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

Physical Review A

Uniform, rapidly convergent algorithm for quantum optimal control of objectives with a positive semidefinite Hessian matrix

Wusheng Zhu, HERSCHEL A. RABITZ

Published Dec 1, 1998• 19 citations
Publisher

Abstract

A uniform iteration method is presented for achieving quantum optimal control over any real objective with a positive semidefinite Hessian matrix. Theoretical analysis shows that this uniform algorithm exhibits quadratic and monotonic convergence. Numerical calculations verify that for this uniform algorithm, within a few steps, the optimized objective functional comes close to its converged limit. For some optimal control purposes, the objective itself is not required to be directly a physical observable, but it is only necessary that the objective have a suitable association with some desired physical observables. As an illustration of the algorithm, the control objective is chosen to achieve maximum population in a target state as well as minimum phase mismatch with the target state.

DOI
10.1103/physreva.58.4741

Related papers

A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator

Wusheng Zhu, HERSCHEL A. RABITZ

A new iteration method is presented for achieving quantum optimal control over the expectation value of a positive definite operator. Theoretical analysis shows that this new algorithm exhibits quadratic and monotonic convergence. Numerical calculations verify that for this new algorithm, within a few steps, the optimized objective functional comes close to its converged limit.

The Journal of Chemical Physics•Jul 8, 1998•321 citations

Rapidly convergent iteration methods for quantum optimal control of population

Wusheng Zhu, Jair Botina, HERSCHEL A. RABITZ

A family of new iteration methods is presented for designing quantum optimal controls to manipulate the transition probability. Theoretical analysis shows that these new methods exhibit quadratic and monotonic convergence. Numerical calculations verify that for these new methods, within very few steps, the optimized objective functional comes close to its convergent limit.

The Journal of Chemical Physics•Feb 1, 1998•437 citations

Noniterative algorithms for finding quantum optimal controls

Wusheng Zhu, HERSCHEL A. RABITZ

Iterative methods are generally necessary for solving the design equations to identify optimal quantum controls. Since iteration can be computationally intensive, it is significant to develop good approximate noniterative methods. In this paper we present noniterative techniques for achieving quantum optimal control over the expectation value of positive semidefinite operators. The noniterative methods are characterized by an order index. Zeroth-order methods involving no feedback from the objective are generally found to be inadequate. A proposed first-order noniterative algorithm is expected to often be a good approximation. Numerical tests verify the noniterative capabilities of the algorithm.

The Journal of Chemical Physics•Apr 15, 1999•37 citations

Efficient algorithm for optimal control of mixed-state quantum systems

S. G. Schirmer, Marvin D. Girardeau, J. V. Leahy

Zhu and Rabitz [J. Chem. Phys. 109, 385 (1998)] presented a rapidly convergent iterative algorithm for optimal control of the expectation value of a positive-definite observable in a pure-state quantum system. In this paper we generalize this algorithm to a quantum-statistical mechanics setting and show that it is both efficient in the mixed-state case and effective in achieving the control objective of maximizing the ensemble average of arbitrary observables in the cases studied.

arXiv:quant-ph/9906089•Physical Review A•Dec 8, 1999•51 citations

Optimal control with accelerated convergence: Combining the Krotov and quasi-Newton methods

Reuven Eitan, Michael D. Mundt, David J. Tannor

One of the most popular methods for solving numerical optimal control problems is the Krotov method, adapted for quantum control by Tannor and coworkers. The Krotov method has the following three appealing properties: (1) monotonic increase of the objective with iteration number, (2) no requirement for a line search, leading to a significant savings over gradient (first-order) methods, and (3) macrosteps at each iteration, resulting in significantly faster growth of the objective at early iterations than in gradient methods where small steps are required. The principal drawback of the Krotov method is slow convergence at later iterations, which is particularly problematic when high fidelity is desired. We show here that, near convergence, the Krotov method degenerates to a first-order gradient method. We then present a variation on the Krotov method that has all the advantages of the original Krotov method but with significantly enhanced convergence (second-order or quasi-Newton) as the optimal solution is approached. We illustrate the method by controlling the three-dimensional dynamics of the valence electron in the Na atom.

Physical Review A•May 24, 2011•106 citations

Monotonically convergent algorithm for quantum optimal control with dissipation

Yukiyoshi Ohtsuki, Wusheng Zhu, HERSCHEL A. RABITZ

This paper extends a monotonically convergent algorithm for quantum optimal control to treat systems with dissipation. The algorithm working with the density matrix is proved to exhibit quadratic and monotonic convergence. Several numerical tests are implemented in three-level model systems. The algorithm is exploited to control various targets, including the expectation value of a Hermitian operator, the modulus square of the expectation value of a non-Hermitian operator, and off-diagonal elements of the density matrix.

The Journal of Chemical Physics•May 22, 1999•172 citations
A uniform iteration method is presented for achieving quantum optimal control over any real objective with a positive semidefinite Hessian matrix. Theoretical analysis shows that this uniform algorithm exhibits quadratic and monotonic convergence. Numerical calculations verify that for this uniform algorithm, within a few steps, the optimized objective functional comes close to its converged limit. For some optimal control purposes, the objective itself is not required to be directly a physical observable, but it is only necessary that the objective have a suitable association with some desired physical observables. As an illustration of the algorithm, the control objective is chosen to achieve maximum population in a target state as well as minimum phase mismatch with the target state.
Iterative methods are generally necessary for solving the design equations to identify optimal quantum controls. Since iteration can be computationally intensive, it is significant to develop good approximate noniterative methods. In this paper we present noniterative techniques for achieving quantum optimal control over the expectation value of positive semidefinite operators. The noniterative methods are characterized by an order index. Zeroth-order methods involving no feedback from the objective are generally found to be inadequate. A proposed first-order noniterative algorithm is expected to often be a good approximation. Numerical tests verify the noniterative capabilities of the algorithm.
One of the most popular methods for solving numerical optimal control problems is the Krotov method, adapted for quantum control by Tannor and coworkers. The Krotov method has the following three appealing properties: (1) monotonic increase of the objective with iteration number, (2) no requirement for a line search, leading to a significant savings over gradient (first-order) methods, and (3) macrosteps at each iteration, resulting in significantly faster growth of the objective at early iterations than in gradient methods where small steps are required. The principal drawback of the Krotov method is slow convergence at later iterations, which is particularly problematic when high fidelity is desired. We show here that, near convergence, the Krotov method degenerates to a first-order gradient method. We then present a variation on the Krotov method that has all the advantages of the original Krotov method but with significantly enhanced convergence (second-order or quasi-Newton) as the optimal solution is approached. We illustrate the method by controlling the three-dimensional dynamics of the valence electron in the Na atom.
This paper extends a monotonically convergent algorithm for quantum optimal control to treat systems with dissipation. The algorithm working with the density matrix is proved to exhibit quadratic and monotonic convergence. Several numerical tests are implemented in three-level model systems. The algorithm is exploited to control various targets, including the expectation value of a Hermitian operator, the modulus square of the expectation value of a non-Hermitian operator, and off-diagonal elements of the density matrix.