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arXiv:1507.08785 · Physical Review A

Role of control constraints in quantum optimal control

Dmitry V. Zhdanov, Tamar Seideman

Published Nov 13, 2015• 13 citations
arXivHTMLPDF

Abstract

The problems of optimizing the value of an arbitrary observable of a two-level system at both a fixed time and the shortest possible time is theoretically explored. Complete identification and classification along with comprehensive analysis of globally optimal control policies and traps (i.e., policies which are locally but not globally optimal) are presented. The central question addressed is whether the control landscape remains trap-free if control constraints of the inequality type are imposed. The answer is astonishingly controversial: Although the traps are proven always to exist in this case, in practice they become trivially escapable once the control time is fixed and chosen long enough.

DOI
10.1103/physreva.92.052109

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The problems of optimizing the value of an arbitrary observable of a two-level system at both a fixed time and the shortest possible time is theoretically explored. Complete identification and classification along with comprehensive analysis of globally optimal control policies and traps (i.e., policies which are locally but not globally optimal) are presented. The central question addressed is whether the control landscape remains trap-free if control constraints of the inequality type are imposed. The answer is astonishingly controversial: Although the traps are proven always to exist in this case, in practice they become trivially escapable once the control time is fixed and chosen long enough.
The success of quantum optimal control for both experimental and theoretical objectives is connected to the topology of the corresponding control landscapes, which are free from local traps if three conditions are met: (1) the quantum system is controllable, (2) the Jacobian of the map from the control field to the evolution operator is of full rank, and (3) there are no constraints on the control field. This paper investigates how the violation of assumption (3) affects gradient searches for globally optimal control fields. The satisfaction of assumptions (1) and (2) ensures that the control landscape lacks fundamental traps, but certain control constraints can still introduce artificial traps. Proper management of these constraints is an issue of great practical importance for numerical simulations as well as optimization in the laboratory. Using optimal control simulations, we show that constraints on quantities such as the number of control variables, the control duration, and the field strength are potentially severe enough to prevent successful optimization of the objective. For each such constraint, we show that exceeding quantifiable limits can prevent gradient searches from reaching a globally optimal solution. These results demonstrate that careful choice of relevant control parameters helps to eliminate artificial traps and facilitates successful optimization.
Abstract Quantum control is a necessary tool for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems for various tasks. Traps are points of local but not global optimum of the objective functional for a given quantum control problem. In a more general sense, traps are critical points of the objective functional which are hard to escape by local search algorithms. Here a review of some results of the analysis of possibility of having traps in landscapes of coherently controlled closed quantum systems is given. In one-qubit case, there are no traps. For special multilevel quantum systems, higher-order traps may appear.
There is a strong interest in optimal manipulating of quantum systems by external controls. Traps are controls which are optimal only locally but not globally. If they exist, they can be serious obstacles to the search of globally optimal controls in numerical and laboratory experiments, and for this reason the analysis of traps attracts considerable attention. In this paper we prove that for a wide range of control problems for two-level quantum systems all locally optimal controls are also globally optimal. Hence we conclude that two-level systems in general are trap-free. In particular, manipulating qubits—two-level quantum systems forming a basic building block for quantum computation—is free of traps for fundamental problems such as the state preparation and gate generation.
We present a comprehensive analysis of the landscape for full quantum-quantum control associated with the expectation value of an arbitrary observable of one quantum system controlled by another quantum system. It is shown that such full quantum-quantum control landscapes are convex, and hence devoid of local suboptima and saddle points that may exist in landscapes for quantum systems controlled by time-dependent classical fields. There is no controllability requirement for the full quantum-quantum landscape to be trap-free, although the forms of Hamiltonians, the flexibility in choosing initial state of the controller, as well as the control duration, can infulence the reachable optimal value on the landscape. All level sets of the full quantum-quantum landscape are connected convex sets. Finally, we show that the optimal solution of the full quantum-quantum control landscape can be readily determined numerically, which is demonstrated using the Jaynes-Cummings model depicting a two-level atom interacting with a quantized radiation field.
The analysis of traps, that is, locally but not globally optimal controls, for quantum control systems has attracted great interest in recent years. The central problem that has remained open is to demonstrate for a given system either the existence or the absence of traps. We prove the absence of traps and hence completely solve this problem for the important tasks of unconstrained manipulation of the transition probability and unitary gate generation in the Landau-Zener system, a system with a wide range of applications across physics, chemistry, and biochemistry. This finding provides an example of a controlled quantum system which is completely free of traps. We also discuss the impact of laboratory constraints due to decoherence, noise in the control pulse, and restrictions on the available controls, which, when being sufficiently severe, can produce traps.
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.