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Physical Review E

Efficient algorithms for the laboratory discovery of optimal quantum controls

Gabriel Turinici, Claude Le Bris, HERSCHEL A. RABITZ

Published Jul 20, 2004• 20 citations
Publisher

Abstract

The laboratory closed-loop optimal control of quantum phenomena, expressed as minimizing a suitable cost functional, is currently implemented through an optimization algorithm coupled to the experimental apparatus. In practice, the most commonly used search algorithms are variants of genetic algorithms. As an alternative choice, a direct search deterministic algorithm is proposed in this paper. For the simple simulations studied here, it outperforms the existing approaches. An additional algorithm is introduced in order to reveal some properties of the cost functional landscape.

DOI
10.1103/physreve.70.016704

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The laboratory closed-loop optimal control of quantum phenomena, expressed as minimizing a suitable cost functional, is currently implemented through an optimization algorithm coupled to the experimental apparatus. In practice, the most commonly used search algorithms are variants of genetic algorithms. As an alternative choice, a direct search deterministic algorithm is proposed in this paper. For the simple simulations studied here, it outperforms the existing approaches. An additional algorithm is introduced in order to reveal some properties of the cost functional landscape.
This paper explores the use of laboratory closed loop learning control to suppress the effects of decoherence in quantum dynamics. Simulations of the process are performed in multilevel quantum systems strongly interacting with the environment. A genetic algorithm is used to find an optimal control field which seeks out transition pathways to achieve a minimum influence of decoherence upon the system at a target time. The simulations suggest that decoherence may be optimally managed in the laboratory through closed loop operations with a suitable cost that is sensitive to the coherence of the dynamics. The case studies of dimension N=4 and N=10 with strong system–environment coupling indicate that the additional complexity with increasing system dimension can make it more difficult to manage decoherence.
Existing algorithms for the optimal control of quantum observables are based on locally optimal steps in the space of control fields, or as in the case of genetic algorithms, operate on the basis of heuristics that do not explicitly take into account details pertaining to the geometry of the search space. We present globally efficient algorithms for quantum observable control that follow direct or close-to-direct paths in the domain of unitary dynamical propagators, based on partial reconstruction of these propagators at successive points along the search trajectory through orthogonal observable measurements. These algorithms can be implemented experimentally and offer an alternative to the adaptive learning control approach to optimal control experiments (OCE). Their performance is compared to that of local gradient-based control optimization.
The exploration of a quantum control landscape, which is the physical observable as a function of the control variables, is fundamental for understanding the ability to perform observable optimization in the laboratory. For high control variable dimensions, trajectory-based methods provide a means for performing such systematic explorations by exploiting the measured gradient of the observable with respect to the control variables. This paper presents a practical, robust, easily implemented statistical method for obtaining the gradient on a general quantum control landscape in the presence of noise. In order to demonstrate the method's utility, the experimentally measured gradient is utilized as input in steepest-ascent trajectories on the landscapes of three model quantum control problems: spectrally filtered and integrated second harmonic generation as well as excitation of atomic rubidium. The gradient algorithm achieves efficiency gains of up to approximately three times that of the standard genetic algorithm and, as such, is a promising tool for meeting quantum control optimization goals as well as landscape analyses. The landscape trajectories directed by the gradient should aid in the continued investigation and understanding of controlled quantum phenomena.
Optimization of quantum systems by closed-loop adaptive pulse shaping offers a rich domain for the development and application of specialized evolutionary algorithms. Derandomized evolution strategies (DESs) are presented here as a robust class of optimizers for experimental quantum control. The combination of stochastic and quasi-local search embodied by these algorithms is especially amenable to the inherent topology of quantum control landscapes. Implementation of DES in the laboratory results in efficiency gains of up to $\ensuremath{\sim}9$ times that of the standard genetic algorithm, and thus is a promising tool for optimization of unstable or fragile systems. The statistical learning upon which these algorithms are predicated also provide the means for obtaining a control problem's Hessian matrix with no additional experimental overhead. The forced optimal covariance adaptive learning (FOCAL) method is introduced to enable retrieval of the Hessian matrix, which can reveal information about the landscape's local structure and dynamic mechanism. Exploitation of such algorithms in quantum control experiments should enhance their efficiency and provide additional fundamental insights.
This paper presents a general self-guided algorithm for direct laboratory learning of controls to manipulate quantum-mechanical systems. The primary focus is on an algorithm based on the learning of a linear laboratory input–output map from a sequence of controls, and their observed impact on the quantum-mechanical system. This map is then employed in an iterative fashion, to sequentially home in on the desired objective. The objective may be a target state at a final time, or a continuously weighted observational trajectory. The self-guided aspects of the algorithm are based on implementing a cost functional that only contains laboratory-accessible information. Through choice of the weights in this functional, the algorithm can automatically stay within the bounds of each local linear map and indicate when a new map is necessary for additional iterative improvement. Finally, these concepts can be generalized to include the possibility of employing nonlinear maps, as well as just the laboratory control instrument settings, rather than observation of the control itself. An illustrative simulation of the concepts is presented for the control of a four-level quantum system.
The laboratory implementation of molecular optimal control has to overcome the problem caused by the changing environmental parameters, such as the temperature of the laser rod, the resonator parameters, the mechanical parameters of the laboratory equipment and other dependent parameters such as the time delay between pulses or the pulse amplitudes. In this paper a solution is proposed: instead of trying to set the parameter(s) with very high precision, their changes are monitored and the control is adjusted to the current values. The optimization in the laboratory can then be run at several values of the parameter(s) with an extended genetic algorithm (GA) which is tailored to such parametric optimization. The extended GA does not presuppose, but can take advantage, and in fact, explores whether the mapping from the parameter(s) to optimal control field is continuous. Then the op1 timization for the different values of the parameter(s) is done cooperatively which reduces the optimizat...