  In this work, we study the detailed structure of quantum control landscape
for the problem of single-qubit phase shift gate generation on the fast time
scale. In previous works, the absence of traps for this problem was proven on
various time scales. A special critical point which was known to exist in
quantum control landscapes was shown to be either a saddle or a global
extremum, depending on the parameters of the control system. However, in the
case of saddle the numbers of negative and positive eigenvalues of Hessian at
this point and their magnitudes have not been studied. At the same time, these
numbers and magnitudes determine the relative ease or difficulty for practical
optimization in a vicinity of the critical point. In this work, we compute the
numbers of negative and positive eigenvalues of Hessian at this saddle point
and moreover, give estimates on magnitude of these eigenvalues. We also
significantly simplify our previous proof of the theorem about this saddle
point of the Hessian [Theorem~3 in B.O.~Volkov, O.V.~Morzhin, A.N.~Pechen,
J.~Phys.~A: Math. Theor. {\bf 54}, 215303 (2021)].
