Pechen and Tannor Reply:

Pechen, Alexander; Tannor, David J.

doi:10.1103/physrevlett.108.198902

Cited by 39 publications

(68 citation statements)

References 6 publications

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“…. > λ n , and the dipole moment satisfies ĩ |µ|k = 0 for all i < k. This important result was proved in [30]. However, previously it was not known if these second order traps are true traps (local maxima).…”

Section: Theoremmentioning

confidence: 92%

“…The simplest example is a three-level Λ-system, where zero control field was shown to be a second order trap. In this work we continue the analysis of [30] and show that this second order trap is not a local maximum; there exist a direction in which the objective grows. We also perform a numerical study of the landscape in a vicinity of the ε(t) = 0 second order trap.…”

Section: Introductionmentioning

confidence: 99%

“…Second order traps were shown to exist for a general class of quantum systems [30]. The simplest example is a three-level Λ-system, where zero control field was shown to be a second order trap.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Control Landscape for a Λ‐atom in the Vicinity of Second‐Order Traps

Pechen

Tannor

2012

Israel Journal of Chemistry

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We show that the second order traps in the control landscape for a three-level Λ-system found in our previous work Phys. Rev. Lett. 106, 120402 (2011) are not local maxima: there exist directions in the space of controls in which the objective grows. The growth of the objective is slow -at best 4th order for weak variations of the control. This implies that simple gradient methods would be problematic in the vicinity of second order traps, where more sophisticated algorithms that exploit the higher order derivative information are necessary to climb up the control landscape efficiently. The theory is supported by a numerical investigation of the landscape in the vicinity of the ε(t) = 0 second order trap, performed using the GRAPE and BFGS algorithms.

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Section: Theoremmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

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Quantum Control Landscape for a Λ‐atom in the Vicinity of Second‐Order Traps

Pechen

Tannor

2012

Israel Journal of Chemistry

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“…Such a trap is not necessarily a local maximum of the landscape, since higher-order functional derivatives may be indefinite [160], but it can in principle prevent a simple gradient search from finding a globally maximal solution. However, a subsequent computational study [161] examined the same control problems as [158,159] and found that the second-order traps only attract search trajectories that originate very close to them (i.e., at fields which are several orders of magnitude weaker than the optimal ones) and thus are very unlikely to affect gradient-based optimizations under realistic searching conditions. In this work, we nonetheless assume, for the sake of simplicity, that condition (2) is satisfied and that there are no singular critical points on the control landscape.…”

Section: Introductionmentioning

confidence: 99%

“…This result indicates that the overwhelming majority of singular critical points are not local optima. Another pair of recent works [158,159] showed that, for several specially constructed combinations of control objective and Hamiltonian, a singular critical point at ε(t) = 0 is a second-order trap. For a maximization problem, a critical point is a second-order trap if the Hessian matrix of the second functional derivatives of J with respect to the field,…”

Section: Introductionmentioning

confidence: 99%

Searching for quantum optimal controls under severe constraints

Riviello¹,

Tibbetts²,

Brif³

et al. 2015

Phys. Rev. A

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The success of quantum optimal control for both experimental and theoretical objectives is connected to the topology of the corresponding control landscapes, which are free from local traps if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is of full rank, and (3) there are no constraints on the control field. This paper investigates how the violation of assumption (3) affects gradient searches for globally optimal control fields. The satisfaction of assumptions (1) and (2) ensures that the control landscape lacks fundamental traps, but certain control constraints can still introduce artificial traps. Proper management of these constraints is an issue of great practical importance for numerical simulations as well as optimization in the laboratory. Using optimal control simulations, we show that constraints on quantities such as the number of control variables, the control duration, and the field strength are potentially severe enough to prevent successful optimization of the objective. For each such constraint, we show that exceeding quantifiable limits can prevent gradient searches from reaching a globally optimal solution. These results demonstrate that careful choice of relevant control parameters helps to eliminate artificial traps and facilitate successful optimization.

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The Surprising Ease of Finding Optimal Solutions for Controlling Nonlinear Phenomena in Quantum and Classical Complex Systems

Rabitz

Russell

2023

J. Phys. Chem. A

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This Perspective addresses the often observed surprising ease of achieving optimal control of nonlinear phenomena in quantum and classical complex systems. The circumstances involved are wide-ranging, with scenarios including manipulation of atomic scale processes, maximization of chemical and material properties or synthesis yields, Nature's optimization of species' populations by natural selection, and directed evolution. Natural evolution will mainly be discussed in terms of laboratory experiments with microorganisms, and the field is also distinct from the other domains where a scientist specifies the goal(s) and oversees the control process. We use the word "control" in reference to all of the available variables, regardless of the circumstance. The empirical observations on the ease of achieving at least good, if not excellent, control in diverse domains of science raise the question of why this occurs despite the generally inherent complexity of the systems in each scenario. The key to addressing the question lies in examining the associated control landscape, which is defined as the optimization objective as a function of the control variables that can be as diverse as the phenomena under consideration. Controls may range from laser pulses, chemical reagents, chemical processing conditions, out to nucleic acids in the genome and more. This Perspective presents a conjecture, based on present findings, that the systematics of readily finding good outcomes from controlled phenomena may be unified through consideration of control landscapes with the same common set of three underlying assumptions�the existence of an optimal solution, the ability for local movement on the landscape, and the availability of sufficient control resources�whose validity needs assessment in each scenario. In practice, many cases permit using myopic gradient-like algorithms while other circumstances utilize algorithms having some elements of stochasticity or introduced noise, depending on whether the landscape is locally smooth or rough. The overarching observation is that only relatively short searches are required despite the common high dimensionality of the available controls in typical scenarios.

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Pechen and Tannor Reply:

Cited by 39 publications

References 6 publications

Quantum Control Landscape for a Λ‐atom in the Vicinity of Second‐Order Traps

Quantum Control Landscape for a Λ‐atom in the Vicinity of Second‐Order Traps

Searching for quantum optimal controls under severe constraints

The Surprising Ease of Finding Optimal Solutions for Controlling Nonlinear Phenomena in Quantum and Classical Complex Systems

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