  The two-dimensional cluster state, a universal resource for measurement-based
quantum computation, is also the gapped ground state of a short-ranged
Hamiltonian. Here, we examine the effect of perturbations to this Hamiltonian.
We prove that, provided the perturbation is sufficiently small and respects a
certain symmetry, the perturbed ground state remains a universal resource. We
do this by characterising the operation of an adaptive measurement protocol
throughout a suitable symmetry-protected quantum phase, relying on generic
properties of the phase rather than any analytic control over the ground state.
