  We investigate the generation of quantum operations for one-qubit systems
under classical noise with 1/f^\alpha power spectrum, where 2>\alpha > 0. We
present an efficient way to approximate the noise with a discrete multi-state
Markovian fluctuator. With this method, the average temporal evolution of the
qubit density matrix under 1/f^\alpha noise can be feasibly determined from
recently derived deterministic master equations. We obtain qubit operations
such as quantum memory and the NOT}gate to high fidelity by a gradient based
optimization algorithm. For the NOT gate, the computed fidelities are
qualitatively similar to those obtained earlier for random telegraph noise. In
the case of quantum memory however, we observe a nonmonotonic dependency of the
fidelity on the operation time, yielding a natural access rate of the memory.
