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.

Neural-network quantum state tomography

Nature Physics volume 14, pages 447–450 (2018)Cite this article

Subjects

Abstract

The experimental realization of increasingly complex synthetic quantum systems calls for the development of general theoretical methods to validate and fully exploit quantum resources. Quantum state tomography (QST) aims to reconstruct the full quantum state from simple measurements, and therefore provides a key tool to obtain reliable analytics1,2,3. However, exact brute-force approaches to QST place a high demand on computational resources, making them unfeasible for anything except small systems4,5. Here we show how machine learning techniques can be used to perform QST of highly entangled states with more than a hundred qubits, to a high degree of accuracy. We demonstrate that machine learning allows one to reconstruct traditionally challenging many-body quantities—such as the entanglement entropy—from simple, experimentally accessible measurements. This approach can benefit existing and future generations of devices ranging from quantum computers to ultracold-atom quantum simulators6,7,8.

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

Get just this article for as long as you need it

$39.95

Learn more

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

Fig. 1: Benchmarking the neural-network tomography of the W state.
Fig. 2: Tomography of ground and dynamically evolved states of many-body Hamiltonians.
Fig. 3: Reconstruction of the entanglement entropy for 1D lattice spin models.

References

  1. Vogel, K. & Risken, H. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A. 40, 2847 (1989).

    Article  ADS  Google Scholar 

  2. Leonhardt, U. Quantum-state tomography and discrete Wigner function. Phys. Rev. Lett. 74, 4101–4105 (1995).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. White, A. G., James, D. F. V., Eberhard, P. H. & Kwiat, P. G. Nonmaximally entangled states: production, characterization, and utilization. Phys. Rev. Lett. 83, 3103–3107 (1999).

    Article  ADS  Google Scholar 

  4. Häffner, H. et al. Scalable multiparticle entanglement of trapped ions. Nature 438, 643–646 (2005).

    Article  ADS  Google Scholar 

  5. Lu, C.-Y. et al. Experimental entanglement of six photons in graph states. Nat. Phys. 3, 91–95 (2007).

    Article  Google Scholar 

  6. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).

    Article  ADS  Google Scholar 

  7. Blatt, R. & Roos, C. F. Quantum simulations with trapped ions. Nat. Phys. 8, 277–284 (2012).

    Article  Google Scholar 

  8. Shulman, M. D. et al. Demonstration of entanglement of electrostatically coupled singlet-triplet qubits. Science 336, 202–205 (2012).

    Article  ADS  Google Scholar 

  9. Hinton, G. E. & Salakhutdinov, R. R. Reducing the dimensionality of data with neural networks. Science 313, 504–507 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. LeCun, Y., Bengio, Y. & Hinton, G. Deep learning. Nature 521, 436–444 (2015).

    Article  ADS  Google Scholar 

  11. Wang, L. Discovering phase transitions with unsupervised learning. Phys. Rev. B 94, 195105 (2016).

    Article  ADS  Google Scholar 

  12. Carrasquilla, J. & Melko, R. G. Machine learning phases of matter. Nat. Phys. 13, 431–434 (2017).

    Article  Google Scholar 

  13. van Nieuwenburg, E. P. L., Liu, Y.-H. & Huber, S. D. Learning phase transitions by confusion. Nat. Phys. 13, 435–439 (2017).

    Article  Google Scholar 

  14. Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science 355, 602–606 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  15. Gross, D., Liu, Y.-K., Flammia, S. T., Becker, S. & Eisert, J. Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105, 150401 (2010).

    Article  ADS  Google Scholar 

  16. Tóth, G. et al. Permutationally invariant quantum tomography. Phys. Rev. Lett. 105, 250403 (2010).

    Article  ADS  Google Scholar 

  17. Cramer, M. et al. Efficient quantum state tomography. Nat. Commun. 1, 149 (2009).

    Article  Google Scholar 

  18. Lanyon, B. P. et al. Efficient tomography of a quantum many-body system. Nat. Phys. 13, 1158–1162 (2017).

    Article  Google Scholar 

  19. Deng, D.-L., Li, X. & Sarma, S. D. Machine learning topological states. Phys. Rev. B 96, 195145 (2017).

    Article  ADS  Google Scholar 

  20. Torlai, G. & Melko, R. G. Neural decoder for topological codes. Phys. Rev. Lett. 119, 030501 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  21. Deng, D.-L., Li, X. & Das Sarma, S. Quantum entanglement in neural network states. Phys. Rev. X 7, 021021 (2017).

    Google Scholar 

  22. Gao, X. & Duan, L.-M. Efficient representation of quantum many-body states with deep neural networks. Nat. Commun. 8, 662 (2017).

    Article  ADS  Google Scholar 

  23. Chen, J., Cheng, S., Xie, H., Wang, L. & Xiang, T. On the equivalence of restricted Boltzmann machines and tensor network states. Preprint at http://arxiv.org/abs/1701.04831 (2017).

  24. Huang, Y. & Moore, J. E. Neural network representation of tensor network and chiral states. Preprint at http://arxiv.org/abs/1701.06246 (2017)

  25. Torlai, G. & Melko, R. G. Learning thermodynamics with Boltzmann machines. Phys. Rev. B 94, 165134 (2016).

    Article  ADS  Google Scholar 

  26. Wang, X.-L. et al. Experimental ten-photon entanglement. Phys. Rev. Lett. 117, 210502 (2016).

    Article  ADS  Google Scholar 

  27. Richerme, P. et al. Non-local propagation of correlations in quantum systems with long-range interactions. Nature 511, 198–201 (2014).

    Article  ADS  Google Scholar 

  28. Islam, R. et al. Measuring entanglement entropy in a quantum many-body system. Nature 528, 77–83 (2015).

    Article  ADS  Google Scholar 

  29. Hastings, M. B., González, I., Kallin, A. B. & Melko, R. G. Measuring Renyi entanglement entropy in quantum Monte Carlo simulations. Phys. Rev. Lett. 104, 157201 (2010).

    Article  ADS  Google Scholar 

  30. Bakr, W. S. et al. Probing the superfluid-to-Mott insulator transition at the single-atom level. Science 329, 547–550 (2010).

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  32. Hinton, G. E. Training products of experts by minimizing contrastive divergence. Neural Comput. 14, 1771–1800 (2002).

    Article  MATH  Google Scholar 

  33. Goodfellow, I., Bengio, Y. & Courville, A. Deep Learning (MIT Press, Cambridge, MA, 2016).

    MATH  Google Scholar 

  34. Amari, S.-i Natural gradient works efficiently in learning. Neural Comput. 10, 251–276 (1998).

    Article  Google Scholar 

  35. Sorella, S. Green function Monte Carlo with stochastic reconfiguration. Phys. Rev. Lett. 80, 4558 (1998).

    Article  ADS  Google Scholar 

  36. Becca, F. & Sorella, S. Quantum Monte Carlo Approaches for Correlated Systems (Cambridge Univ. Press, Cambridge, 2017).

    Book  MATH  Google Scholar 

Download references

Acknowledgements

We thank L. Aolita, H. Carteret, G. Tóth and B. Kulchytskyy for useful discussions. G.T. thanks the Institute for Theoretical Physics, ETH Zurich, for hospitality during various stages of this work. G.T. and R.M. acknowledge support from NSERC, the Canada Research Chair programme, the Ontario Trillium Foundation and the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. G.C., G.M. and M.T. acknowledge support from the European Research Council through ERC Advanced Grant SIMCOFE, and the Swiss National Science Foundation through NCCR QSIT and MARVEL. Simulations were performed on resources provided by SHARCNET, and by the Swiss National Supercomputing Centre CSCS.

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada

    Giacomo Torlai & Roger Melko

  2. Perimeter Institute of Theoretical Physics, Waterloo, Ontario, Canada

    Giacomo Torlai & Roger Melko

  3. Theoretische Physik, ETH Zurich, Zurich, Switzerland

    Guglielmo Mazzola, Matthias Troyer & Giuseppe Carleo

  4. Vector Institute, Toronto, Ontario, Canada

    Juan Carrasquilla

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

    Juan Carrasquilla

  6. Quantum Architectures and Computation Group, Station Q, Microsoft Research, Redmond, WA, USA

    Matthias Troyer

  7. Center for Computational Quantum Physics, Flatiron Institute, New York, NY, USA

    Giuseppe Carleo

Authors
  1. Giacomo Torlai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guglielmo Mazzola
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Juan Carrasquilla
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Matthias Troyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Roger Melko
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Giuseppe Carleo
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

G.C. designed the research. G.T. devised the machine learning methods. G.T., G.M. and J.C. performed the machine learning numerical experiments. G.M. performed QMC simulations. All authors contributed to the analysis of the results and writing of the manuscript.

Corresponding author

Correspondence to Giuseppe Carleo.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

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

Supplementary information

Supplementary Information

Supplementary Figures 1–5, Supplementary Notes, Supplementary References.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Torlai, G., Mazzola, G., Carrasquilla, J. et al. Neural-network quantum state tomography. Nature Phys 14, 447–450 (2018). https://doi.org/10.1038/s41567-018-0048-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-018-0048-5

This article is cited by

Search

Advanced 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