  We consider a one-dimensional spin chain for which the ground state is the
cluster state, capable of functioning as a quantum computational wire when
subjected to local adaptive measurements of individual qubits, and investigate
the robustness of this property to local and coupled (Ising-type)
perturbations. We investigate the ground state both by identifying suitable
correlation functions as order parameters, as well as numerically using a
variational method based on matrix product states. We find that the model
retains an infinite localizable entanglement length for Ising and local fields
up to a quantum phase transition, but that the resulting entangled state is not
simply characterized by a Pauli correction based on the measurement results.
