Quantum observable homotopy tracking control

Rothman, Adam E.; Ho, Tak‐San; Rabitz, Herschel

doi:10.1063/1.2042456

Cited by 61 publications

(52 citation statements)

References 24 publications

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“…We will utilize the gradient algorithm to prescribe the path of steepest ascent, thereby forming a natural "rule" to climb the landscape and consequently to navigate the underlying control space. The D-MORPH procedure will be used to implement the gradient search trajectory [30,31]. The process of climbing the landscape may be confounded by the presence of saddle submanifolds located at certain regions of specific intermediate heights on the landscape.…”

Section: Negotiating the Quantum Control Landscapementioning

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: Negotiating the Quantum Control Landscapementioning

confidence: 99%

Exploring the complexity of quantum control optimization trajectories

Nanduri

Shir

Donovan

et al. 2015

Phys. Chem. Chem. Phys.

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“…Readers are referred to [13] for further details and to [14][15][16][17] for background. When the number m of unknown parameters (w i 's) is larger than the number of observation data points N (i.e., m > N ) with the provision that y lies in Ran( ) when does not have a full row rank, Eq.…”

Section: D-morph Regressionmentioning

confidence: 99%

D-MORPH regression for modeling with fewer unknown parameters than observation data

Rey-de-Castro

Rabitz

2012

J Math Chem

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D-MORPH regression is a procedure for the treatment of a model prescribed as a linear superposition of basis functions with less observation data than the number of expansion parameters. In this case, there is an infinite number of solutions exactly fitting the data. D-MORPH regression provides a practical systematic means to search over the solutions seeking one with desired ancillary properties while preserving fitting accuracy. This paper extends D-MORPH regression to consider the common case where there is more observation data than unknown parameters. This situation is treated by utilizing a proper subset of the normal equation of least-squares regression to judiciously reduce the number of linear algebraic equations to be less than the number of unknown parameters, thereby permitting application of D-MORPH regression. As a result, no restrictions are placed on model complexity, and the model with the best prediction accuracy can be automatically and efficiently identified. Ignition data for a H 2 /air combustion model as well as laboratory data for quantum-control-mechanism identification are used to illustrate the method.

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“…Various algorithms can be used to search for an optimal control solution [18][19][20][21]. In QOCT, most numerical optimizations employ gradient-based methods [41][42][43][44][45][46][47][48][49][50][51][52][53] (second-order methods use, in addition to the gradient, the Hessian matrix, see appendix B). In order to implement such an optimization, one needs to compute the functional derivative of the objective with respect to each control field: δ δ…”

Section: Formulating Optimal Control Theory For Aqcmentioning

confidence: 99%

Exploring adiabatic quantum trajectories via optimal control

et al. 2014

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Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring the properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time.

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Quantum observable homotopy tracking control

Cited by 61 publications

References 24 publications

Exploring the complexity of quantum control optimization trajectories

Exploring the complexity of quantum control optimization trajectories

D-MORPH regression for modeling with fewer unknown parameters than observation data

Exploring adiabatic quantum trajectories via optimal control

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