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Neural-network quantum state tomography

Nature Physicsvolume 14pages447450 (2018) | Download Citation


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.

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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


  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


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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.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Giuseppe Carleo.

Supplementary information

  1. Supplementary Information

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

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