Quantum Control and Manipulation of Systems and Processes at Molecular Scale - Advanced Models of Neural Networks - page 255

where H 0 is the interaction Hamiltonian of the quantum system, H 1 is the control
Hamiltonian of the quantum system, and f.t/is the real-valued control field for the
quantum system. The control problem consists of calculating the control function
f.t/ such that the system's state (probability transition matrix .t/) with initial
conditions .0/ D 0 converges to the desirable final state d for t!1.Itis
considered that the initial state 0 and the final state d have the same spectrum
and this is a condition needed for reachability of the final state through unitary
evolutions.
Because of the existence of the interaction Hamiltonian H 0 it is also considered
that the desirable target state also evolves in time according to the Lindblad
equation, i.e.

P d .t/ DiŒH 0 ; d .t/ (13.24)

The target state is considered to be stationary if it holds ŒH 0 ; d .t/ D 0, therefore
in such a case it also holds P d .t/ D 0. When ŒH 0 ; d .t/¤0 then one has P d ¤0 and
the control problem of the quantum system is a tracking problem. The requirement
.t/! d .t/ for t!1 implies a trajectory tracking problem, while the requirement
.t/!O. d /.t/ for t!1 is an orbit tracking problem.
It will be shown that the calculation of the control function f.t/which assures
that .t/ converges to d .t/ can be performed with the use of the Lyapunov method.
To this end, a suitable Lyapunov function V.; d / will be chosen and it will be
shown that there exists a gradient-based control law f.t/such that V.; d /0.
The dynamics of the state of the quantum system, as well as the dynamics of the
target state are jointly described by

P .t/ DiŒH 0 C f.t/H 1 ;.t/
P d .t/ DiŒH 0 ; d .t/

(13.25)

A potential Lyapunov function for the considered quantum system is taken to be

V D 1 Tr . d / (13.26)

It holds that V>0if ¤ d . The Lyapunov function given in Eq. ( 13.26 ) can
be also considered as equivalent to the Lyapunov function V. ; d / D 1 j <
d .t/j .t/ > j

2 , which results from the description of the quantum system with
the use of Schrödinger's equation given in Eq. ( 13.1 ). The term < d .t/j .t/ >
expresses an internal product which takes value 1 if d .t/ and .t/ are aligned.
The first derivative of the Lyapunov function defined in Eq. ( 13.26 )is

V D Tr . P d / Tr . d P /

(13.27)