Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Quantum Metropolis sampling

Abstract

The original motivation to build a quantum computer came from Feynman1, who imagined a machine capable of simulating generic quantum mechanical systems—a task that is believed to be intractable for classical computers. Such a machine could have far-reaching applications in the simulation of many-body quantum physics in condensed-matter, chemical and high-energy systems. Part of Feynman’s challenge was met by Lloyd2, who showed how to approximately decompose the time evolution operator of interacting quantum particles into a short sequence of elementary gates, suitable for operation on a quantum computer. However, this left open the problem of how to simulate the equilibrium and static properties of quantum systems. This requires the preparation of ground and Gibbs states on a quantum computer. For classical systems, this problem is solved by the ubiquitous Metropolis algorithm3, a method that has basically acquired a monopoly on the simulation of interacting particles. Here we demonstrate how to implement a quantum version of the Metropolis algorithm. This algorithm permits sampling directly from the eigenstates of the Hamiltonian, and thus evades the sign problem present in classical simulations. A small-scale implementation of this algorithm should be achievable with today’s technology.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

Figure 1: Building blocks for the quantum algorithm.
Figure 2: Quantum Metropolis stochastic map.
Figure 3: Decision tree for unwinding the measurement.
Figure 4: Inverse spectral gap of the completely positive map for the quantum Ising model.

References

  1. Feynman, R. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982)

    MathSciNet  Article  Google Scholar 

  2. Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996)

    ADS  MathSciNet  CAS  Article  Google Scholar 

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

    ADS  CAS  Article  Google Scholar 

  4. Durr, S. et al. Ab initio determination of light hadron masses. Science 322, 1224–1227 (2008)

    ADS  CAS  Article  Google Scholar 

  5. Geman, S. & Geman, D. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984)

    CAS  Article  Google Scholar 

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

    ADS  MathSciNet  CAS  Article  Google Scholar 

  7. Suzuki, M. (ed.) Quantum Monte Carlo Methods in Equilibrium and Nonequilibrium Systems (Springer Ser. Solid-State Sci. 74, Springer, 1987)

  8. Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Phys. Rev. Lett. 83, 5162–5165 (1999)

    ADS  CAS  Article  Google Scholar 

  9. Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005)

    ADS  CAS  Article  Google Scholar 

  10. Verstraete, F., Wolf, M. M. & Cirac, J. I. Quantum computation and quantum-state engineering driven by dissipation. Nature Phys. 5, 633–636 (2009)

    ADS  CAS  Article  Google Scholar 

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

    ADS  MathSciNet  CAS  Article  Google Scholar 

  12. Terhal, B. M. & DiVincenzo, D. P. Problem of equilibration and the computation of correlation functions on a quantum computer. Phys. Rev. A 61, 022301 (2000)

    ADS  Article  Google Scholar 

  13. Binder, K. Monte Carlo and Molecular Dynamics Simulations in Polymer Science (Oxford Univ. Press, 1995)

    Google Scholar 

  14. Liu, J. & Luijten, E. Rejection-free geometric cluster algorithm for complex fluids. Phys. Rev. Lett. 92, 035504 (2004)

    ADS  Article  Google Scholar 

  15. Swendsen, R. H. & Wang, J.-S. Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett. 58, 86–88 (1987)

    ADS  CAS  Article  Google Scholar 

  16. Evertz, H. G. The loop algorithm. Adv. Phys. 52, 1–66 (2003)

    ADS  Article  Google Scholar 

  17. Kitaev, A. Y., Shen, A. H. & Vyalyi, M. N. Classical and Quantum Computation (American Mathematical Society, 2002)

    Book  Google Scholar 

  18. Aharonov, D. & Naveh, T. Quantum NP - a survey. Preprint at 〈http://arxiv.org/abs/quant-ph/0210077〉 (2002)

  19. Kitaev, A. Y. Quantum computations: algorithms and error correction. Russ. Math. Surv. 52, 1191–1249 (1997)

    MathSciNet  Article  Google Scholar 

  20. Cleve, R., Ekert, A., Macchiavello, C. & Mosca, M. Quantum algorithms revisited. Proc. R. Soc. Lond. A 454, 339–354 (1998)

    ADS  MathSciNet  Article  Google Scholar 

  21. Wootters, W. K. & Zurek, W. H. A single quantum cannot be cloned. Nature 299, 802–803 (1982)

    ADS  CAS  Article  Google Scholar 

  22. Marriott, C. & Watrous, J. Quantum Arthur-Merlin games. Comput. Complex. 14, 122–152 (2005)

    MathSciNet  Article  Google Scholar 

  23. Oliveira, R. & Terhal, B. M. The complexity of quantum spin systems on a two-dimensional square lattice. Quant. Inf. Comp. 8, 900–924 (2008)

    MathSciNet  MATH  Google Scholar 

  24. Aharonov, D., Gottesman, D., Irani, D. & Kempe, J. The power of quantum systems on a line. Commun. Math. Phys. 287, 41–65 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  25. Schuch, N. & Verstraete, F. Computational complexity of interacting electrons and fundamental limitations of density functional theory. Nature Phys. 5, 732–735 (2009)

    ADS  CAS  Article  Google Scholar 

  26. Jordan, P. & Wigner, E. Über das Paulische Äquivalenzverbot. Zeit. Phys. A 47, 631–651 (1928)

    ADS  CAS  Article  Google Scholar 

  27. Abrams, D. S. & Lloyd, S. Simulation of many-body Fermi systems on a universal quantum computer. Phys. Rev. Lett. 79, 2586–2589 (1997)

    ADS  CAS  Article  Google Scholar 

  28. Szegedy, M. in Proc. Annu. IEEE Symp. Found. Comput. Sci. 32–41 (IEEE, 2004)

    Google Scholar 

  29. Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008)

    ADS  CAS  Article  Google Scholar 

  30. Poulin, D. & Wocjan, P. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Phys. Rev. Lett. 103, 220502 (2009)

    ADS  MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

We would like to thank S. Bravyi, C. Dellago, J. Kempe and H. Verschelde for discussions. Part of this work was done during a workshop at the Erwin Schrödinger Institute for Mathematical Physics. K.T. was supported by the FWF programme CoQuS. T.J.O. was supported, in part, by EPSRC. K.G.V. is supported by DFG FG 635. D.P. is partly funded by NSERC, MITACS and FQRNT. F.V. is supported by the FWF grants FoQuS and ViCoM, by the European grant QUEVADIS and by the ERC grant QUERG.

Author information

Authors and Affiliations

Authors

Contributions

This project was started by T.J.O. and F.V. five years ago and rejuvenated by an inspiring visit of K.G.V. with K.T. and F.V. in 2009. The connection to the QMA amplification scheme of Marriott and Watrous was made by D.P. at the Erwin Schrödinger Institute, and the project was finalized by K.T. and F.V. by proving quantum detailed balance.

Corresponding author

Correspondence to F. Verstraete.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

The file contains Supplementary Text, additional references and Supplementary Figures 1-5 with legends. (PDF 2051 kb)

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Temme, K., Osborne, T., Vollbrecht, K. et al. Quantum Metropolis sampling. Nature 471, 87–90 (2011). https://doi.org/10.1038/nature09770

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature09770

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing