Quantum Information Processing

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

Differential topology of adiabatically controlled quantum processes

  • Edmond A. Jonckheere
  • Ali T. Rezakhani
  • Farooq Ahmad
Article

Abstract

It is shown that in a controlled adiabatic homotopy between two Hamiltonians, H0 and H1, 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 H0 + i H1 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.

Keywords

Adiabatic theorem Numerical range Cusps Swallow tails Gap Anti-crossing 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Edmond A. Jonckheere
    • 1
  • 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

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