  We study reliable quantum information processing (QIP) under two different
types of environment. First type is Markovian exponential decay, and the
appropriate elementary strategy of protection of qubit is to apply fast gates.
The second one is strongly non-Markovian and occurs solely during operations on
the qubit. The best strategy is then to work with slow gates. If the two types
are both present, one has to optimize the speed of gate. We show that such a
trade-off is present for a single-qubit operation in a semiconductor quantum
dot implementation of QIP, where recombination of exciton (qubit) is Markovian,
while phonon dressing gives rise to the non-Markovian contribution.
