  Measurement-based quantum computation is an efficient model to perform
universal computation. Nevertheless, theoretical questions have been raised,
mainly with respect to realistic noise conditions. In order to shed some light
on this issue, we evaluate the exact dynamics of some single qubit gate
fidelities using the measurement-based quantum computation scheme when the
qubits which are used as resource interact with a common dephasing environment.
We report a necessary condition for the fidelity dynamics of a general pure
N-qubit state, interacting with this type of error channel, to present an
oscillatory behavior and we show that for the initial canonical cluster state
the fidelity oscillates as a function of time. This state fidelity oscillatory
behavior brings significant variations to the values of the computational
results of a generic gate acting on that state depending on the instants we
choose to apply our set of projective measurements. As we shall see,
considering some specific gates that are frequently found in the literature,
neither fast application of the set of projective measurements necessarily
implies high gate fidelity, nor slow application thereof necessarily implies
low gate fidelity. Our condition for the occurrence of the fidelity oscillatory
behavior shows that the oscillation presented by the cluster state is due
exclusively to its initial geometry. Other states that can be used as resources
for measurement-based quantum computation can present the same initial
geometrical condition. Therefore, it is very important for the present scheme
to know when the fidelity of a particular resource state will oscillate in time
and, if this is the case, what are the best times to perform the measurements.
