  In this work, we consider an environment formed by incoherent photons as a
resource for controlling open quantum systems via an incoherent control. We
exploit a coherent control in the Hamiltonian and an incoherent control in the
dissipator which induces the time-dependent decoherence rates $\gamma_k(t)$
(via time-dependent spectral density of incoherent photons) for generation of
single-qubit gates for a two-level open quantum system which evolves according
to the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation with
time-dependent coefficients determined by these coherent and incoherent
controls. The control problem is formulated as minimization of the objective
functional, which is the sum of Hilbert-Schmidt norms between four fixed basis
states evolved under the GKSL master equation with controls and the same four
states evolved under the ideal gate transformation. The exact expression for
the gradient of the objective functional with respect to piecewise constant
controls is obtained. Subsequent optimization is performed using a gradient
type algorithm with an adaptive step size that leads to oscillating behaviour
of the gradient norm vs iterations. Optimal trajectories in the Bloch ball for
various initial states are computed. A relation of quantum gate generation with
optimization on complex Stiefel manifolds is discussed. We develop methodology
and apply it here for unitary gates as a testing example. The next step is to
apply the method for generation of non-unitary processes and to multi-level
quantum systems.
