Quantum Information Processing

, Volume 12, Issue 3, pp 1515–1538 | Cite as

Differential topology of adiabatically controlled quantum processes

  • Edmond A. JonckheereEmail author
  • Ali T. Rezakhani
  • Farooq Ahmad


It is shown that in a controlled adiabatic homotopy between two Hamiltonians, H 0 and H 1, the gap or “anti-crossing” phenomenon can be viewed as the development of cusps and swallow tails in the region of the complex plane where two critical value curves of the quadratic map associated with the numerical range of H 0 + i H 1 come close. The “near crossing” in the energy level plots happens to be a generic situation, in the sense that a crossing is a manifestation of the quadratic numerical range map being unstable in the sense of differential topology. The stable singularities that can develop are identified and it is shown that they could occur near the gap, making those singularities of paramount importance. Various applications, including the quantum random walk, are provided to illustrate this theory.


Adiabatic theorem Numerical range Cusps Swallow tails Gap Anti-crossing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Sarandy M.S., Wu L.A., Lidar D.A.: Consistency of the adiabatic theorem. Quantum Inf. Process. 3, 331 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Lidar D.A., Rezakhani A.T., Hamma A.: Adiabatic approximation with exponential accuracy or many-body systems and quantum computation. J. Math. Phys. 50, 102106 (2009)MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Rezakhani A.T., Pimachev A.K., Lidar D.A.: Accuracy versus run time in an adiabatic quantum search. Phys. Rev. A 82, 052305 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    Jonckheere E.A., Ahmad F., Gutkin E.: Differential topology of numerical range. Linear Algebra Appl. 279(1-3), 227 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    von Neumann J., Wigner E.: Uber das Verhalten von Eigenwerten bei adia-balischen Prozessen. Phys. Zschr. 30, 467 (1929)zbMATHGoogle Scholar
  6. 6.
    Gutkin E., Jonckheere E.A., Karow M.: Convexity of the joint numerical range: topological and differential geometric viewpoints. Linear Algebra Appl. LAA 376, 143 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jonckheere, E., Shabani, A., Rezakhani, A.: 4th International Symposium on Communications, Control, and Signal Processing (ISCCSP’2010) (Limassol, Cyprus, 2010), Special Session on Quantum Control I. ArXiv:1002.1515v2Google Scholar
  8. 8.
    Golubitsky M., Guillemin V.: Stable Mappings and Their Singularities, Graduate Texts in Mathematics, vol. 14. Springer, New York (1973)CrossRefGoogle Scholar
  9. 9.
    Ahmad, F.: Differential topology of numerical range and robust control analysis. Ph.D. thesis, Department of Electrical Engineering, University of Southern California (1999)Google Scholar
  10. 10.
    Arnold V.I., Gusein-Zade S.M., Varchenko A.N.: Singularities of Differentiable Maps—The Classification of Critical Points, Caustics and Wave Fronts, vol. 1. Birkhäuser, Boston (1985)Google Scholar
  11. 11.
    Arnold V.I., Gusein-Zade S.M., Varchenko A.N.: Singularities of Differentiable Maps —Monodromy and Asymptotics of Integrals, vol. 2. Birkhäuser, Boston (1988)CrossRefGoogle Scholar
  12. 12.
    Arnold, V.I.: The theory of singularities and its applications. Accademia Nationale Dei Lincei; Secuola Normale Superiore Lezioni Fermiane. Press Syndicate of the University of Cambridge, Pisa, Italy (1993)Google Scholar
  13. 13.
    Arnold, V.: Topological invariants of plane curves and caustics, University Lecture Series; Dean Jacqueline B. Lewis Memorial Lectures, Rutgers University, vol. 5. American Mathematical Society, Providence, RI (1994)Google Scholar
  14. 14.
    Rezakhani A.T., Kuo W.J., Hamma A., Lidar D.A., Zanardi P.: Quantum adiabatic brachistochrone. Phys. Rev. Lett. 103, 080502 (2009)ADSCrossRefGoogle Scholar
  15. 15.
    Cerf, J.: Publications Mathématiques, Institut des Hautes Etudes Scientifiques (I.H.E.S.) (39), p. 5 (1970)Google Scholar
  16. 16.
    Jonckheere E.A.: Algebraic and Differential Topology of Robust Stability. Oxford University Press, New York (1997)Google Scholar
  17. 17.
    O’Neill B.: Elementary Differential Geometry. Academic Press, London (1997)zbMATHGoogle Scholar
  18. 18.
    do Carmo M.P.: Riemannian Geometry. Birkhauser, Boston (1992)zbMATHCrossRefGoogle Scholar
  19. 19.
    Diaz F.S., Nuno-Ballestros J.: Plane curve diagrams and geometrical applications. Q. J. Math. 59, 1 (2007).  10.1093/qmath/ham039 CrossRefGoogle Scholar
  20. 20.
    Harris J.: Algebraic Geometry–A First Course. Springer, New York (1992)zbMATHGoogle Scholar
  21. 21.
    Kempe J.: Quantum random walks—an introductory overview. Contemp. Phys. 44(4), 307 (2003)MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Krovi H., Ozols M., Roland J.: Adiabatic condition and the quantum hitting time of Markov chains. Phys. Rev. A 82, 1–022333 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Edmond A. Jonckheere
    • 1
    Email author
  • Ali T. Rezakhani
    • 2
  • Farooq Ahmad
    • 3
  1. 1.USC Center for Quantum Information Science & TechnologyLos AngelesUSA
  2. 2.Sharif University of TechnologyTeheranIran
  3. 3.Delta Tau Data Systems, Inc.ChatsworthUSA

Personalised recommendations