Evidence for quantum annealing with more than one hundred qubits

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Abstract

Quantum technology is maturing to the point where quantum devices, such as quantum communication systems, quantum random number generators and quantum simulators may be built with capabilities exceeding classical computers. A quantum annealer, in particular, solves optimization problems by evolving a known initial configuration at non-zero temperature towards the ground state of a Hamiltonian encoding a given problem. Here, we present results from tests on a 108 qubit D-Wave One device based on superconducting flux qubits. By studying correlations we find that the device performance is inconsistent with classical annealing or that it is governed by classical spin dynamics. In contrast, we find that the device correlates well with simulated quantum annealing. We find further evidence for quantum annealing in the form of small-gap avoided level crossings characterizing the hard problems. To assess the computational power of the device we compare it against optimized classical algorithms.

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Figure 1: Success probability distributions.
Figure 2: Correlations.
Figure 3: Correlations of gauge-averaged data.
Figure 4: Evolution of the lowest spectral gap.
Figure 5: Correlation of success probability and the Hamming distance from excited states to the nearest ground state.
Figure 6: Scaling with problem size.

References

  1. 1

    Muhly, J. in The beginning of the use of metals and alloys (ed Maddin, R.) The beginnings of metallurgy in the Old World. 2–20 (MIT Press, 1988).

  2. 2

    Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Optimization by simulated annealing. Science 220, 671–680 (1983).

  3. 3

    Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953).

  4. 4

    Ray, P., Chakrabarti, B. K. & Chakrabarti, A. Sherrington–Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations. Phys. Rev. B 39, 11828–11832 (1989).

  5. 5

    Finnila, A., Gomez, M., Sebenik, C., Stenson, C. & Doll, J. Quantum annealing: A new method for minimizing multidimensional functions. Chem. Phys. Lett. 219, 343–348 (1994).

  6. 6

    Kadowaki, T. & Nishimori, H. Quantum annealing in the transverse Ising model. Phys. Rev. E 58, 5355–5363 (1998).

  7. 7

    Martoňák, R., Santoro, G. E. & Tosatti, E. Quantum annealing by the path-integral Monte Carlo method: The two-dimensional random Ising model. Phys. Rev. B 66, 094203 (2002).

  8. 8

    Santoro, G. E., Martoňák, R., Tosatti, E. & Car, R. Theory of quantum annealing of an Ising spin glass. Science 295, 2427–2430 (2002).

  9. 9

    Battaglia, D. A., Santoro, G. E. & Tosatti, E. Optimization by quantum annealing: Lessons from hard satisfiability problems. Phys. Rev. E 71, 066707 (2005).

  10. 10

    Brooke, J., Bitko, D., Rosenbaum, F. T. & Aeppli, G. Quantum annealing of a disordered magnet. Science 284, 779–781 (1999).

  11. 11

    Johnson, M. W. et al. Quantum annealing with manufactured spins. Nature 473, 194–198 (2011).

  12. 12

    Boixo, S., Albash, T., Spedalieri, F. M., Chancellor, N. & Lidar, D. A. Experimental signature of programmable quantum annealing. Nature Commun. 4, 2067 (2013).

  13. 13

    Dickson, N. G. et al. Thermally assisted quantum annealing of a 16-qubit problem. Nature Commun. 4, 1903 (2013).

  14. 14

    Barahona, F. On the computational complexity of Ising spin glass models. J. Phys. A 15, 3241–3253 (1982).

  15. 15

    Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472–475 (2001).

  16. 16

    Jörg, T., Krzakala, F., Semerjian, G. & Zamponi, F. First-order transitions and the performance of quantum algorithms in random optimization problems. Phys. Rev. Lett. 104, 207206 (2010).

  17. 17

    Hen, I. & Young, A. P. Exponential complexity of the quantum adiabatic algorithm for certain satisfiability problems. Phys. Rev. E 84, 061152 (2011).

  18. 18

    Farhi, E. et al. Performance of the quantum adiabatic algorithm on random instances of two optimization problems on regular hypergraphs. Phys. Rev. A 86, 052334 (2012).

  19. 19

    Bian, Z., Chudak, F., Macready, W. G., Clark, L. & Gaitan, F. Experimental determination of Ramsey numbers. Phys. Rev. Lett. 111, 130505 (2013).

  20. 20

    Perdomo-Ortiz, A., Dickson, N., Drew-Brook, M., Rose, G. & Aspuru-Guzik, A. Finding low-energy conformations of lattice protein models by quantum annealing. Sci. Rep. 2, 571 (2012).

  21. 21

    Kashurnikov, V. A., Prokof’ev, N. V., Svistunov, B. V. & Troyer, M. Quantum spin chains in a magnetic field. Phys. Rev. B 59, 1162–1167 (1999).

  22. 22

    Young, A. P., Knysh, S. & Smelyanskiy, V. N. Size dependence of the minimum excitation gap in the quantum adiabatic algorithm. Phys. Rev. Lett. 101, 170503 (2008).

  23. 23

    Altshuler, B., Krovi, H. & Roland, J. Anderson localization makes adiabatic quantum optimization fail. Proc. Natl Acad. Sci. USA 107, 12446–12450 (2010).

  24. 24

    Dechter, R. Bucket elimination: A unifying framework for reasoning. Artif. Intell. 113, 41–85 (1999).

  25. 25

    McGeoch, C. C. & Wang, C. Proc. 2013 ACM Conf. Comput. Frontiers (ACM, 2013).

  26. 26

    Choi, V. Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. Quant. Inform. Process. 10, 343–353 (2011).

  27. 27

    Harris, R. et al. Experimental demonstration of a robust and scalable flux qubit. Phys. Rev. B 81, 134510 (2010).

  28. 28

    Harris, R. et al. Experimental investigation of an eight-qubit unit cell in a superconducting optimization processor. Phys. Rev. B 82, 024511 (2010).

  29. 29

    Berkley, A. J. et al. A scalable readout system for a superconducting adiabatic quantum optimization system. Supercond. Sci. Tech. 23, 105014 (2010).

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Acknowledgements

We acknowledge useful discussions with M. H. Amin, M. H. Freedman, H. G. Katzgraber, C. Marcus, B. Smith and K. Svore. We thank L. Wang for providing data of spin dynamics simulations, G. Wagenbreth for help in optimizing the belief propagation code and P. Messmer for help in optimizing the GPU codes. We are grateful to J. Smolin and G. Smith for suggesting that we test classical spin dynamics. Simulations were performed on the Brutus cluster at ETH Zurich and on computing resources of Microsoft Research with the help of J. Jernigan. This work was supported by the Swiss National Science Foundation through the NCCR QSIT, by ARO grant number W911NF-12-1-0523, by ARO MURI grant number W911NF-11-1-0268, by Lockheed Martin Corporation, by DARPA grant number FA8750-13-2-0035, and by NSF grant number CHE-1037992. MT acknowledges the hospitality of the Aspen Center for Physics, supported by NSF grant PHY-1066293. The initial planning of the tests by MT was funded by Microsoft Research.

Author information

M.T., J.M.M. and D.A.L. designed the tests and wrote the manuscript, with input from all other authors. S.B. and Z.W. performed the tests on D-Wave One. S.V.I., T.F.R. and M.T. wrote the simulated classical and quantum annealing codes and T.F.R., S.V.I., M.T. and D.W. performed the simulations. S.B. and T.F.R. wrote the bucket sort code and the divide-and-conquer codes. T.F.R., S.V.I., M.T., S.B., Z.W. and D.A.L. evaluated the data. All authors contributed to the discussion and presentation of the results.

Correspondence to Matthias Troyer.

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Boixo, S., Rønnow, T., Isakov, S. et al. Evidence for quantum annealing with more than one hundred qubits. Nature Phys 10, 218–224 (2014). https://doi.org/10.1038/nphys2900

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