We investigate the fidelity of the measurement-based quantum computation (MBQC) when it is coupled with boson environment, by measuring cluster state fidelity and gate fidelity. Two different schemes of cluster state preparation are studied. In the C
ontrolled-Z (CZ) creation scheme, cluster states are prepared by entangling all qubits in $|+rangle$ state with CZ gates on all neighboring sites. The fidelity shows an oscillation pattern over time evolution. The influence of environment temperature is evaluated, and suggestions are given to enhance the performance of MBQC realized in this way. In the Hamiltonian creation scheme, cluster states are made by cooling a system with cluster Hamiltonians, of which ground states are cluster states. The fidelity sudden drop phenomenon is discovered. When the coupling is below a threshold, MBQC systems are highly robust against the noise. Our main environment model is the one with a single collective bosonic mode.
