Quantum Control and Manipulation of Systems and Processes at Molecular Scale - Advanced Models of Neural Networks - page 248

The main approaches to the control of quantum systems are: (a) open-loop
control and (b) measurement-based feedback control [ 210 ]. In open-loop control,
the control signal is obtained using prior knowledge about the quantum system
dynamics and assuming a model that describes its evolution in time. Some open-
loop control schemes for quantum systems have been studied in [ 153 , 154 ]. Previous
work on quantum open-loop control includes flatness-based control on a single
qubit gate [ 41 ]. On the other hand, measurement-based quantum feedback control
provides more robustness to noise and model uncertainty [ 30 ]. In measurement-
based quantum feedback control, the overall system dynamics are described by
the estimation equation called stochastic master equation or Belavkin's equation
[ 20 ]. An equivalent approach can be obtained using Lindblad's differential equation
[ 210 ]. Several researchers have presented results on measurement-based feedback
control of quantum systems using the stochastic master equation or the Lindblad
differential equation, while theoretical analysis of the stability for the associated
control loop has been also attempted in several cases [ 18 , 124 , 203 ].
In this chapter, a gradient-based approach to the control of quantum systems
will be examined. Previous results on control laws which are derived through the
calculation of the gradient of an energy function of the quantum system can be
found in [ 4 , 9 , 155 , 163 ]. Convergence properties of gradient algorithms have been
associated with Lyapunov stability theory in [ 22 ]. The chapter considers a quantum
system confined in a cavity that is weakly coupled to a probe laser. The spin model is
used to define the eigenstates of the quantum system. The dynamics of the quantum
model are described by Lindblad's differential equation and thus an estimate of the
system's state can be obtained. Using Lyapunov's stability theory a gradient-based
control law is derived. Furthermore, by applying LaSalle's invariance principle
it can be assured that under the proposed gradient-based control the quantum
system's state will track the desirable state within acceptable accuracy levels. The
performance of the control loop is studied through simulation experiments for the
case of a two-qubit quantum system.

13.2 The Spin as a Two-Level Quantum System

13.2.1 Description of a Particle in Spin Coordinates

As explained in Chap. 7 , the basic equation of quantum mechanics is Schrödinger's
equation , i.e.

i @

@t D H .x;t/

(13.1)

2 is the probability density function of finding the particle at position
x at time instant t, and H is the system's Hamiltonian, i.e. the sum of its kinetic

where j .x;t/j