  We apply quantum optimal control theory (QOCT) to an exactly solvable
non-Markovian open quantum bit (qubit) system to achieve state-independent
quantum control and construct high-fidelity quantum gates for moderate qubit
decaying parameters. An important quantity, improvement $\mathcal{I}$, is
proposed and defined to quantify the correction of gate errors due to the QOCT
iteration when the environment effects are taken into account. With the help of
the exact dynamics, we explore how the gate error is corrected in the open
qubit system and determine the conditions for significant improvement. The
model adopted in this paper can be implemented experimentally in realistic
systems such as the circuit QED system.
