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 systems 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.
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