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.