The European Physical Journal Special Topics

, Volume 224, Issue 1, pp 111–129 | Cite as

Reexamining classical and quantum models for the D-Wave One processor

The role of excited states and ground state degeneracy
  • T. Albash
  • T.F. Rønnow
  • M. Troyer
  • D.A. LidarEmail author
Part of the following topical collections:
  1. Quantum Annealing: The Fastest Route to Quantum Computation?


We revisit the evidence for quantum annealing in the D-Wave One device (DW1) based on the study of random Ising instances. Using the probability distributions of finding the ground states of such instances, previous work found agreement with both simulated quantum annealing (SQA) and a classical rotor model. Thus the DW1 ground state success probabilities are consistent with both models, and a different measure is needed to distinguish the data and the models. Here we consider measures that account for ground state degeneracy and the distributions of excited states, and present evidence that for these new measures neither SQA nor the classical rotor model correlate perfectly with the DW1 experiments. We thus provide evidence that SQA and the classical rotor model, both of which are classically efficient algorithms, do not satisfactorily explain all the DW1 data. A complete model for the DW1 remains an open problem. Using the same criteria we find that, on the other hand, SQA and the classical rotor model correlate closely with each other. To explain this we show that the rotor model can be derived as the semiclassical limit of the spin-coherent states path integral. We also find differences in which set of ground states is found by each method, though this feature is sensitive to calibration errors of the DW1 device and to simulation parameters.


European Physical Journal Special Topic Success Probability Random Ising Instance Annealing Schedule Total Variation Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    M.W. Johnson, P. Bunyk, F. Maibaum, E. Tolkacheva, A.J. Berkley, E.M. Chapple, R. Harris, J. Johansson, T. Lanting, I. Perminov, et al., Superconductor Sci. Technol. 23(6), 065004 (2010)ADSCrossRefGoogle Scholar
  2. 2.
    A.J. Berkley, M.W. Johnson, P. Bunyk, R. Harris, J. Johansson, T. Lanting, E. Ladizinsky, E. Tolkacheva, M.H.S. Amin, G. Rose, Superconductor Sci. Technol. 23, 105014 (2010)ADSCrossRefGoogle Scholar
  3. 3.
    R. Harris, M.W. Johnson, T. Lanting, A.J. Berkley, J. Johansson, P. Bunyk, E. Tolkacheva, E. Ladizinsky, N. Ladizinsky, T. Oh et al., Phys. Rev. B 82, 024511 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    P.I. Bunyk, E. Hoskinson, M.W. Johnson, E. Tolkacheva, F. Altomare, A.J. Berkley, R. Harris, J.P. Hilton, T. Lanting, J. Whittaker, [arXiv:1401.5504] (2014),
  5. 5.
    P. Ray, B.K. Chakrabarti, A. Chakrabarti, Phys. Rev. B 39, 11828 (1989)ADSCrossRefGoogle Scholar
  6. 6.
    A.B. Finnila, M.A. Gomez, C. Sebenik, C. Stenson, J.D. Doll, Chem. Phys. Lett. 219, 343 (1994)ADSCrossRefGoogle Scholar
  7. 7.
    T. Kadowaki, H. Nishimori, Phys. Rev. E 58(5), 5355 (1998)ADSCrossRefGoogle Scholar
  8. 8.
    J. Brooke, D. Bitko, T.F. Rosenbaum, G. Aeppli, Science 284, 779 (1999)ADSCrossRefGoogle Scholar
  9. 9.
    J. Brooke, T.F. Rosenbaum, G. Aeppli, Nature 413, 610 (2001)ADSCrossRefGoogle Scholar
  10. 10.
    G.E. Santoro, E. Tosatti, J. Phys. A: Math. General 39, R393 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    S. Morita, H. Nishimori, J. Math. Phys. 49, 125210 (2008)ADSCrossRefMathSciNetGoogle Scholar
  12. 12.
    A. Das, B.K. Chakrabarti, Rev. Mod. Phys. 80, 1061 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    V. Bapst, L. Foini, F. Krzakala, G. Semerjian, F. Zamponi, Phys. Reports 523, 127 (2013)ADSCrossRefMathSciNetGoogle Scholar
  14. 14.
    M.W. Johnson, M.H.S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A.J. Berkley, J. Johansson, P. Bunyk, et al., Nature 473, 194 (2011)ADSCrossRefGoogle Scholar
  15. 15.
    T. Lanting, A.J. Przybysz, A.Y. Smirnov, F.M. Spedalieri, M.H. Amin, A.J. Berkley, R. Harris, F. Altomare, S. Boixo, P. Bunyk, et al., Phys. Rev. X 4, 021041 (2014)Google Scholar
  16. 16.
    S. Boixo, T.F. Rønnow, S.V. Isakov, Z. Wang, D. Wecker, D.A. Lidar, J.M. Martinis, M. Troyer, Nat. Phys. 10, 218 (2014)CrossRefGoogle Scholar
  17. 17.
    S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Science 220, 671 (1983)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    J.A. Smolin, G. Smith, [arXiv:1305.4904] (2013),
  19. 19.
    L. Wang, T.F. Rønnow, S. Boixo, S.V. Isakov, Z. Wang, D. Wecker, D.A. Lidar, J.M. Martinis, M. Troyer, [arXiv:1305.5837] (2013),
  20. 20.
    T. Gilbert, IEEE Trans. Magn. 40(6), 3443 (2004)ADSCrossRefGoogle Scholar
  21. 21.
    R. Martoňák, G.E. Santoro, E. Tosatti, Phys. Rev. B 66, 094203 (2002)ADSCrossRefGoogle Scholar
  22. 22.
    G.E. Santoro, R. Martoňák, E. Tosatti, R. Car, Science 295(5564), 2427 (2002)ADSCrossRefGoogle Scholar
  23. 23.
    S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, [arXiv:1401.7087] (2014),
  24. 24.
    S. Boixo, T. Albash, F.M. Spedalieri, N. Chancellor, D.A. Lidar, Nat. Commun. 4 (2013)Google Scholar
  25. 25.
    W. Vinci, T. Albash, A. Mishra, P.A. Warburton, D.A. Lidar, [arXiv:1403.4228] (2014),
  26. 26.
    S.W. Shin, G. Smith, J.A. Smolin, U. Vazirani, [arXiv:1404.6499] (2014),
  27. 27.
    T. Albash, S. Boixo, D.A. Lidar, P. Zanardi, New J. Phys. 14, 123016 (2012)ADSCrossRefMathSciNetGoogle Scholar
  28. 28.
    V. Smelyanskiy, lecture presented at AQC14 (2014)Google Scholar
  29. 29.
    T. Lanting, D-Wave Inc. (private communications) (2013)Google Scholar
  30. 30.
    M. Suzuki, Progr. Theor. Phys. 56, 1454 (1976)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    F.T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, Phys. Rev. A 6, 2211 (1972)ADSCrossRefGoogle Scholar
  32. 32.
    E. Lieb, Commun. Math. Phys. 31, 327 (1973)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    S. Kirchner, J. Low Temp. Phys. 161, 282 (2010)ADSCrossRefGoogle Scholar
  34. 34.
    H.G. Katzgraber, F. Hamze, R.S. Andrist, Phys. Rev. X 4, 021008 (2014)Google Scholar
  35. 35.
    Y. Matsuda, H. Nishimori, H.G. Katzgraber, New J. Phys. 11, 073021 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    P.J.D. Crowley, T. Duric, W. Vinci, P.A. Warburton, A.G. Green, [arXiv:1405.5185] (2014),
  37. 37.
    T.F. Rønnow, Z. Wang, J. Job, S. Boixo, S.V. Isakov, D. Wecker, J.M. Martinis, D.A. Lidar, M. Troyer, Science 345, 420 (2014)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences and Springer 2015

Authors and Affiliations

  • T. Albash
    • 1
    • 2
  • T.F. Rønnow
    • 3
  • M. Troyer
    • 3
  • D.A. Lidar
    • 1
    • 2
    • 4
    • 5
    Email author
  1. 1.Department of Physics and AstronomyUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Center for Quantum Information Science & Technology, University of Southern CaliforniaLos AngelesUSA
  3. 3.Theoretische Physik, ETH ZurichZurichSwitzerland
  4. 4.Department of ChemistryUniversity of Southern CaliforniaLos AngelesUSA
  5. 5.Department of Electrical EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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