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Coherent quantum annealing in a programmable 2,000 qubit Ising chain

Nature Physics volume 18, pages 1324–1328 (2022)Cite this article

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Abstract

Quantum simulation has emerged as a valuable arena for demonstrating and understanding the capabilities of near-term quantum computers1,2,3. Quantum annealing4,5 has been successfully used in simulating a range of open quantum systems, both at equilibrium6,7,8 and out of equilibrium9,10,11. However, in all previous experiments, annealing has been too slow to coherently simulate a closed quantum system, due to the onset of thermal effects from the environment. Here we demonstrate coherent evolution through a quantum phase transition in the paradigmatic setting of a one-dimensional transverse-field Ising chain, using up to 2,000 superconducting flux qubits in a programmable quantum annealer. In large systems, we observe the quantum Kibble–Zurek mechanism with theoretically predicted kink statistics, as well as characteristic positive kink–kink correlations, independent of temperature. In small chains, excitation statistics validate the picture of a Landau–Zener transition at a minimum gap. In both cases, the results are in quantitative agreement with analytical solutions to the closed-system quantum model. For slower anneals, we observe anti-Kibble–Zurek scaling in a crossover to the open quantum regime. The coherent dynamics of large-scale quantum annealers demonstrated here can be exploited to perform approximate quantum optimization, machine learning and simulation tasks.

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Fig. 1: QPT in an annealed Ising chain.
Fig. 2: Kink density scaling and distribution.
Fig. 3: Normalized kink–kink correlations.
Fig. 4: Crossover to adiabaticity.

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Data availability

Data supporting the findings of this paper are available from the corresponding author upon request. Source data are provided with this paper.

Code availability

The TEBD code used in this paper is available from the corresponding author upon reasonable request. An open-source version of the PIMC code used in the Supplementary Information is available via GitHub at https://github.com/dwavesystems/dwave-pimc. The version for this work is archived in Zenodo at https://doi.org/10.5281/zenodo.6842260.

References

  1. Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

    Article  ADS  Google Scholar 

  2. Zhang, J. et al. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature 551, 601–604 (2017).

    Article  ADS  Google Scholar 

  3. Keesling, A. et al. Quantum Kibble–Zurek mechanism and critical dynamics on a programmable Rydberg simulator. Nature 568, 207–211 (2019).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  6. Harris, R. et al. Phase transitions in a programmable quantum spin glass simulator. Science 361, 162–165 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  7. King, A. D. et al. Observation of topological phenomena in a programmable lattice of 1,800 qubits. Nature 560, 456–460 (2018).

    Article  ADS  Google Scholar 

  8. Nishimura, K., Nishimori, H. & Katzgraber, H. G. Griffiths-McCoy singularity on the diluted Chimera graph: Monte Carlo simulations and experiments on quantum hardware. Phys. Rev. A 102, 042403 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  9. Gardas, B., Dziarmaga, J., Zurek, W. H. & Zwolak, M. Defects in quantum computers. Sci. Rep. 8, 4539 (2018).

    Article  ADS  Google Scholar 

  10. Bando, Y. et al. Probing the universality of topological defect formation in a quantum annealer: Kibble-Zurek mechanism and beyond. Phys. Rev. Res. 2, 033369 (2020).

    Article  Google Scholar 

  11. Weinberg, P. et al. Scaling and diabatic effects in quantum annealing with a D-Wave device. Phys. Rev. Lett. 124, 090502 (2020).

    Article  ADS  Google Scholar 

  12. Dutta, A. et al. Quantum Phase Transitions in Transverse Field Spin Models: From Statistical Physics to Quantum Information (Cambridge Univ. Press, 2015).

  13. Albash, T. & Lidar, D. A. Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  14. Zurek, W. H. Cosmological experiments in superfluid helium? Nature 317, 505–508 (1985).

    Article  ADS  Google Scholar 

  15. Anquez, M. et al. Quantum Kibble-Zurek mechanism in a spin-1 Bose-Einstein condensate. Phys. Rev. Lett. 116, 155301 (2016).

  16. Clark, L. W., Feng, L. & Chin, C. Universal space-time scaling symmetry in the dynamics of bosons across a quantum phase transition. Science 354, 606–610 (2016).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Browaeys, A. & Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nat. Phys. 16, 132–142 (2020).

    Article  Google Scholar 

  18. Cui, J. M. et al. Experimentally testing quantum critical dynamics beyond the Kibble–Zurek mechanism. Commun. Phys. 3, 44 (2020).

    Article  Google Scholar 

  19. Zurek, W. H., Dorner, U. & Zoller, P. Dynamics of a quantum phase transition. Phys. Rev. Lett. 95, 105701 (2005).

    Article  ADS  Google Scholar 

  20. Dziarmaga, J. Dynamics of a quantum phase transition: exact solution of the quantum Ising model. Phys. Rev. Lett. 95, 245701 (2005).

    Article  ADS  Google Scholar 

  21. Polkovnikov, A. Universal adiabatic dynamics in the vicinity of a quantum critical point. Phys. Rev. B 72, 161201 (2005).

    Article  ADS  Google Scholar 

  22. Cincio, L., Dziarmaga, J., Meisner, J. & Rams, M. M. Dynamics of a quantum phase transition with decoherence: quantum Ising chain in a static spin environment. Phys. Rev. B 79, 094421 (2009).

    Article  ADS  Google Scholar 

  23. Arceci, L., Barbarino, S., Rossini, D. & Santoro, G. E. Optimal working point in dissipative quantum annealing. Phys. Rev. B 98, 064307 (2018).

    Article  ADS  Google Scholar 

  24. del Campo, A. Universal statistics of topological defects formed in a quantum phase transition. Phys. Rev. Lett. 121, 200601 (2018).

    Article  Google Scholar 

  25. Cherng, R. W. & Levitov, L. S. Entropy and correlation functions of a driven quantum spin chain. Phys. Rev. A 73, 043614 (2006).

    Article  ADS  Google Scholar 

  26. Mayo, J. J., Fan, Z., Chern, G.-W. & del Campo, A. Distribution of kinks in an Ising ferromagnet after annealing and the generalized Kibble-Zurek mechanism. Phys. Rev. Res. 3, 033150 (2021).

    Article  Google Scholar 

  27. Krebs, K., Pfannmüller, M. P., Wehefritz, B. & Hinrichsen, H. Finite-size scaling studies of one-dimensional reaction-diffusion systems. Part I. Analytical results. J. Stat. Phys. 78, 1429–1470 (1995).

    Article  ADS  MATH  Google Scholar 

  28. Roychowdhury, K., Moessner, R. & Das, A. Dynamics and correlations at a quantum phase transition beyond Kibble-Zurek. Phys. Rev. B 104, 014406 (2021).

    Article  ADS  Google Scholar 

  29. Nowak, R. J. & Dziarmaga, J. Quantum Kibble-Zurek mechanism: kink correlations after a quench in the quantum Ising chain. Phys. Rev. B 104, 075448 (2021).

    Article  ADS  Google Scholar 

  30. Oshiyama, H., Shibata, N. & Suzuki, S. Kibble–Zurek mechanism in a dissipative transverse Ising chain. J. Phys. Soc. Jpn 89, 104002 (2020).

    Article  ADS  Google Scholar 

  31. Schuch, N., Wolf, M. M., Verstraete, F. & Cirac, J. I. Entropy scaling and simulability by matrix product states. Phys. Rev. Lett. 100, 030504 (2008).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Dziarmaga, J. Dynamics of a quantum phase transition and relaxation to a steady state. Adv. Phys. 59, 1063–1189 (2010).

    Article  ADS  Google Scholar 

  33. Zanca, T. & Santoro, G. E. Quantum annealing speedup over simulated annealing on random Ising chains. Phys. Rev. B 93, 224431 (2016).

    Article  ADS  Google Scholar 

  34. Zener, C. Non-adiabatic crossing of energy levels. Proc. R. Soc. Lond. A 137, 696–702 (1932).

    Article  ADS  MATH  Google Scholar 

  35. Kato, T. On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Jpn 5, 435–439 (1950).

    Article  Google Scholar 

  36. Isakov, S. V. et al. Understanding quantum tunneling through quantum Monte Carlo simulations. Phys. Rev. Lett. 117, 180402 (2016).

    Article  ADS  Google Scholar 

  37. Mazzola, G., Smelyanskiy, V. N. & Troyer, M. Quantum Monte Carlo tunneling from quantum chemistry to quantum annealing. Phys. Rev. B 96, 134305 (2017).

    Article  ADS  Google Scholar 

  38. Suzuki, M. Relationship between d-dimensional quantal spin systems and (d+1)-dimensional Ising systems: equivalence, critical exponents and systematic approximants of the partition function and spin correlations. Prog. Theor. Phys. 56, 1454–1469 (1976).

    Article  ADS  MATH  Google Scholar 

  39. Liu, C. W., Polkovnikov, A. & Sandvik, A. W. Quantum versus classical annealing: insights from scaling theory and results for spin glasses on 3-regular graphs. Phys. Rev. Lett. 114, 147203 (2015).

    Article  ADS  MathSciNet  Google Scholar 

  40. King, A. D. et al. Scaling advantage over path-integral Monte Carlo in quantum simulation of geometrically frustrated magnets. Nat. Commun. 12, 1113 (2021).

    Article  ADS  Google Scholar 

  41. Bando, Y. & Nishimori, H. Simulated quantum annealing as a simulator of nonequilibrium quantum dynamics. Phys. Rev. A 104, 022607 (2021).

    Article  ADS  Google Scholar 

  42. Yip, K. W., Albash, T. & Lidar, D. A. Quantum trajectories for time-dependent adiabatic master equations. Phys. Rev. A 97, 022116 (2018).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We thank H. Oshiyama, N. Shibata, A. del Campo, L. Addario-Berry, T. Albash and A. W. Sandvik for fruitful discussions, and acknowledge the contributions of both technical and non-technical staff at D-Wave. We acknowledge the Center for Advanced Research Computing (CARC) at the University of Southern California for providing computing resources that have contributed to the research results reported within this publication. URL: https://carc.usc.edu. DAL acknowledges support from the National Science Foundation ‘the Quantum Leap Big Idea’ under Grant No. OMA-1936388, and by DARPA under the RQMLS program, Agreement No. HR00112190071.

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Authors and Affiliations

  1. D-Wave Systems, Burnaby, British Columbia, Canada

    Andrew D. King, Jack Raymond, Alex Zucca, Trevor Lanting, Fabio Altomare, Andrew J. Berkley, Sara Ejtemaee, Emile Hoskinson, Shuiyuan Huang, Eric Ladizinsky, Allison J. R. MacDonald, Gaelen Marsden, Travis Oh, Gabriel Poulin-Lamarre, Mauricio Reis, Chris Rich, Yuki Sato, Jed D. Whittaker, Jason Yao, Richard Harris & Mohammad H. Amin

  2. Department of Liberal Arts, Saitama Medical University, Moroyama, Japan

    Sei Suzuki

  3. Departments of Electrical and Computer Engineering, Chemistry, Physics & Astronomy, University of Southern California, Los Angeles, CA, USA

    Daniel A. Lidar

  4. Center for Quantum Information Science & Technology (CQIST), University of Southern California, Los Angeles, CA, USA

    Daniel A. Lidar

  5. International Research Frontiers Initiative, Tokyo Institute of Technology, Tokyo, Japan

    Hidetoshi Nishimori

  6. Graduate School of Information Sciences, Tohoku University, Sendai, Japan

    Hidetoshi Nishimori

  7. Interdisciplinary Theoretical and Mathematical Sciences, RIKEN, Wako, Japan

    Hidetoshi Nishimori

  8. Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada

    Mohammad H. Amin

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Contributions

A.D.K., S.S., J.R., A.Z., T.L. F.A., A.J.B., S.E., R.H., D.A.L., H.N. and M.H.A. conceived and designed the experiments and analysed the data. A.D.K., S.S., J.R., A.Z., T.L. and R.H. performed the experiments. T.L., F.A., A.J.B., S.E., E.H., S.H., E.L., A.J.R.M., G.M., T.O., G.P.-L., M.R., C.R., Y.S., J.D.W., J.Y., R.H. and M.H.A. contributed to the design, fabrication, deployment and calibration of the quantum annealing system. A.D.K., S.S., J.R., D.A.L., H.N. and M.H.A. wrote the manuscript.

Corresponding author

Correspondence to Andrew D. King.

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Competing interests

S.S., D.A.L. and H.N. declare no competing interests. A.D.K., J.R., A.Z., T.L., F.A., A.J.B., S.E., E.H., S.H., E.L., A.J.R.M., G.M., T.O., G.P.-L., M.R., C.R., Y.S., J.D.W., J.Y., R.H. and M.H.A. affiliated with D-Wave hold stock options in D-Wave and declare a competing financial interest on that basis.

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Nature Physics thanks Guglielmo Mazzola, David Bernal Neira and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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King, A.D., Suzuki, S., Raymond, J. et al. Coherent quantum annealing in a programmable 2,000 qubit Ising chain. Nat. Phys. 18, 1324–1328 (2022). https://doi.org/10.1038/s41567-022-01741-6

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