16 research outputs found
Optimization search effort over the control landscapes for open quantum systems with Kraus-map evolution
A quantum control landscape is defined as the expectation value of a target
observable as a function of the control variables. In this work
control landscapes for open quantum systems governed by Kraus map evolution are
analyzed. Kraus maps are used as the controls transforming an initial density
matrix into a final density matrix to maximize the expectation
value of the observable . The absence of suboptimal local maxima for
the relevant control landscapes is numerically illustrated. The dependence of
the optimization search effort is analyzed in terms of the dimension of the
system , the initial state , and the target observable
. It is found that if the number of nonzero eigenvalues in remains constant, the search effort does not exhibit any significant
dependence on . If has no zero eigenvalues, then the
computational complexity and the required search effort rise with . The
dimension of the top manifold (i.e., the set of Kraus operators that maximizes
the objective) is found to positively correlate with the optimization search
efficiency. Under the assumption of full controllability, incoherent control
modelled by Kraus maps is found to be more efficient in reaching the same value
of the objective than coherent control modelled by unitary maps. Numerical
simulations are also performed for control landscapes with linear constraints
on the available Kraus maps, and suboptimal maxima are not revealed for these
landscapes.Comment: 29 pages, 8 figure
Unified analysis of terminal-time control in classical and quantum systems
Many phenomena in physics, chemistry, and biology involve seeking an optimal
control to maximize an objective for a classical or quantum system which is
open and interacting with its environment. The complexity of finding an optimal
control for maximizing an objective is strongly affected by the possible
existence of sub-optimal maxima. Within a unified framework under specified
conditions, control objectives for maximizing at a terminal time physical
observables of open classical and quantum systems are shown to be inherently
free of sub-optimal maxima. This attractive feature is of central importance
for enabling the discovery of controls in a seamless fashion in a wide range of
phenomena transcending the quantum and classical regimes.Comment: 10 page
Control landscapes for two-level open quantum systems
A quantum control landscape is defined as the physical objective as a
function of the control variables. In this paper the control landscapes for
two-level open quantum systems, whose evolution is described by general
completely positive trace preserving maps (i.e., Kraus maps), are investigated
in details. The objective function, which is the expectation value of a target
system operator, is defined on the Stiefel manifold representing the space of
Kraus maps. Three practically important properties of the objective function
are found: (a) the absence of local maxima or minima (i.e., false traps); (b)
the existence of multi-dimensional sub-manifolds of optimal solutions
corresponding to the global maximum and minimum; and (c) the connectivity of
each level set. All of the critical values and their associated critical
sub-manifolds are explicitly found for any initial system state. Away from the
absolute extrema there are no local maxima or minima, and only saddles may
exist, whose number and the explicit structure of the corresponding critical
sub-manifolds are determined by the initial system state. There are no saddles
for pure initial states, one saddle for a completely mixed initial state, and
two saddles for other initial states. In general, the landscape analysis of
critical points and optimal manifolds is relevant to the problem of explaining
the relative ease of obtaining good optimal control outcomes in the laboratory,
even in the presence of the environment.Comment: Minor editing and some references adde