  In this work, we consider the problem of ultrafast controlled generation of
single-qubit phase shift quantum gates. Globally optimal control is a control
which realizes the gate with maximal possible fidelity. Trap is a control which
is optimal only locally but not globally. It was shown before that traps do not
exist for controlled generation of arbitrary single-qubit quantum gates for
sufficiently long times, as well as for fast control of quantum gates other
than phase shift gates. Ultrafast generation of phase-shift gates was missed in
the previous analysis. In this work we show, combining analytical and numerical
optimization methods such as Gradient Ascent Pulse Engineering (GRAPE),
differential evolution, and dual annealing, that control landscape for
ultrafast generation of phase shift gates is also free of traps. Mathematical
analysis of quantum control landscapes, which aims to prove either absence or
existence of traps for quantum control objective functionals, is an important
topic in quantum control. In this work, we provide a rigorous analysis of
quantum control landscapes for ultrafast generation of single-qubit quantum
gates and show, combining analytical methods based on a sophisticated analysis
of spectrum of the Hessian, and numerical optimization methods such as Gradient
Ascent Pulse Engineering (GRAPE), differential evolution, and dual annealing,
that control landscape for ultrafast generation of phase shift gates is free of
traps.
