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Reconstructing quantum states with generative models


A major bottleneck in the development of scalable many-body quantum technologies is the difficulty in benchmarking state preparations, which suffer from an exponential ‘curse of dimensionality’ inherent to the classical description of quantum states. We present an experimentally friendly method for density matrix reconstruction based on neural network generative models. The learning procedure comes with a built-in approximate certificate of the reconstruction and makes no assumptions about the purity of the state under scrutiny. It can efficiently handle a broad class of complex systems including prototypical states in quantum information, as well as ground states of local spin models common to condensed matter physics. The key insight is to reduce state tomography to an unsupervised learning problem of the statistics of an informationally complete quantum measurement. This constitutes a modern machine learning approach to the validation of complex quantum devices, which may in addition prove relevant as a neural-network ansatz over mixed states suitable for variational optimization.

A preprint version of the article is available at ArXiv.

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The numerically generated measurements used to produce Fig. 5, the implementation of the generative models and the code to numerically generate the data sets used in this manuscript are available at

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The authors thank G. Vidal, L. Cincio and M. Stoudenmire for discussions and encouragement, and N. Berkovits, A. Reily Rocha and P. Vieira for organizing the ICTP-SAIFR/IFT-UNESP Minicourse on Machine Learning for Many-Body Physics, where this work was started. This research was supported by the Perimeter Institute for Theoretical Physics and the Shared Hierarchical Academic Research Computing Network (SHARCNET). Research at the Perimeter Institute is supported by the Government of Canada through Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade. R.G.M. acknowledges support from NSERC of Canada and a Canada Research Chair. J.C. acknowledges financial and computational support from the AI grant and Canada CIFAR AI (CCAI) Chairs Program. L.A. acknowledges financial support from the Brazilian agencies CNPq (PQ grant no. 311416/2015-2 and INCT-IQ), FAPERJ (JCN E-26/202.701/2018), CAPES (PROCAD2013), FAPESP and the Brazilian Serrapilheira Institute (grant no. Serra-1709-17173).

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All authors contributed significantly to this work.

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The authors declare no competing interests.

Correspondence to Juan Carrasquilla.

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Fig. 1: Tensor-network schematics of the formalism.
Fig. 2: Learning a Bell state under local depolarizing noise.
Fig. 3: Sample complexity of learning locally depolarized GHZ states with RNN models.
Fig. 4: Direct estimation of local observables from Ns model samples.
Fig. 5: Learning ground states of local Hamiltonians in one and two dimensions.