  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 Controlled-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.
