Skip to main content

Thank you for visiting 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.

Sample-efficient learning of interacting quantum systems


Learning the Hamiltonian that describes interactions in a quantum system is an important task in both condensed-matter physics and the verification of quantum technologies. Its classical analogue arises as a central problem in machine learning known as learning Boltzmann machines. Previously, the best known methods for quantum Hamiltonian learning with provable performance guarantees required a number of measurements that scaled exponentially with the number of particles. Here we prove that only a polynomial number of local measurements on the thermal state of a quantum system are necessary and sufficient for accurately learning its Hamiltonian. We achieve this by establishing that the absolute value of the finite-temperature free energy of quantum many-body systems is strongly convex with respect to the interaction coefficients. The framework introduced in our work provides a theoretical foundation for applying machine learning techniques to quantum Hamiltonian learning, achieving a long-sought goal in quantum statistical learning.

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

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Dependence of Hamiltonian learning on the inverse temperature β.
Fig. 2: Strong convexity of the log-partition function.
Fig. 3: Dependence of the strong convexity parameter on the inverse temperature β.

Data availability

The data presented in the figures are available at

Code availability

The codes used to generate the figures are available at


  1. Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (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. Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).

    Article  ADS  Google Scholar 

  4. Simon, J. et al. Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature 472, 307–312 (2011).

    Article  ADS  Google Scholar 

  5. Brandao, F. G. S. L. & Svore, K. M. Quantum speed-ups for solving semidefinite programs. In Proc. 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) 415–426 (IEEE, 2017).

  6. van Apeldoorn, J., Gilyén, A., Gribling, S. & de Wolf, R. Quantum SDP-solvers: better upper and lower bounds. Quantum 4, 230 (2020).

    Article  Google Scholar 

  7. Brandão, F. G. S. L., Kueng, R. & França, D. S. Faster quantum and classical SDP approximations for quadratic binary optimization. Preprint at (2019).

  8. Montanaro, A. Quantum speedup of Monte Carlo methods. Proc. R. Soc. A Math. Phys. Eng. Sci. 471, 20150301 (2015).

    ADS  MathSciNet  MATH  Google Scholar 

  9. Harrow, A. W. & Wei, A. Y. Adaptive quantum simulated annealing for Bayesian inference and estimating partition functions. In Proc. Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (ed. Chawla, S.) 193–212 (SIAM, 2020).

  10. Wiebe, N., Kapoor, A. & Svore, K. M. Quantum deep learning. Quantum Inf. Comput. 16, 541–587 (2016).

    MathSciNet  Google Scholar 

  11. Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nat. Phys. 16, 205–210 (2020).

    Article  Google Scholar 

  12. Wainwright, M. J. & Jordan, M. I. Graphical models, exponential families and variational inference. Found. Trends Mach. Learn. 1, 1–305 (2008).

    Article  Google Scholar 

  13. Chow, C. & Liu, C. Approximating discrete probability distributions with dependence trees. IEEE Trans. Inf. Theory 14, 462–467 (1968).

    Article  Google Scholar 

  14. Hinton, G. E. et al. Learning and relearning in Boltzmann machines. Parallel Distrib. Process. Explorations Microstruct. Cognition 1, 282–317 (1986).

    Google Scholar 

  15. Bresler, G. Efficiently learning Ising models on arbitrary graphs. In Proc. 2015 ACM Symposium on Theory of Computing 771–782 (ACM, 2015).

  16. Klivans, A. R. & Meka, R. Learning graphical models using multiplicative weights. In Proc. 58th Annual IEEE Symposium on Foundations of Computer Science—FOCS 343–354 (IEEE, 2017).

  17. Vuffray, M., Misra, S., Lokhov, A. & Chertkov, M. Interaction screening: efficient and sample-optimal learning of Ising models. In Advances in Neural Information Processing Systems (eds Lee, D. et al.) 2595–2603 (NIPS, 2016).

  18. Bairey, E., Arad, I. & Lindner, N. H. Learning a local Hamiltonian from local measurements. Phys. Rev. Lett. 122, 020504 (2019).

    Article  ADS  Google Scholar 

  19. Swingle, B. & Kim, I. H. Reconstructing quantum states from local data. Phys. Rev. Lett. 113, 260501 (2014).

    Article  ADS  Google Scholar 

  20. Qi, X.-L. & Ranard, D. Determining a local Hamiltonian from a single eigenstate. Quantum 3, 159 (2019).

    Article  Google Scholar 

  21. Wang, J. et al. Experimental quantum Hamiltonian learning. Nat. Phys. 13, 551–555 (2017).

    Article  Google Scholar 

  22. Senko, C. et al. Coherent imaging spectroscopy of a quantum many-body spin system. Science 345, 430–433 (2014).

    Article  ADS  MathSciNet  Google Scholar 

  23. Evans, T. J., Harper, R. & Flammia, S. T. Scalable bayesian Hamiltonian learning. Preprint at (2019).

  24. Bairey, E., Guo, C., Poletti, D., Lindner, N. H. & Arad, I. Learning the dynamics of open quantum systems from their steady states. New J. Phys. 22, 032001 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  25. Shabani, A., Mohseni, M., Lloyd, S., Kosut, R. L. & Rabitz, H. Estimation of many-body quantum Hamiltonians via compressive sensing. Phys. Rev. A 84, 012107 (2011).

    Article  ADS  Google Scholar 

  26. Wiebe, N., Granade, C., Ferrie, C. & Cory, D. G. Hamiltonian learning and certification using quantum resources. Phys. Rev. Lett. 112, 190501 (2014).

    Article  ADS  Google Scholar 

  27. Wiebe, N., Granade, C., Ferrie, C. & Cory, D. Quantum Hamiltonian learning using imperfect quantum resources. Phys. Rev. A 89, 042314 (2014).

    Article  ADS  Google Scholar 

  28. Leifer, M. S. & Poulin, D. Quantum graphical models and belief propagation. Ann. Phys. 323, 1899–1946 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  29. Jaynes, E. T. On the rationale of maximum-entropy methods. Proc. IEEE 70, 939–952 (1982).

    Article  ADS  Google Scholar 

  30. Jaynes, E. T. Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957).

    Article  ADS  MathSciNet  Google Scholar 

  31. Cotler, J. & Wilczek, F. Quantum overlapping tomography. Phys. Rev. Lett. 124, 100401 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  32. Bonet-Monroig, X., Babbush, R. & O’Brien, T. E. Nearly optimal measurement scheduling for partial tomography of quantum states. Phys. Rev. X 10, 031064 (2020).

    Google Scholar 

  33. Huang, H.-Y., Kueng, R. & Preskill, J. Predicting many properties of a quantum system from very few measurements. Nat. Phys. 16, 1050–1057 (2020).

    Article  Google Scholar 

  34. Santhanam, N. P. & Wainwright, M. J. Information-theoretic limits of selecting binary graphical models in high dimensions. IEEE Trans. Inf. Theory 58, 4117–4134 (2012).

    Article  MathSciNet  Google Scholar 

  35. Montanari, A. Computational implications of reducing data to sufficient statistics. Electron. J. Stat. 9, 2370–2390 (2015).

    Article  MathSciNet  Google Scholar 

  36. Kuwahara, T., Kato, K. & L. Brandão, F. G. S. Clustering of conditional mutual information for quantum Gibbs states above a threshold temperature. Phys. Rev. Lett. 124, 220601 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  37. Harrow, A. W., Mehraban, S. & Soleimanifar, M. Classical algorithms, correlation decay and complex zeros of partition functions of quantum many-body systems. In Proc. 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC) 378–386 (ACM, 2020).

  38. Poulin, D. & Hastings, M. B. Markov entropy decomposition: a variational dual for quantum belief propagation. Phys. Rev. Lett. 106, 080403 (2011).

    Article  ADS  Google Scholar 

  39. Ferris, A. J. & Poulin, D. Algorithms for the Markov entropy decomposition. Phys. Rev. B 87, 205126 (2013).

    Article  ADS  Google Scholar 

  40. Verdon, G., Marks, J., Nanda, S., Leichenauer, S. & Hidary, J. Quantum Hamiltonian-based models and the variational quantum thermalizer algorithm. Preprint at (2019).

  41. Wiebe, N. & Wossnig, L. Generative training of quantum Boltzmann machines with hidden units. Preprint at (2019).

  42. G.S.L. Brandão, F. & Kastoryano, M. J. Finite correlation length implies efficient preparation of quantum thermal states. Commun. Math. Phys. 365, 1–16 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  43. Brydges, T. Probing Rényi entanglement entropy via randomized measurements. Science 364, 260–263 (2019).

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

Download references


We thank A. Harrow, Y. Huang, R. La Placa, S. Subramanian, J. Wright and H. Yuen for helpful discussions. Part of this work was done when S.A. and T.K. were visiting Perimeter Institute. S.A. was supported in part by the Army Research Laboratory and the Army Research Office under grant no. W911NF-20-1-0014. The work was done when A.A. was affiliated to the Institute for Quantum Computing and Department of Combinatorics and Optimization, University of Waterloo and the Perimeter Institute for Theoretical Physics. A.A. was supported by the Canadian Institute for Advanced Research, through funding provided to the Institute for Quantum Computing by the Government of Canada and the Province of Ontario. Perimeter Institute is also supported in part by the Government of Canada and the Province of Ontario. T.K. was supported by the RIKEN Center for AIP and JSPS KAKENHI grant no. 18K13475. M.S. was supported by NSF grant no. CCF-1729369, a Samsung Advanced Institute of Technology Global Research Cluster and grant no. FXQi-RFP-1811A from the Foundational Questions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Community Foundation.

Author information

Authors and Affiliations



A.A., S.A., T.K. and M.S. contributed equally in developing the main ideas, technical aspects and the writing of this work. The authors are arranged alphabetically based on the last name.

Corresponding authors

Correspondence to Anurag Anshu, Srinivasan Arunachalam, Tomotaka Kuwahara or Mehdi Soleimanifar.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Vedran Dunjko and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–3 and Discussion.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Anshu, A., Arunachalam, S., Kuwahara, T. et al. Sample-efficient learning of interacting quantum systems. Nat. Phys. 17, 931–935 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


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