Topology of the quantum control landscape for observables

Hsieh, Michael; Wu, Rongbo; Rabitz, Herschel

doi:10.1063/1.2981796

Cited by 40 publications

(52 citation statements)

References 58 publications

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“…In the kinematic formulation, it has also been shown that the sufficient and necessary condition for U T to be a critical point of the landscape is that the final-time density matrix ρ(T ) = U T ρ 0 U † T commutes with the target observable θ [98,105,106], i.e.,…”

Section: B Formulation and Landscape Topology Of The Control Objectivementioning

confidence: 99%

“…Several recent studies [97][98][99] strongly indicate that the success of numerous OCEs and OCT simulations is related to the favorable topology of the quantum control landscape defined by the functional dependence of J on ε(t) [7,9,100]. In particular, it has been shown that the control landscape lacks local optima (referred to as traps) if three conditions are satisfied: (i) the quantum system is controllable, i.e., any unitary evolution operator can be produced by some admissible control field beyond some finite time; (ii) the Jacobian matrix mapping the control field ε(t) to the final-time evolution operator U (T, 0) is of full rank everywhere on the landscape; (iii) there are no constraints on the control field [98,[101][102][103][104][105][106][107][108]. The absence of local suboptimal extrema is of central importance to optimization; numerical studies have described the appearance of local traps on the control landscape due to the violation of assumption (i) [109] and shown that the violation of assumption (ii) [110] can, in special cases, prevent a gradient search from identifying globally optimal controls.…”

Section: Introductionmentioning

confidence: 99%

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Searching for an optimal control in the presence of saddles on the quantum-mechanical observable landscape

Riviello

Sun

et al. 2017

Phys. Rev. A

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The broad success of theoretical and experimental quantum optimal control is intimately connected to the topology of the underlying control landscape. For several common quantum control goals, including the maximization of an observable expectation value, the landscape has been shown to lack local optima if three assumptions are satisfied: (i) the quantum system is controllable, (ii) the Jacobian of the map from the control field to the evolution operator is full-rank, and (iii) the control field is not constrained. In the case of the observable objective, this favorable analysis shows that the associated landscape also contains saddles, i.e., critical points that are not local suboptimal extrema. In this paper, we investigate whether the presence of these saddles affects the trajectories of gradient-based searches for an optimal control. We show through simulations that both the detailed topology of the control landscape and the parameters of the system Hamiltonian influence whether the searches are attracted to a saddle. For some circumstances with a special initial state and target observable, optimizations may approach a saddle very closely, reducing the efficiency of the gradient algorithm. Encounters with such attractive saddles are found to be quite rare. Neither the presence of a large number of saddles on the control landscape nor a large number of system states increase the likelihood that a search will closely approach a saddle. Even for applications that encounter a saddle, well-designed gradient searches with carefully chosen algorithmic parameters will readily locate optimal controls.

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Section: B Formulation and Landscape Topology Of The Control Objectivementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Searching for an optimal control in the presence of saddles on the quantum-mechanical observable landscape

Riviello

Sun

et al. 2017

Phys. Rev. A

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“…The success of a growing number of optimal control experiments, as well as vast numbers of simulations, led to the formulation of a key theorem referred to here as the landscape principle. The principle states that upon satisfaction of assumptions (sufficient conditions) about * hrabitz@princeton.edu the controllability, surjectivity, and available resources, the topology of quantum control landscapes for systems with a finite number of states allows for facile determination of optimal controls [6][7][8].…”

Section: = [H 0 + V (Xt) + G(xt)|ψ(xt)| 2 ]ψ(Xt) (1)mentioning

confidence: 99%

Optimal nonlinear coherent mode transitions in Bose-Einstein condensates utilizing spatiotemporal controls

2016

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Bose-Einstein condensates (BECs) offer the potential to examine quantum behavior at large length and time scales, as well as forming promising candidates for quantum technology applications. Thus, the manipulation of BECs using control fields is a topic of prime interest. We consider BECs in the mean-field model of the Gross-Pitaevskii equation (GPE), which contains linear and nonlinear features, both of which are subject to control. In this work we report successful optimal control simulations of a one-dimensional GPE by modulation of the linear and nonlinear terms to stimulate transitions into excited coherent modes. The linear and nonlinear controls are allowed to freely vary over space and time to seek their optimal forms. The determination of the excited coherent modes targeted for optimization is numerically performed through an adaptive imaginary time propagation method. Numerical simulations are performed for optimal control of mode-to-mode transitions between the ground coherent mode and the excited modes of a BEC trapped in a harmonic well. The results show greater than 99% success for nearly all trials utilizing reasonable initial guesses for the controls, and analysis of the optimal controls reveals primarily direct transitions between initial and target modes. The success of using solely the nonlinearity term as a control opens up further research toward exploring novel control mechanisms inaccessible to linear Schrödinger-type systems.

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“…The quantum control landscape defined by this functional dependence has been depicted in experimental studies for various control problems [119][120][121][122][123][124][125][126], and its favorable topology [9,127] has been correlated [128][129][130] to the success of OCEs and OCT simulations. Specifically, it has been shown [129,[131][132][133][134][135][136][137][138] that the landscapes for N -level closed quantum systems lack local optima if three conditions are satisfied: (1) the quantum system is controllable, i.e., any given unitary evolution can be generated by some control field in finite time; (2) the Jacobian of the map from the control field ε(t) to the final-time evolution operator U (T, 0) is of full rank; (3) the control field is unconstrained. We discuss these conditions in more detail in Sec.…”

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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Topology of the quantum control landscape for observables

Cited by 40 publications

References 58 publications

Searching for an optimal control in the presence of saddles on the quantum-mechanical observable landscape

Searching for an optimal control in the presence of saddles on the quantum-mechanical observable landscape

Optimal nonlinear coherent mode transitions in Bose-Einstein condensates utilizing spatiotemporal controls

Searching for quantum optimal controls under severe constraints

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