  We consider protecting a Hadamard operation from independent dephasing, bit
flipping, and dissipation. These environment-induced errors are represented by
three uncorrelated reservoirs of thermalized bosons and we show that the
protection is achievable through continuous dynamical decoupling. We find that,
to decouple the Hadamard evolution from the environmental influence, we need a
control field of higher frequency if the boson spectral density is super-ohmic
than if it is ohmic. We also study the relevance of bit flipping and
dissipation to the gate fidelity when it is protected from dephasing, showing
how robust this partial protection is against these other perturbations.
Finally, we calculate an efficient field arrangement capable of protecting
simultaneously the gate operation from these three error classes.
