  The dynamical evolution of a quantum register of arbitrary length coupled to
an environment of arbitrary coherence length is predicted within a relevant
model of decoherence. The results are reported for quantum bits (qubits)
coupling individually to different environments (`independent decoherence') and
qubits interacting collectively with the same reservoir (`collective
decoherence'). In both cases, explicit decoherence functions are derived for
any number of qubits. The decay of the coherences of the register is shown to
strongly depend on the input states: we show that this sensitivity is a
characteristic of $both$ types of coupling (collective and independent) and not
only of the collective coupling, as has been reported previously. A non-trivial
behaviour ("recoherence") is found in the decay of the off-diagonal elements of
the reduced density matrix in the specific situation of independent
decoherence. Our results lead to the identification of decoherence-free states
in the collective decoherence limit. These states belong to subspaces of the
system's Hilbert space that do not get entangled with the environment, making
them ideal elements for the engineering of ``noiseless'' quantum codes. We also
discuss the relations between decoherence of the quantum register and
computational complexity based on the new dynamical results obtained for the
register density matrix.
