Control landscapes for observable preparation with open quantum systems

Wu, Rongbo; Pechen, Alexander; Rabitz, Herschel; Hsieh, Michael; Tsou, Benjamin K.

doi:10.1063/1.2883738

Cited by 66 publications

(68 citation statements)

References 43 publications

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“…This framework opens up the possibility to explore the landscape of more complex quantum systems. We have applied it to the control landscape of open quantum systems [13], and another potential application in the future is the analyses of control landscapes for an infinite dimensional quantum system as the limit of a series of finite dimensional systems, which can be dealt with using the techniques developed here.…”

Section: Discussionmentioning

confidence: 99%

Characterization of the critical submanifolds in quantum ensemble control landscapes

Wu¹,

Rabitz²,

Hsieh³

2007

J. Phys. A: Math. Theor.

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111

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Abstract. The quantum control landscape is defined as the functional that maps the control variables to the expectation value of an observable over the ensemble of quantum systems. Analyzing the topology of such landscapes is important for understanding the origins of the increasing number of laboratory successes in the optimal control of quantum processes. This paper proposes a simple scheme to compute the characteristics of the critical topology of the quantum ensemble control landscapes, showing that the set of disjoint critical submanifolds one-to-one corresponds to a finite number of contingency tables that solely depend on the degeneracy structure of the eigenvalues of the initial system density matrix and the observable whose expectation value is to be maximized. The landscape characteristics can be calculated as functions of the table entries, including the dimensions and the numbers of positive and negative eigenvalues of the Hessian quadratic form of each of the connected components of the critical submanifolds. Typical examples are given to illustrate the effectiveness of this method.

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Section: Discussionmentioning

confidence: 99%

Characterization of the critical submanifolds in quantum ensemble control landscapes

Wu¹,

Rabitz²,

Hsieh³

2007

J. Phys. A: Math. Theor.

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111

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“…Local optima can potentially trap a gradient search, so their absence from the control landscape facilitates identification of a globally optimal control field. Although the control landscape topology [139][140][141][142] and optimization search effort [143] for open quantum systems have been studied, we do not consider issues related to open-system control in this work.…”

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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“…Control landscapes for closed quantum systems were studied in [10,11], where the absence of traps was shown under some assumptions. The analysis for open quantum systems was performed in [12,13] and later was extended to a unified analysis of control landscapes for open classical and quantum systems [14].…”

Section: Introductionmentioning

confidence: 99%

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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Control landscapes for observable preparation with open quantum systems

Cited by 66 publications

References 43 publications

Characterization of the critical submanifolds in quantum ensemble control landscapes

Characterization of the critical submanifolds in quantum ensemble control landscapes

Searching for quantum optimal controls under severe constraints

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

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