  We obtain an upper bound on the time available for quantum computation for a
given quantum computer and decohering environment with quantum error correction
implemented. First, we derive an explicit quantum evolution operator for the
logical qubits and show that it has the same form as that for the physical
qubits but with a reduced coupling strength to the environment. Using this
evolution operator, we find the trace distance between the real and ideal
states of the logical qubits in two cases. For a super-Ohmic bath, the trace
distance saturates, while for Ohmic or sub-Ohmic baths, there is a finite time
before the trace distance exceeds a value set by the user.
