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Title: Landscape for optimal control of quantum-mechanical unitary transformations

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= parallel W-U 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=-W. The other distinct N-1 extrema values of J are saddle points. The results of this analysis have significance for the practical construction of unitary transformations.

Authors:
;  [1];  [2]
  1. Department of Chemistry, Princeton University, Princeton, New Jersey 08544 (United States)
  2. Department of Chemistry, Drexel University, Philadelphia, Pennsylvania 19104 (United States)
Publication Date:
OSTI Identifier:
20786489
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 72; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.72.052337; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; OPTIMAL CONTROL; QUANTUM COMPUTERS; QUANTUM MECHANICS; TRANSFORMATIONS; UNITARITY

Citation Formats

Rabitz, Herschel, Hsieh, Michael, and Rosenthal, Carey. Landscape for optimal control of quantum-mechanical unitary transformations. United States: N. p., 2005. Web. doi:10.1103/PHYSREVA.72.0.
Rabitz, Herschel, Hsieh, Michael, & Rosenthal, Carey. Landscape for optimal control of quantum-mechanical unitary transformations. United States. doi:10.1103/PHYSREVA.72.0.
Rabitz, Herschel, Hsieh, Michael, and Rosenthal, Carey. Tue . "Landscape for optimal control of quantum-mechanical unitary transformations". United States. doi:10.1103/PHYSREVA.72.0.
@article{osti_20786489,
title = {Landscape for optimal control of quantum-mechanical unitary transformations},
author = {Rabitz, Herschel and Hsieh, Michael and Rosenthal, Carey},
abstractNote = {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= parallel W-U 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=-W. The other distinct N-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.72.0},
journal = {Physical Review. A},
number = 5,
volume = 72,
place = {United States},
year = {Tue Nov 15 00:00:00 EST 2005},
month = {Tue Nov 15 00:00:00 EST 2005}
}
  • 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.more » 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.« less
  • Abstract not provided.
  • 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 structuremore » 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.« less
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