  An important problem in quantum computation is generation of single-qubit
quantum gates such as Hadamard ($H$) and $\pi/8$ ($T$) gates which are
components of a universal set of gates. Qubits in experimental realizations of
quantum computing devices are interacting with their environment. While the
environment is often considered as an obstacle leading to decrease of the gate
fidelity, in some cases it can be used as a resource. Here we consider the
problem of optimal generation of $H$ and $T$ gates using coherent control and
the environment as a resource acting on the qubit via incoherent control. For
this problem, we study quantum control landscape which represents the behaviour
of the infidelity as a functional of the controls. We consider three
landscapes, with infidelities defined by steering between two, three (via
Goerz-Reich-Koch approach), and four matrices in the qubit Hilbert space. We
observe that for the $H$ gate, which is Clifford gate, for all three
infidelities the distributions of minimal values obtained with gradient search
have a simple form with just one peak. However, for $T$ gate which is a
non-Clifford gate, the situation is surprisingly different - this distribution
for the infidelity defined by two matrices also has one peak, whereas
distributions for the infidelities defined by three and four matrices have two
peaks, that might indicate possible existence of two isolated minima in the
control landscape. Important is that among these three infidelities only those
defined with three and four matrices guarantee closeness of generated gate to a
target and can be used as a good measure of closeness. We study sets of
optimized solutions for this most general and not treated before case of
coherent and incoherent controls acting together, and discover that they form
submanifolds in the control space, and unexpected, in some cases two isolated
submanifolds.
