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Journal of Mathematical Chemistry

Local topology at limited resource induced suboptimal traps on the quantum control landscape

Ashley Donovan, Vincent Beltrani, HERSCHEL A. RABITZ

Published Oct 17, 2013• 7 citations
Publisher
DOI
10.1007/s10910-013-0269-x

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A search for the variables that control a quantum system's dynamics occurs over a landscape, defined as the target objective as a function of the variables. Prior studies show that upon satisfaction of three specific assumptions, the topology of the landscape is free of suboptimal traps that could prematurely halt the search for an optimal control. One key assumption is free access to all necessary control variables; however, in practice, the controls are always limited in some fashion which may result in constraint-induced traps on the landscape. This paper aims to introduce the means to systematically explore the nature of constrained controls that yield suboptimal outcomes. The procedure utilizes kinematic controls, which comprise a simple set of time-independent variables, and then performs a landscape topology-preserving transformation into corresponding dynamic controls. The equivalent landscape topology of these two formulations permits the study of a family of dynamic controls that reflect constrained control landscape behavior. In particular, constrained dynamic controls are identified as isolated points on the landscape or as suboptimal level sets. The wide range of such dynamic controls indicates the richness and complexity of constraint-induced features on the landscape.
A broad class of quantum control problems entails optimizing the expectation value of an observable operator through tailored unitary propagation of the system density matrix. Such optimization processes can be viewed as a directed search over a quantum control landscape. The attainment of the global extrema of this landscape is the goal of quantum control. Local optima will generally exist, and their enumeration is shown to scale factorially with the system's effective Hilbert space dimension. A Hessian analysis reveals that these local optima have saddlepoint topology and cannot behave as suboptimal extrema traps. The implications of the landscape topology for practical quantum control efforts are discussed, including in the context of nonideal operating conditions.
The broad success of optimally controlling quantum systems with external fields has been attributed to the favorable topology of the underlying control landscape, where the landscape is the physical observable as a function of the controls. The control landscape can be shown to contain no suboptimal trapping extrema upon satisfaction of reasonable physical assumptions, but this topological analysis does not hold when significant constraints are placed on the control resources. This work employs simulations to explore the topology and features of the control landscape for pure-state population transfer with a constrained class of control fields. The fields are parameterized in terms of a set of uniformly spaced spectral frequencies, with the associated phases acting as the controls. This restricted family of fields provides a simple illustration for assessing the impact of constraints upon seeking optimal control. Optimization results reveal that the minimum number of phase controls necessary to assure a high yield in the target state has a special dependence on the number of accessible energy levels in the quantum system, revealed from an analysis of the first- and second-order variation of the yield with respect to the controls. When an insufficient number of controls and/or a weak control fluence are employed, trapping extrema and saddle points are observed on the landscape. When the control resources are sufficiently flexible, solutions producing the globally maximal yield are found to form connected "level sets" of continuously variable control fields that preserve the yield. These optimal yield level sets are found to shrink to isolated points on the top of the landscape as the control field fluence is decreased, and further reduction of the fluence turns these points into suboptimal trapping extrema on the landscape. Although constrained control fields can come in many forms beyond the cases explored here, the behavior found in this paper is illustrative of the impacts that constraints can introduce.