Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1004.3492 · Infinite Dimensional Analysis Quantum Probability and Related Topics

A CLOSER LOOK AT QUANTUM CONTROL LANDSCAPES AND THEIR IMPLICATION FOR CONTROL OPTIMIZATION

Pierre de Fouquieres, S. G. Schirmer

Published Sep 1, 2013• 73 citations
arXivHTMLPDF

Abstract

The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in L2(0, T), however, there are critical points for which the fidelity can assume any value in (0, 1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over U(N), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in L2(0, T) is larger and includes traps, some of which remain traps even when the target time is allowed to vary.

DOI
10.1142/s0219025713500215

Related papers

Optimal control landscape for the generation of unitary transformations

Michael Hsieh, HERSCHEL A. RABITZ

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.

Physical Review A•Apr 7, 2008•46 citations

Characterization of the Critical Sets of Quantum Unitary Control Landscapes

Jason Dominy, Tak‐San Ho, HERSCHEL A. RABITZ

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.

arXiv:1102.3502•IEEE Transactions on Automatic Control•Apr 29, 2014•11 citations

Quantum Control Landscapes and Traps

Boris Olegovich Volkov, Alexander Nikolaevich Pechen

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.

Russian Microelectronics•Dec 1, 2023•1 citation

Exploring constrained quantum control landscapes

Katharine W. Moore, HERSCHEL A. RABITZ

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.

arXiv:1111.1269•The Journal of Chemical Physics•Oct 5, 2012•76 citations

Landscape of unitary transformations in controlled quantum dynamics

Tak‐San Ho, Jason Dominy, HERSCHEL A. RABITZ

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.

Physical Review A•Jan 27, 2009•65 citations

Topology of the quantum control landscape for observables

Michael Hsieh, Re-Bing Wu, HERSCHEL A. RABITZ

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 Journal of Chemical Physics•Mar 11, 2009•35 citations
The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in L2(0, T), however, there are critical points for which the fidelity can assume any value in (0, 1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over U(N), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in L2(0, T) is larger and includes traps, some of which remain traps even when the target time is allowed to vary.
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.
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.
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.
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.
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.
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.