Maintaining coherence in quantum computers

The effects of the inevitable coupling to external degrees of freedom of a quantum computer are examined. It is found that for quantum calculations (in which the maintenance of coherence over a large number of states is important), not only must the coupling be small, but the time taken in the quantum calculation must be less than the thermal time scale \ensuremath{\Elzxh}/${\mathit{k}}_{\mathit{B}}$T. For longer times the condition on the strength of the coupling to the external world becomes much more stringent.