Exploring quantum control landscapes: Topology, features, and optimization scaling

Moore, Katharine W.; Rabitz, Herschel

doi:10.1103/physreva.84.012109

Cited by 48 publications

(109 citation statements)

References 67 publications

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“…Quantum optimal control theory (OCT) provides a basis to explore applied field tuning of quantum system dynamics [1,2,3,4,5]. Recent successes include achieving control of electrons in quantum dots through time-varying gate voltages [6], controlling electronic correlations in molecules [7], manipulating the dynamics of quantum many-body systems [8], cooling ultracold atomic gases [9], eliciting quantum revivals [10], and generating adiabatic time evolution [11].…”

Section: Introductionmentioning

confidence: 99%

Exploring the complexity of quantum control optimization trajectories

Nanduri

Shir

Donovan

et al. 2015

Phys. Chem. Chem. Phys.

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The control of quantum system dynamics is generally performed by seeking a suitable applied field. The physical objective as a functional of the field forms the quantum control landscape, whose topology, under certain conditions, has been shown to contain no critical point suboptimal traps, thereby enabling effective searches for fields that give the global maximum of the objective. This paper addresses the structure of the landscape as a complement to topological critical point features. Recent work showed that landscape structure is highly favorable for optimization of state-to-state transition probabilities, in that gradient-based control trajectories to the global maximum value are nearly straight paths. The landscape structure is codified in the metric R ≥ 1.0, defined as the ratio of the length of the control trajectory to the Euclidean distance between the initial and optimal controls. A value of R = 1 would indicate an exactly straight trajectory to the optimal observable value. This paper extends the state-to-state transition probability results to the quantum ensemble and unitary transformation control landscapes. Again, nearly straight trajectories predominate, and we demonstrate that R can take values approaching 1.0 with high precision. However, the interplay of optimization trajectories with critical saddle submanifolds is found to influence landscape structure. A fundamental relationship necessary for perfectly straight gradient-based control trajectories is derived, wherein the gradient on the quantum control landscape must be an eigenfunction of the Hessian. This relation is an indicator of landscape structure and may provide a means to identify physical conditions when control trajectories can achieve perfect linearity. The collective favorable landscape topology 1 and structure provide a foundation to understand why optimal quantum control can be readily achieved.

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Section: Introductionmentioning

confidence: 99%

Exploring the complexity of quantum control optimization trajectories

Nanduri

Shir

Donovan

et al. 2015

Phys. Chem. Chem. Phys.

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“…Traps can strongly influence on both theoretical and experimental quantum control studies-they determine the level of difficulty of controlling the system and can significantly slow down or even completely prevent finding globally optimal controls. Whereas the analysis of traps in manipulation by quantum systems has attracted high attention [20][21][22][23][24][25][26][27][28], no examples of trap-free quantum systems have been known. Only partial theoretical results have been obtained stating the absence of traps at special regular controls.…”

Section: Introductionmentioning

confidence: 99%

Trap-free manipulation in the Landau-Zener system

Pechen

Ильин

2012

Phys. Rev. A

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The analysis of traps, i.e., locally but not globally optimal controls, for quantum control systems has attracted a great interest in recent years. The central problem that has been remained open is to demonstrate for a given system either existence or absence of traps. We prove the absence of traps and hence completely solve this problem for the important tasks of unconstrained manipulation of the transition probability and unitary gate generation in the Landau-Zener system-a system with a wide range of applications across physics, chemistry and biochemistry. This finding provides the first example of a controlled quantum system which is completely free of traps. We also discuss the impact of laboratory constraints due to decoherence, noise in the control pulse, and restrictions on the available controls which when being sufficiently severe can produce traps.

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“…The landscape for to state-transition control contains no saddles, so such problems are not considered in this paper; see Ref. [117] for a numerical study of state-transition landscapes. For a particular ρ 0 and θ, each permutation Π leads to the construction of a (not necessarily unique) contingency table C, as described above.…”

Section: B Formulation and Landscape Topology Of The Control Objectivementioning

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 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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Exploring quantum control landscapes: Topology, features, and optimization scaling

Cited by 48 publications

References 67 publications

Exploring the complexity of quantum control optimization trajectories

Exploring the complexity of quantum control optimization trajectories

Trap-free manipulation in the Landau-Zener system

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

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