Science China Physics, Mechanics and Astronomy

, Volume 55, Issue 9, pp 1630–1634 | Cite as

Speedup in adiabatic evolution based quantum algorithms

  • Jie Sun
  • SongFeng Lu
  • Fang Liu


In this context, we study three different strategies to improve the time complexity of the widely used adiabatic evolution algorithms when solving a particular class of quantum search problems where both the initial and final Hamiltonians are one-dimensional projector Hamiltonians on the corresponding ground state. After some simple analysis, we find the time complexity improvement is always accompanied by the increase of some other “complexities” that should be considered. But this just gives the implication that more feasibilities can be achieved in adiabatic evolution based quantum algorithms over the circuit model, even though the equivalence between the two has been shown. In addition, we also give a rough comparison between these different models for the speedup of the problem.


adiabatic evolution evolution paths quantum computing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Farhi E, Goldstone J, Gutmann S, et al. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science, 2001, 292: 472–476MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Messiah A. Quantum Mechanics. New York: Dover Publications, 1999Google Scholar
  3. 3.
    Amin M H S. Consistency of the adiabatic theorem. Phys Rev Lett, 2009, 102: 220401MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Comparat D. General conditions for quantum adiabatic evolution. Phys Rev A, 2009, 80: 012106MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Yukalov V I. Adiabatic theorems for linear and nonlinear Hamiltonians. Phys Rev A, 2009, 79: 052117ADSCrossRefGoogle Scholar
  6. 6.
    Lidar D A, Rezakhani A T, Hamma A. Adiabatic approximation with exponential accuracy for many-body systems and quantum computation. J Math Phys, 2009, 50: 102106MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Tong D M, Singh K, Kwek L C, et al. Sufficiency criterion for the validity of the adiabatic approximation. Phys Rev Lett, 2007, 98: 150402ADSCrossRefGoogle Scholar
  8. 8.
    Wei Z H, Ying MS. Quantum adiabatic computation and adiabatic conditions. Phys Rev A, 2007, 76: 024304ADSCrossRefGoogle Scholar
  9. 9.
    Zhao Y. Reexamination of the quantum adiabatic theorem. Phys Rev A, 2008, 76: 032109ADSCrossRefGoogle Scholar
  10. 10.
    MacKenzie R, Morin-Duchesne A, Paquette H, et al. Validity of the adiabatic approximation in quantum mechanics. Phys Rev A, 2007, 76: 044102ADSCrossRefGoogle Scholar
  11. 11.
    Jansen S, Seiler R, Ruskai M B. Bounds for the adiabatic approximation with applications to quantum computation. J Math Phys, 2007, 48: 102111MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Farhi E, Goldstone J, Gutmann S. Quantum adiabatic evolution algorithms with different paths. arXiv:quant-ph/0208135Google Scholar
  13. 13.
    van Dam W, Mosca M, Vazirani U. How powerful is adiabatic quantum computation? In: Proceedings of the 42th Ann. Symp. Foundations of Computer Science (FOCS’01). Las Vegas: IEEE Computer Society, 2001. 279–287Google Scholar
  14. 14.
    van Dam W, Vazirani U. Limits on quantum adiabatic optimization. Available at
  15. 15.
    Choi V. Avoid first order quantum phase transition by changing problem Hamiltonians. arXiv:quant-ph/1010.1220Google Scholar
  16. 16.
    Altshuler B, Krovi H, Roland J. Anderson localization makes adiabatic quantum optimization fail. Proc Natl Acad Sci USA, 2010, 107: 12446–12450ADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Roland J, Cerf N J. Quantum search by local adiabatic evolution. Phys Rev A, 2002, 65: 042308ADSCrossRefGoogle Scholar
  18. 18.
    Tulsi A. Adiabatic quantum computation with a one-dimensional projector Hamiltonian. Phys Rev A, 2009, 80: 052328ADSCrossRefGoogle Scholar
  19. 19.
    Andrecut M, Ali M K. On the adiabatic quantum evolution of a single qubit. Int J Quant Inf, 2004, 2: 447–452zbMATHCrossRefGoogle Scholar
  20. 20.
    Das S, Kobes R, Kunstatter G. Energy and efficiency of adiabatic quantum search algorithms. J Phys A-Math Gen, 2003, 36: 2839–2845MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Grover L K. Quantum mechanics helps in searching for a needle in a haystack. Phys Rev Lett, 1997, 79: 325–328ADSCrossRefGoogle Scholar
  22. 22.
    Mizel A, Lidar D A, Mitchell M. Simple proof of equivalence between adiabatic quantum computation and the circuit model. Phys Rev Lett, 2007, 99: 070502ADSCrossRefGoogle Scholar
  23. 23.
    Aharonov D, van Dam W, Kempe J, et al. Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J Comput, 2007, 37: 166–194MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.School of Computer Science and TechnologyHuazhong University of Science and TechnologyWuhanChina

Personalised recommendations