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arXiv:1911.07105 · Physical Review A

Exploiting landscape geometry to enhance quantum optimal control

Martín Larocca, Esteban Calzetta, Diego A. Wisniacki

Published Feb 18, 2020• 27 citations
arXivHTMLPDF

Abstract

The successful application of quantum optimal control (QOC) over the past decades has unlocked the possibility of directing the dynamics of quantum systems. Nevertheless, solutions obtained from QOC algorithms are usually highly irregular, making them unsuitable for direct experimental implementation. In this paper, we propose a method to reshape those unattractive optimal controls. The approach is based on the fact that solutions to QOC problems are not isolated policies but constitute multidimensional submanifolds of control space. This was originally shown for finite-dimensional systems. Here, we analytically prove that this property is still valid in a continuous variable system. The degenerate subspace can be effectively traversed by moving in the null subspace of the Hessian of the cost function, allowing for the pursuit of secondary objectives. To demonstrate the usefulness of this procedure, we apply the method to smooth and compress optimal protocols in order to meet laboratory demands.

DOI
10.1103/physreva.101.023410

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The successful application of quantum optimal control (QOC) over the past decades has unlocked the possibility of directing the dynamics of quantum systems. Nevertheless, solutions obtained from QOC algorithms are usually highly irregular, making them unsuitable for direct experimental implementation. In this paper, we propose a method to reshape those unattractive optimal controls. The approach is based on the fact that solutions to QOC problems are not isolated policies but constitute multidimensional submanifolds of control space. This was originally shown for finite-dimensional systems. Here, we analytically prove that this property is still valid in a continuous variable system. The degenerate subspace can be effectively traversed by moving in the null subspace of the Hessian of the cost function, allowing for the pursuit of secondary objectives. To demonstrate the usefulness of this procedure, we apply the method to smooth and compress optimal protocols in order to meet laboratory demands.
The quantum control landscape is defined as the functional that maps the control variables to the expectation values 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.
The generation of specific unitary transformations is central to a variety of quantum control problems. Given a target unitary transformation, the optimal control landscape is defined as the Hilbert-Schmidt distance between the target and controlled unitary transformation as a function of the control variables. The critical topology of the landscape is analyzed for controllable quantum systems evolving under unitary dynamics over a finite dimensional Hilbert space. It is found that the critical regions of the landscape corresponding to global optima are isolated points, and the local optima are Grassmannian submanifolds. The volumes of the critical submanifolds corresponding to suboptimal critical values asymptotically vanish in the limit of large Hilbert space dimension. Furthermore, these critical submanifolds have saddle-point topology, which cannot act as traps when searching for optimal controls. These favorable properties of the local optima suggest that the landscape topology is generally amenable to optimization. The analysis is independent of the particular structure of the system Hamiltonian, except for the assumption of full controllability, and the results are universal to the control of unitary transformations of any quantum system.
Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrödinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.
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.
Quantum optimal control (QOC) is the field devoted to the production of external control protocols that actively guide quantum dynamics. Solutions to QOC problems were shown to constitute continuous submanifolds of control space. A solution navigation method exploiting this property to achieve secondary features in the control protocols was proposed [Larocca, Calzetta, and Wisniacki, Phys. Rev. A 101, 023410 (2020)]. Here, we present a navigation-powered protocol postprocessing mechanism allowing naturally inflexible optimization algorithms to produce user-customized solutions, for example, in terms of a Fourier decomposition. We test the performance of the method in two inherently different models: the two-level Landau-Zener system and the quantum harmonic oscillator.
Quantum control landscape theory was formulated to assess the ease of finding optimal control fields in simulations and in the laboratory. The landscape is the observable as a function of the controls, and a primary goal of the theory is the analysis of landscape features. In what is referred to as the kinematic picture of the landscape, prior work showed that the landscapes are generally free of traps that could halt the search for an optimal control at a suboptimal observable value. The present paper considers the dynamical picture of the landscape, seeking the existence of singular controls, especially of a nonkinematic nature along with an assessment of whether they correspond to traps. We analyze the necessary and sufficient conditions for singular controls to be kinematic or nonkinematic critical solutions and the likelihood of their being encountered while maximizing an observable. An algorithm is introduced to seek singular controls on the landscape in simulations along with an associated Hessian landscape analysis. Simulations are performed for a large number of model finite-level quantum systems, showing that all the numerically identified kinematic and nonkinematic singular critical controls are not traps, in support of the prior empirical observations on the ease of finding high-quality optimal control fields.