On the quantum adiabatic evolution with the most general system Hamiltonian

  • Jie Sun
  • Songfeng LuEmail author


In this paper, we study the problem that when quantum adiabatic evolution with the most general form of system Hamiltonian will get failed. Here the most general form means that the initial and final Hamiltonians are just designed according to the adiabatic theorem in quantum mechanics. As we will see, even in this most general model of quantum adiabatic evolution, it still exists the possibility that the quantum adiabatic computation can fail totally if some condition is satisfied, which implies the time complexity of the quantum algorithm is infinity. That is, here we propose a rather general criterion for judging whether a quantum adiabatic evolution is successful. This result largely extends the authors’ previous research on this topic, and it may be seen as a further important clue for us when designing quantum algorithms in the framework of adiabatic evolution for some practical problems.


General system Hamiltonian Quantum adiabatic evolution Quantum computation 



Jie Sun gratefully acknowledges the support from the China Postdoctoral Science Foundation under Grant No. 2017M620322, the support from the National Natural Science Foundation of China under Grant No. 61402188, the fund by Priority for the Postdoctoral Scientific and Technological Program of Hubei Province in 2017, and the Seed Foundation of Huazhong University of Science and Technology under Grant No. 2017KFYXJJ070. This work is also supported by the Science and Technology Program of Shenzhen of China under Grant Nos. JCYJ 20170818160208570 and JCYJ 20180306124612893. Finally, the authors should appreciate greatly the anonymous reviewer for helpful comments and advice on the revision of the paper which make it be in its present form.


  1. 1.
    Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472–475 (2001)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J. Comput. 37, 166–194 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Mizel, A., Lidar, D.A., Mitchell, M.: Simple proof of equivalence between adiabatic quantum computation and the circuit model. Phys. Rev. Lett. 99, 070502 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    Nagaj, D., Mozes, S.: New construction for a QMA complete three-local Hamiltonian. J. Math. Phys. 48, 072104 (2007)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Nagaj, D.: Fast universal quantum computation with railroad-switch local Hamiltonians. J. Math. Phys. 51, 062201 (2010)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Breuckmann, N.P., Terhal, B.M.: Space-time circuit-to-Hamiltonian construction and its applications. J. Phys. A Math. Theor. 47, 195304 (2014)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Gosset, D., Terhal, B.M., Vershynina, A.: Universal adiabatic quantum computation via the space-time circuit-to-Hamiltonian construction. Phys. Rev. Lett. 114, 140501 (2015)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Lloyd, S., Terhal, B.M.: Adiabatic and Hamiltonian computing on a 2D lattice with simple two-qubit interactions. New J. Phys. 18, 023042 (2016)ADSCrossRefGoogle Scholar
  9. 9.
    Bausch, J., Crosson, E.: Analysis and limitations of modified circuit-to-Hamiltonian constructions. Quantum 2, 94 (2018)CrossRefGoogle Scholar
  10. 10.
    Žnidarič, M., Horvat, M.: Exponential complexity of an adiabatic algorithm for an NP-complete problem. Phys. Rev. A 73, 022329 (2006)ADSCrossRefGoogle Scholar
  11. 11.
    Altshuler, B., Krovi, H., Roland, J.: Anderson localization makes adiabatic quantum optimization fail. Proc. Natl. Acad. Sci. USA 107, 12446–12450 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Hen, I., Young, A.P.: Exponential complexity of the quantum adiabatic algorithm for certain satisfiability problems. Phys. Rev. E 84, 061152 (2011)ADSCrossRefGoogle Scholar
  13. 13.
    Dickson, N.G., Amin, M.H.: Algorithmic approach to adiabatic quantum optimization. Phys. Rev. A 85, 032303 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Zhuang, Q.T.: Increase of degeneracy improves the performance of the quantum adiabatic algorithm. Phys. Rev. A 90, 052317 (2014)ADSCrossRefGoogle Scholar
  15. 15.
    Zeng, L.S., Zhang, J., Sarovar, M.: Schedule path optimization for adiabatic quantum computing and optimization. J. Phys. A Math. Theor. 49, 165305 (2016)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Bringewatt, J., Dorland, W., Jordan, S.P., Mink, A.: Diffusion Monte Carlo approach versus adiabatic computation for local Hamiltonians. Phys. Rev. A 97, 022323 (2018)ADSCrossRefGoogle Scholar
  17. 17.
    Mahasinghe, A., Hua, R., Dinneen, M.J., Goyal, R.: Solving the Hamiltonian cycle problem using a quantum computer. In: Proceedings of the Australasian Computer Science Week Multiconference (ACSW’19) (2019)Google Scholar
  18. 18.
    Messiah, A.: Quantum Mechanics. Dover, New York (1999)zbMATHGoogle Scholar
  19. 19.
    Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325–328 (1997)ADSCrossRefGoogle Scholar
  20. 20.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution (2000). arXiv:quant-ph/0001106
  21. 21.
    Roland, J., Cerf, N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65, 042308 (2002)ADSCrossRefGoogle Scholar
  22. 22.
    Das, S., Kobes, R., Kunstatter, G.: Energy and efficiency of adiabatic quantum search algorithms. J. Phys. A Math. Gen. 36, 2839–2845 (2003)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Sun, J., Lu, S., Liu, F.: Speedup in adiabatic evolution based quantum algorithms. Sci. China Phys. Mech. Astron. 55, 1630–1634 (2012)ADSCrossRefGoogle Scholar
  24. 24.
    Sun, J., Lu, S., Liu, F.: On the general class of models of adiabatic evolution. Open Syst. Inf. Dyn. 23, 1650016 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sun, J., Lu, S.: On the adiabatic evolution of one-dimensional projector Hamiltonians. Int. J. Quantum Inf. 10, 1250046 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sun, J., Lu, S., Braunstein, S.L.: On models of nonlinear evolution paths in adiabatic quantum algorithms. Commun. Theor. Phys. 59, 22–26 (2013)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Computer Science and TechnologyHuazhong University of Science and TechnologyWuhanChina
  2. 2.College of Educational Information and TechnologyHubei Normal UniversityHuangshiChina
  3. 3.Shenzhen Huazhong University of Science and Technology Research InstituteShenzhenChina

Personalised recommendations