  Motivated by the possibility of universal quantum computation under noise
perturbations, we compute the phase diagram of the 2d cluster state Hamiltonian
in the presence of Ising terms and magnetic fields. Unlike in previous analysis
of perturbed 2d cluster states, we find strong evidence of a very well defined
cluster phase, separated from a polarized phase by a line of 1st and 2nd order
transitions compatible with the 3d Ising universality class and a tricritical
end point. The phase boundary sets an upper bound for the amount of
perturbation in the system so that its ground state is still useful for
measurement-based quantum computation purposes. Moreover, we also compute the
local fidelity with the unperturbed 2d cluster state. Besides a classical
approximation, we determine the phase diagram by combining series expansions
and variational infinite Projected entangled-Pair States (iPEPS) methods. Our
work constitutes the first analysis of the non-trivial effect of few-body
perturbations in the 2d cluster state, which is of relevance for experimental
proposals.
