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Critical Landscape Topology for Optimization on the Symplectic Group

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Abstract

Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over a noncompact symplectic Lie group Sp(2N,ℝ), i.e., minimizing the Frobenius distance from a target symplectic transformation, which can be used to assess the fidelity function over dynamical transformations in classical mechanics and quantum optics. The topology of the set of critical points is proven to have a unique local minimum and a number of saddlepoint submanifolds, exhibiting the absence of local suboptima that may hinder the search for ultimate optimal solutions. Compared with those of previously studied problems on compact Lie groups, such as the orthogonal and unitary groups, the topology is more complicated due to the significant nonlinearity brought by the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group.

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References

  1. Helmke, U., Moore, J.B.: Optimization and Dynamical Systems. Springer, London (1994)

    Google Scholar 

  2. Rabitz, H., Hsieh, H., Rosenthal, C.: Quantum optimally controlled transition landscapes. Science 303, 1998–2001 (2004)

    Article  Google Scholar 

  3. Frankel, T.: Critical submanifolds of the classical groups and Stiefel manifolds. In: Cairns, S.S. (ed.) Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse, vol. 1, pp. 37–53. Princeton University Press, Princeton (1962)

    Google Scholar 

  4. Dynnikov, I., Veselov, A.: Integrable gradient flows and Morse theory. St. Petersb. Math. J. 8, 429–446 (1997)

    MathSciNet  Google Scholar 

  5. Brockett, R.: Least squares matching problems. Linear Algebra Appl. 123–124, 761–777 (1989)

    Article  MathSciNet  Google Scholar 

  6. Glaser, S.J., Schulte-Herbruggen, T., Sieveking, M., Scheletzky, O., Nielsen, N.C., Sorensen, O.W., Griesinger, C.: Unitary control in quantum ensembles: maximizing signal intensity in coherent spectroscopy. Science 280, 421–424 (1998)

    Article  Google Scholar 

  7. Khaneja, N., Brockett, R.W., Glaser, S.J.: Time optimal control in spin systems. Phys. Rev. A 63, 032308 (2001)

    Article  Google Scholar 

  8. Khaneja, N., Glaser, S.J., Brockett, R.W.: Sub-Riemannian geometry and time optimal control in three spin systems: quantum gates and coherence transfer. Phys. Rev. A 65, 032301 (2002)

    Article  MathSciNet  Google Scholar 

  9. Rabitz, H., Hsieh, M., Rosenthal, C.: The landscape for optimal control of quantum-mechanical unitary transformations. Phys. Rev. A 72, 52337 (2005)

    Article  Google Scholar 

  10. Mahony, R., Manton, J.: The geometry of the Newton method on noncompact Lie groups. J. Glob. Optim. 23, 309–327 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Wu, R.-B., Chakrabarti, R., Rabitz, H.: Optimal control theory for continuous-variable quantum gates. Phys. Rev. A 77, 052303 (2008)

    Article  Google Scholar 

  12. Dragt, A., Neri, F., Rangarajan, G.: Lie algebraic treatment of linear and nonlinear beam dynamics. Ann. Rev. Nucl. Part. Sci. 38, 455–496 (1988)

    Article  Google Scholar 

  13. Han, Y., Park, F.: Least squares tracking, on the Euclidean group. IEEE Trans. Automat. Control 46, 1127–1131 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cardoso, J., Silva-Leite, F.: Extending results from orthogonal to P-orthogonal matrices. http://www2.isec.pt/~jocar/Ficheiros/Pmatrizes.pdf (2007)

  15. Bartlett, S.D., Sanders, B., Braunstein, S., Nemoto, K.: Efficient classical simulation of continuous variable quantum information processes. Phys. Rev. Lett. 88, 097904 (2002)

    Article  Google Scholar 

  16. Gottesman, D., Kitaev, A., Preskill, J.: Encoding a qubit in an oscillator. Phys. Rev. A 64, 012310 (2001)

    Article  Google Scholar 

  17. Chakrabarti, R., Rabitz, H.: Quantum control landscapes. Int. Rev. Phys. Chem. 26, 671–735 (2007)

    Article  Google Scholar 

  18. Wu, R.-B., Rabitz, H., Hsieh, M.: Characterization of the critical submanifolds in quantum ensemble control landscapes. J. Phys. A 41, 015006 (2008)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Automation, Tsinghua University, Beijing, 100084, P.R. China

    R.-B. Wu

  2. Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing, 100084, P.R. China

    R.-B. Wu

  3. Department of Chemistry, Princeton University, Princeton, NJ, 08544, USA

    R.-B. Wu & H. Rabitz

  4. School of Chemical Engineering, Purdue University, West Lafayette, IN, 47906, USA

    R. Chakrabarti

  5. Department of Chemistry, Princeton University, Princeton, NJ, 08544, USA

    R. Chakrabarti

Authors
  1. R.-B. Wu
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  2. R. Chakrabarti
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  3. H. Rabitz
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Corresponding author

Correspondence to R.-B. Wu.

Additional information

Communicated by Q.C. Zhao.

The authors acknowledge support from NSF. R.B. Wu would also like to acknowledge support from NSFC (Grant No. 60904034 and 60635040).

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Wu, RB., Chakrabarti, R. & Rabitz, H. Critical Landscape Topology for Optimization on the Symplectic Group. J Optim Theory Appl 145, 387–406 (2010). https://doi.org/10.1007/s10957-009-9641-1

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  • DOI: https://doi.org/10.1007/s10957-009-9641-1

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