  We study decoherence produced by a discrete environment on a charge Josephson
qubit by introducing a model of an environment of bistable fluctuators. In
particular we address the effect of $1/f$ noise where memory effects play an
important role. We perform a detailed investigation of various computation
procedures (single shot measurements, repeated measurements) and discuss the
problem of the information needed to characterize the effect of the
environment. Although in general information beyond the power spectrum is
needed, in many situations this results in the knowledge of only one more
microscopic parameter of the environment. This allows to determine which
degrees of freedom of the environment are effective sources of decoherence in
each different physical situation considered.
