191 research outputs found
Optimality of extremals for linear systems with quadratic costs
AbstractIn this paper we consider linear control systems on Rn with integral quadratic cost functionals and investigate optimality properties of extremals. In this context, extremals are curves which satisfy the Pontryagin maximum principle. We extend the Jacobi necessary condition from the calculus of variations to the optimal control setting and describe a relationship between optimality of extremals and the existence of a solution for a certain Riccati differential equation
Quantum Control Landscapes
Numerous lines of experimental, numerical and analytical evidence indicate
that it is surprisingly easy to locate optimal controls steering quantum
dynamical systems to desired objectives. This has enabled the control of
complex quantum systems despite the expense of solving the Schrodinger equation
in simulations and the complicating effects of environmental decoherence in the
laboratory. Recent work indicates that this simplicity originates in universal
properties of the solution sets to quantum control problems that are
fundamentally different from their classical counterparts. Here, we review
studies that aim to systematically characterize these properties, enabling the
classification of quantum control mechanisms and the design of globally
efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry,
Vol. 26, Iss. 4, pp. 671-735 (2007
Limits of control for quantum systems: kinematical bounds on the optimization of observables and the question of dynamical realizability
In this paper we investigate the limits of control for mixed-state quantum
systems. The constraint of unitary evolution for non-dissipative quantum
systems imposes kinematical bounds on the optimization of arbitrary
observables. We summarize our previous results on kinematical bounds and show
that these bounds are dynamically realizable for completely controllable
systems. Moreover, we establish improved bounds for certain partially
controllable systems. Finally, the question of dynamical realizability of the
bounds for arbitary partially controllable systems is shown to depend on the
accessible sets of the associated control system on the unitary group U(N) and
the results of a few control computations are discussed briefly.Comment: 5 pages, orginal June 30, 2000, revised September 28, 200
The role of controllability in optimizing quantum dynamics
This paper discusses the important role of controllability played on the
complexity of optimizing quantum mechanical control systems. The study is based
on a topology analysis of the corresponding quantum control landscape, which is
referred to as the optimization objective as a functional of control fields. We
find that the degree of controllability is closely relevant with the ruggedness
of the landscape, which determines the search efficiency for global optima.
This effect is demonstrated via the gate fidelity control landscape of a system
whose controllability is restricted on a SU(2) dynamic symmetry group. We show
that multiple local false traps (i.e., non-global suboptima) exist even if the
target gate is realizable and that the number of these traps is increased by
the loss of controllability, while the controllable systems are always devoid
of false traps.Comment: 13 pages, 3 figure
Chow's theorem and universal holonomic quantum computation
A theorem from control theory relating the Lie algebra generated by vector
fields on a manifold to the controllability of the dynamical system is shown to
apply to Holonomic Quantum Computation. Conditions for deriving the holonomy
algebra are presented by taking covariant derivatives of the curvature
associated to a non-Abelian gauge connection. When applied to the Optical
Holonomic Computer, these conditions determine that the holonomy group of the
two-qubit interaction model contains . In particular, a
universal two-qubit logic gate is attainable for this model.Comment: 13 page
Optimal Control Theory for Continuous Variable Quantum Gates
We apply the methodology of optimal control theory to the problem of
implementing quantum gates in continuous variable systems with quadratic
Hamiltonians. We demonstrate that it is possible to define a fidelity measure
for continuous variable (CV) gate optimization that is devoid of traps, such
that the search for optimal control fields using local algorithms will not be
hindered. The optimal control of several quantum computing gates, as well as
that of algorithms composed of these primitives, is investigated using several
typical physical models and compared for discrete and continuous quantum
systems. Numerical simulations indicate that the optimization of generic CV
quantum gates is inherently more expensive than that of generic discrete
variable quantum gates, and that the exact-time controllability of CV systems
plays an important role in determining the maximum achievable gate fidelity.
The resulting optimal control fields typically display more complicated Fourier
spectra that suggest a richer variety of possible control mechanisms. Moreover,
the ability to control interactions between qunits is important for delimiting
the total control fluence. The comparative ability of current experimental
protocols to implement such time-dependent controls may help determine which
physical incarnations of CV quantum information processing will be the easiest
to implement with optimal fidelity.Comment: 39 pages, 11 figure
Complete controllability of finite-level quantum systems
Complete controllability is a fundamental issue in the field of control of
quantum systems, not least because of its implications for dynamical
realizability of the kinematical bounds on the optimization of observables. In
this paper we investigate the question of complete controllability for
finite-level quantum systems subject to a single control field, for which the
interaction is of dipole form. Sufficient criteria for complete controllability
of a wide range of finite-level quantum systems are established and the
question of limits of complete controllability is addressed. Finally, the
results are applied to give a classification of complete controllability for
four-level systems.Comment: 14 pages, IoP-LaTe
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