  The method of quantum tomography, which allows us to track with high accuracy
the evolution of multilevel quantum systems (qudits) in Hilbert spaces of
various dimensions is presented. The developed algorithms for quantum control
are based on the use of the spinor representation of the Lorentz transformation
group. In the simplest case of one-qubit states, it turns out that, in addition
to three-dimensional rotations on the Bloch sphere, one can introduce
four-dimensional Lorentz pseudorotations, similar to the transformations of the
special theory of relativity. We show that feedback through weakly perturbing
adaptive quantum measurements turns out to be capable of providing
high-precision control of the quantum system, while introducing only weak
perturbations into the initial quantum state. It turns out that, together with
the control of a quantum system through its weak perturbation, the developed
algorithms for controlling the evolution of the state of a quantum system can
be super-efficient, providing a higher measurement accuracy than any standard
POVM (Positive-Operator Valued Measure) protocols. The results of the study are
important for the development of optimal adaptive methods for quantum states
and operations controlling. The results obtained are essential for the
development of high-precision control methods for quantum information
technologies.
