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Physical Review A

Landscape for optimal control of quantum-mechanical unitary transformations

HERSCHEL A. RABITZ, Michael Hsieh, Carey M. Rosenthal

Published Nov 30, 2005• 89 citations
Publisher

Abstract

The optimal creation of a targeted unitary transformation $W$ is considered under the influence of an external control field. The controlled dynamics produces the unitary transformation $U$ and the goal is to seek a control field that minimizes the cost $J=\ensuremath{\parallel}W\ensuremath{-}U\ensuremath{\parallel}$. The optimal control landscape is the cost $J$ as a functional of the control field. For a controllable quantum system with $N$ states and without restrictions placed on the controls, the optimal control landscape is shown to have extrema with $N+1$ possible distinct values, where the desired transformation at $U=W$ is a minimum and the maximum value is at $U=\ensuremath{-}W$. The other distinct $N\ensuremath{-}1$ extrema values of $J$ are saddle points. The results of this analysis have significance for the practical construction of unitary transformations.

DOI
10.1103/physreva.72.052337

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The optimal creation of a targeted unitary transformation WWW is considered under the influence of an external control field. The controlled dynamics produces the unitary transformation UUU and the goal is to seek a control field that minimizes the cost J=\ensuremath∥W\ensuremath−U\ensuremath∥J=\ensuremath{\parallel}W\ensuremath{-}U\ensuremath{\parallel}J=\ensuremath∥W\ensuremath−U\ensuremath∥. The optimal control landscape is the cost JJJ as a functional of the control field. For a controllable quantum system with NNN states and without restrictions placed on the controls, the optimal control landscape is shown to have extrema with N+1N+1N+1 possible distinct values, where the desired transformation at U=WU=WU=W is a minimum and the maximum value is at U=\ensuremath−WU=\ensuremath{-}WU=\ensuremath−W. The other distinct N\ensuremath−1N\ensuremath{-}1N\ensuremath−1 extrema values of JJJ are saddle points. The results of this analysis have significance for the practical construction of unitary transformations.
We consider the control problem of generating unitary transformations, which is especially relevant to current research in quantum information processing and computing, in contrast to the usual state-to-state or the more general observable expectation value control problems. A previous analysis of optimal control landscapes for unitary transformations from a kinematic perspective in the finite-dimensional unitary matrices is extended to a dynamical one in the infinite-dimensional function space of the time-dependent external field. The underlying dynamical landscape is defined as the Frobenius square norm of the difference between the control unitary matrix and the target matrix. A nonsingular adaptation matrix is introduced to provide additional freedom for exploring and manipulating key features, specifically the slope and curvature, of the control landscapes. The dynamical analysis reveals many essential geometric features of optimal control landscapes for unitary transformations, including bounds on the local landscape slope and curvature. Close examination of the curvatures at the critical points shows that the unitary transformation control landscapes are free of local traps and proper choices of the adaptation matrix may facilitate the search for optimal control fields producing desired unitary transformations, in particular, in the neighborhood of the global extrema.
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.
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.
The optimal control of quantum systems provides the means to achieve the best outcome from redirecting dynamical behavior. Quantum systems for optimal control are characterized by an evolving density matrix and a Hermitian operator associated with the observable of interest. The optimal control landscape is the observable as a functional of the control field. The features of interest over this control landscape consist of the extremum values and their topological character. For controllable finite dimensional quantum systems with no constraints placed on the controls, it is shown that there is only a finite number of distinct values for the extrema, dependent on the spectral degeneracy of the initial and target density matrices. The consequences of these findings for the practical discovery of effective quantum controls in the laboratory is discussed.
Optimal control of quantum phenomena involves the introduction of a cost functional J to characterize the degree of achieving a physical objective by a chosen shaped electromagnetic field. The cost functional dependence upon the control forms a control landscape. Two theoretically important canonical cases are the landscapes associated with seeking to achieve either a physical observable or a unitary transformation. Upon satisfaction of particular assumptions, both landscapes are analytically known to be trap-free, yet possess saddle points at precise suboptimal J values. The presence of saddles on the landscapes can influence the effort needed to find an optimal field. As a foundation to future algorithm development and analyzes, we define metrics that identify the 'distance' from a given saddle based on the sufficient and necessary conditions for the existence of the saddles. Algorithms are introduced utilizing the metrics to find a control such that the dynamics arrive at a targeted saddle. The saddle distance metric and saddle-seeking methodology is tested numerically in several model systems.
This work considers various families of quantum control landscapes (i.e. objective functions for optimal control) for obtaining target unitary transformations as the general solution of the controlled Schrödinger equation. We examine the critical point structure of the kinematic landscapes JF(U) = ||(U - W)A||2and JP(U) = ||A||4- |Tr(AA†W†U)|2defined on the unitary group U(H) of a finite-dimensional Hilbert space H. The parameter operator A E (H) is allowed to be completely arbitrary, yielding an objective function that measures the difference in the actions of U and the target W on a subspace of state space, namely the column space of A. The analysis of this function includes a description of the structure of the critical sets of these kinematic landscapes and characterization of the critical points as maxima, minima, and saddles. In addition, we consider the question of whether these landscapes are Morse-Bott functions on U(H). Landscapes based on the intrinsic (geodesic) distance on U(H) and the projective unitary group PU(H) are also considered. These results are then used to deduce properties of the critical set of the corresponding dynamical landscapes.