to the squared inverse of the
squeezing. Their ground states are the celebrated Gaussian graph states, which
are universal resources for quantum computation in the limit of infinite
squeezing. These Hamiltonians constitute the basic ingredient for the adiabatic
preparation of graph states and thus open new venues for the physical
realization of continuous-variable quantum computing beyond the standard
optical approaches. We characterize the correlations in these systems at
thermal equilibrium. In particular, we prove that the correlations across any
multipartition are contained exactly in its boundary, automatically yielding a
correlation area law.
