  We consider fault-tolerant quantum computation in the context where there are
no fresh ancilla qubits available during the computation, and where the noise
is due to a general quantum channel. We show that there are three classes of
noisy channels: In the first, typified by the depolarizing channel, computation
is only possible for a logarithmic time. In the second class, of which the
dephasing channel is an example, computation is possible for polynomial time.
The amplitude damping channel is an example of the third class, and for this
class of channels, it is possible to compute for an exponential time in the
number of qubits available.
