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Physical Review A

Gradient algorithm applied to laboratory quantum control

Jonathan Roslund, HERSCHEL A. RABITZ

Published May 18, 2009• 56 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.79.053417

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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.
A common goal of quantum control is to maximize a physical observable through the application of a tailored field. The observable value as a function of the field constitutes a quantum-control landscape. Previous papers have shown, under specified conditions, that the quantum-control landscape should be free of suboptimal critical points. This favorable landscape topology is one factor contributing to the efficiency of climbing the landscape. An additional complementary factor is the landscape structure, which constitutes all nontopological features. If the landscape's structure is too complex, then climbs may be forced to take inefficient convoluted routes to find optimal controls. This paper provides a foundation for understanding control-landscape structure by examining the linearity of gradient-based optimization trajectories through the space of control fields. For this assessment, a metric $R\ensuremath{\ge}1$ is defined as the ratio of the path length of the optimization trajectory to the Euclidean distance between the initial control field and the resultant optimal control field that takes an observable from the bottom to the top of the landscape. Computational analyses for simple model quantum systems are performed to ascertain the relative abundance of nearly straight control trajectories encountered when optimizing a state-to-state transition probability. The distribution of RRR values is found to be centered near remarkably low values upon sampling large numbers of randomly chosen initial control fields. Additionally, a stochastic algorithm is used to locate many distinct initial control fields, each of which corresponds to the start of an almost straight control trajectory with $R\ensuremath{\simeq}1.0$. The collected results indicate that quantum-control landscapes have very simple structural features. The favorable topology and the complementary simple structure of the control landscape provide a basis for understanding the generally observed ease of optimizing a state-to-state transition probability.
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 and structure provide a foundation to understand why optimal quantum control can be readily achieved.
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.
Quantum control landscape is generally a physical objective defined as a functional of the control field and plays an important role for the analysis and manipulation of quantum systems. In this paper, we focus on typical learning methods (i.e., gradient decent method, genetic algorithm and deferential evolution) for the landscape control of open quantum systems and explores the characteristics of these different types of learning algorithms. Taking a two-level open quantum system as an example, the optimal value of the control landscape can be obtained under varying Lindblad operators that reflect the system's interactions with the environment. Numerical experiments demonstrate the learning performances to acquire the optimal control strategy by exploring the control landscape of this two-level open quantum system.
In this work, we consider the problem of the analysis of quantum control landscapes in both situations when the controls during optimization are unconstrained and when they should satisfy a given constraint, for example their amplitude should be bounded by some constant value. For the analysis of unconstrained quantum control landscapes, a special approach was developed in [Pechen and Tannor, Israel J. of Chemistry 52 (5), 467–472 (2012)] based on a fixed step gradient ascent pulse engineering (GRAPE) algorithm. Here we also adopt a GRAPE approach based on the Pontryagin maximum principle (PMP). For the analysis of the constrained quantum control landscapes, we adopt the gradient projection method which was developed for quantum systems in [A. Oza et al., J. Phys. A: Math. Theor. 42, 205305 (2009)] (GPM; for our purpose the most suitable is a one-step GPM with a fixed step), both for closed and open quantum systems. As an explicit example, we apply these approaches to study quantum control landscape for optimization of an observable mean value for a three-level degenerate Ξ\XiΞ -system system having a seventh-order trap.
The dynamics of closed quantum systems may be manipulated by using an applied field to achieve a control objective value for a physical goal. The functional relationship between the applied field and the objective value forms a quantum control landscape, and the optimization process consists of a guided climb up the landscape from the bottom to the top. Two classes of landscape features are important for understanding the ease of finding an optimal control field. The first class of topological landscape features has been proven to be especially simple in that no suboptimal local maxima exist (upon satisfaction of certain assumptions), which partially accounts for the ease of finding optimal fields. Complementary to the topology, the second class of features entails the landscape structure, characterizing the sinuous nature of the paths leading to an optimal control field. Previous work found that the landscape structure is also particularly simple, as excursions up the landscape guided by a gradient algorithm correspond to nearly straight paths through the space of control fields. In this paper we take an alternative approach to examining landscape structure by constructing, and then following, exactly straight trajectories in control space. Each trajectory starts at a corresponding point on the bottom of the landscape and ends at an associated point on the top, with the observable values taken either as the state-to-state transition probability, the expectation value of a general observable, or the distance from a desired unitary transformation. In some cases the starting point is at a suboptimal critical-point saddle, with the goal, again, of following a straight field path to the optimal objective yield or another suboptimal critical point. We find that the objective value almost always rises monotonically upon following a straight control path from one critical point to another, which shows that landscape structure is very simple, being devoid of rough bumps and gnarled ``twists and turns''. An analysis reveals that the generally featureless nature of quantum control landscapes can be understood in terms of the occurrence of many interfering quantum pathways contributing while traversing the landscape, essentially smoothing out the terrain. These results also provide a basis for further studies to seek a new efficient algorithm to discover optimal fields by means of taking into account the inherently smooth landscape structure.