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:1006.1829 · Physical Review A

Search complexity and resource scaling for the quantum optimal control of unitary transformations

Katharine W. Moore, Raj Chakrabarti, Gregory Riviello, HERSCHEL A. RABITZ

Published Jan 31, 2011• 58 citations
arXivHTMLPDF

Abstract

The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms, but (ii) the required time for convergence to optimal controls can scale exponentially with the Hilbert space dimension. Depending on the control-system Hamiltonian, the landscape structure and scaling may vary. This work introduces methods for quantifying Hamiltonian-dependent and kinematic effects on control optimization dynamics in order to classify quantum systems according to the search effort and control resources required to implement arbitrary unitary transformations.

DOI
10.1103/physreva.83.012326

Related papers

On the relationship between quantum control landscape structure and optimization complexity

Katharine W. Moore, Michael Hsieh, HERSCHEL A. RABITZ

It has been widely observed in optimal control simulations and experiments that state preparation is surprisingly easy to achieve, regardless of the dimension N of the system Hilbert space. In contrast, simulations for the generation of targeted unitary transformations indicate that the effort increases exponentially with N. In order to understand such behavior, the concept of quantum control landscapes was recently introduced, where the landscape is defined as the physical objective, as a function of the control variables. The present work explores how the local structure of the control landscape influences the effectiveness and efficiency of quantum optimal control search efforts. Optimizations of state and unitary transformation preparation using kinematic control variables (i.e., the elements of the action matrix) are performed with gradient, genetic, and simplex algorithms. The results indicate that the search effort scales weakly, or possibly independently, with N for state preparation, while the search effort for the unitary transformation objective increases exponentially with N. Analysis of the mean path length traversed during a search trajectory through the space of action matrices and the local structure along this trajectory provides a basis to explain the difference in the scaling of the search effort with N for these control objectives. Much more favorable scaling for unitary transformation preparation arises upon specifying an initial action matrix based on state preparation results. The consequences of choosing a reduced number of control variables for state preparation is also investigated, showing a significant reduction in performance for using fewer than 2N-2 variables, which is consistent with the topological analysis of the associated landscape.

The Journal of Chemical Physics•Apr 18, 2008•48 citations

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

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

Optimal control landscape for the generation of unitary transformations with constrained dynamics

Michael Hsieh, Re-Bing Wu, HERSCHEL A. RABITZ +1 more

The reliable and precise generation of quantum unitary transformations is essential for the realization of a number of fundamental objectives, such as quantum control and quantum information processing. Prior work has explored the optimal control problem of generating such unitary transformations as a surface-optimization problem over the quantum control landscape, defined as a metric for realizing a desired unitary transformation as a function of the control variables. It was found that under the assumption of nondissipative and controllable dynamics, the landscape topology is trap free, which implies that any reasonable optimization heuristic should be able to identify globally optimal solutions. The present work is a control landscape analysis, which incorporates specific constraints in the Hamiltonian that correspond to certain dynamical symmetries in the underlying physical system. It is found that the presence of such symmetries does not destroy the trap-free topology. These findings expand the class of quantum dynamical systems on which control problems are intrinsically amenable to a solution by optimal control.

arXiv:0906.4533•Physical Review A•Jun 30, 2010•16 citations

Computational complexity of quantum optimal control landscapes

Raj Chakrabarti, Re-Bing Wu, HERSCHEL A. RABITZ

We study the Hamiltonian-independent contribution to the complexity of quantum optimal control problems. The optimization of controls that steer quantum systems to desired objectives can itself be considered a classical dynamical system that executes an analog computation. The system-independent component of the equations of motion of this dynamical system can be integrated analytically for various classes of discrete quantum control problems. For the maximization of observable expectation values from an initial pure state and the maximization of the fidelity of quantum gates, the time complexity of the corresponding computation belongs to the class continuous log (CLOG), the lowest analog complexity class, equivalent to the discrete complexity class NC. The simple scaling of the Hamiltonian-independent contribution to these problems with quantum system dimension indicates that with appropriately designed search algorithms, quantum optimal control can be rendered efficient even for large systems.

arXiv:0708.3513•Aug 27, 2007•2 citations

Subsystem-based Approach to Scalable Quantum Optimal Control

Jun Li

The development of quantum control methods is an essential task for emerging quantum technologies. In general, the process of optimizing quantum controls scales very unfavorably in system size due to the exponential growth of the Hilbert space dimension. Here, I present a scalable subsystem-based method for quantum optimal control on a large quantum system. The basic idea is to cast the original problem as a robust control problem on subsystems with requirement of robustness in respect of inter-subsystem couplings, thus enabling a drastic reduction of problem size. The method is in particular suitable for target quantum operations that are local, e.g., elementary quantum gates. As an illustrative example, I employ it to tackle the demanding task of pulse searching on a 12-spin coupled system, achieving substantial reduction of memory and time costs. This work has significant implications for coherent quantum engineering on near-term intermediate-scale quantum devices.

arXiv:1910.02061•Oct 4, 2019•0 citations
The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms, but (ii) the required time for convergence to optimal controls can scale exponentially with the Hilbert space dimension. Depending on the control-system Hamiltonian, the landscape structure and scaling may vary. This work introduces methods for quantifying Hamiltonian-dependent and kinematic effects on control optimization dynamics in order to classify quantum systems according to the search effort and control resources required to implement arbitrary unitary transformations.
It has been widely observed in optimal control simulations and experiments that state preparation is surprisingly easy to achieve, regardless of the dimension N of the system Hilbert space. In contrast, simulations for the generation of targeted unitary transformations indicate that the effort increases exponentially with N. In order to understand such behavior, the concept of quantum control landscapes was recently introduced, where the landscape is defined as the physical objective, as a function of the control variables. The present work explores how the local structure of the control landscape influences the effectiveness and efficiency of quantum optimal control search efforts. Optimizations of state and unitary transformation preparation using kinematic control variables (i.e., the elements of the action matrix) are performed with gradient, genetic, and simplex algorithms. The results indicate that the search effort scales weakly, or possibly independently, with N for state preparation, while the search effort for the unitary transformation objective increases exponentially with N. Analysis of the mean path length traversed during a search trajectory through the space of action matrices and the local structure along this trajectory provides a basis to explain the difference in the scaling of the search effort with N for these control objectives. Much more favorable scaling for unitary transformation preparation arises upon specifying an initial action matrix based on state preparation results. The consequences of choosing a reduced number of control variables for state preparation is also investigated, showing a significant reduction in performance for using fewer than 2N-2 variables, which is consistent with the topological analysis of the associated landscape.
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.
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 reliable and precise generation of quantum unitary transformations is essential for the realization of a number of fundamental objectives, such as quantum control and quantum information processing. Prior work has explored the optimal control problem of generating such unitary transformations as a surface-optimization problem over the quantum control landscape, defined as a metric for realizing a desired unitary transformation as a function of the control variables. It was found that under the assumption of nondissipative and controllable dynamics, the landscape topology is trap free, which implies that any reasonable optimization heuristic should be able to identify globally optimal solutions. The present work is a control landscape analysis, which incorporates specific constraints in the Hamiltonian that correspond to certain dynamical symmetries in the underlying physical system. It is found that the presence of such symmetries does not destroy the trap-free topology. These findings expand the class of quantum dynamical systems on which control problems are intrinsically amenable to a solution by optimal control.
We study the Hamiltonian-independent contribution to the complexity of quantum optimal control problems. The optimization of controls that steer quantum systems to desired objectives can itself be considered a classical dynamical system that executes an analog computation. The system-independent component of the equations of motion of this dynamical system can be integrated analytically for various classes of discrete quantum control problems. For the maximization of observable expectation values from an initial pure state and the maximization of the fidelity of quantum gates, the time complexity of the corresponding computation belongs to the class continuous log (CLOG), the lowest analog complexity class, equivalent to the discrete complexity class NC. The simple scaling of the Hamiltonian-independent contribution to these problems with quantum system dimension indicates that with appropriately designed search algorithms, quantum optimal control can be rendered efficient even for large systems.
The development of quantum control methods is an essential task for emerging quantum technologies. In general, the process of optimizing quantum controls scales very unfavorably in system size due to the exponential growth of the Hilbert space dimension. Here, I present a scalable subsystem-based method for quantum optimal control on a large quantum system. The basic idea is to cast the original problem as a robust control problem on subsystems with requirement of robustness in respect of inter-subsystem couplings, thus enabling a drastic reduction of problem size. The method is in particular suitable for target quantum operations that are local, e.g., elementary quantum gates. As an illustrative example, I employ it to tackle the demanding task of pulse searching on a 12-spin coupled system, achieving substantial reduction of memory and time costs. This work has significant implications for coherent quantum engineering on near-term intermediate-scale quantum devices.