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On the relationship between quantum control landscape structure and optimization complexity.

Katharine Moore1, Michael Hsieh, Herschel Rabitz

  • 1Department of Chemistry, Princeton University, Princeton, NJ 08544, USA.

The Journal of Chemical Physics
|April 25, 2008
PubMed
Summary
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Quantum control landscapes reveal why preparing quantum states is easy, but generating unitary transformations is exponentially harder. Optimizing control variables shows landscape structure impacts search efficiency, especially for unitary transformations.

Area of Science:

  • Quantum Control
  • Quantum Information Science
  • Computational Physics

Background:

  • State preparation in quantum systems is often efficient, irrespective of system size (N).
  • Generating targeted unitary transformations in quantum systems typically requires effort that scales exponentially with N.
  • Quantum control landscapes, mapping physical objectives to control variables, offer a framework to understand this discrepancy.

Purpose of the Study:

  • To investigate how the local structure of quantum control landscapes affects the efficiency of quantum optimal control searches.
  • To analyze the scaling of search effort with system dimension (N) for both state preparation and unitary transformation objectives.
  • To explore strategies for improving the efficiency of unitary transformation preparation.

Main Methods:

Related Experiment Videos

  • Simulations employing gradient, genetic, and simplex algorithms for optimal control.
  • Analysis of quantum control landscapes using kinematic control variables (elements of the action matrix).
  • Examination of mean path length and local landscape structure along search trajectories.

Main Results:

  • Search effort for state preparation scales weakly or independently with N.
  • Search effort for unitary transformation preparation scales exponentially with N.
  • Specifying an initial action matrix from state preparation results significantly improves unitary transformation preparation efficiency.
  • Reducing control variables below 2N-2 for state preparation drastically reduces performance.

Conclusions:

  • The local structure of quantum control landscapes explains the differing scaling behaviors of state preparation and unitary transformation generation.
  • Quantum control landscape analysis provides insights into optimizing quantum control protocols.
  • Leveraging state preparation strategies can enhance the efficiency of complex quantum operations like unitary transformations.