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.

  • Article
  • Published:

Variational quantum unsampling on a quantum photonic processor

An Author Correction to this article was published on 04 February 2020

This article has been updated


A promising route towards the demonstration of near-term quantum advantage (or supremacy) over classical systems relies on running tailored quantum algorithms on noisy intermediate-scale quantum machines. These algorithms typically involve sampling from probability distributions that—under plausible complexity-theoretic conjectures—cannot be efficiently generated classically. Rather than determining the computational features of output states produced by a given physical system, we investigate what features of the generating system can be efficiently learnt given direct access to an output state. To tackle this question, here we introduce the variational quantum unsampling protocol, a nonlinear quantum neural network approach for verification and inference of near-term quantum circuit outputs. In our approach, one can variationally train a quantum operation to unravel the action of an unknown unitary on a known input state, essentially learning the inverse of the black-box quantum dynamics. While the principle of our approach is platform independent, its implementation will depend on the unique architecture of a specific quantum processor. We experimentally demonstrate the variational quantum unsampling protocol on a quantum photonic processor. Alongside quantum verification, our protocol has broad applications, including optimal quantum measurement and tomography, quantum sensing and imaging, and ansatz validation.

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

Access options

Buy this article

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

Fig. 1: VQU.
Fig. 2: Optical VQU in a quantum photonic processor.
Fig. 3: Experimental results.
Fig. 4: Monte Carlo numerics.

Similar content being viewed by others

Data availability

The datasets generated during and or analysed during the current study are available from the corresponding author on reasonable request.

Change history


  1. Nielsen, M. & Chuang, I. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2010).

  2. Montanaro, A. Quantum algorithms: an overview. npj Quantum Inf. 2, 15023 (2016).

    ADS  Google Scholar 

  3. Barends, R. et al. Superconducting quantum circuits at the surface code threshold for fault tolerance. Nature 508, 500–503 (2014).

    ADS  Google Scholar 

  4. Gaebler, J. P. et al. High-fidelity universal gate set for 9Be+ ion qubits. Phys. Rev. Lett. 117, 060505 (2016).

    ADS  Google Scholar 

  5. Fowler, A. G., Mariantoni, M., Martinis, J. M. & Cleland, A. N. Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012).

    ADS  Google Scholar 

  6. Mohseni, M. et al. Commercialize quantum technologies in five years. Nature 543, 171–174 (2017).

    ADS  Google Scholar 

  7. Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

    Google Scholar 

  8. Harrow, A. W. & Montanaro, A. Quantum computational supremacy. Nature 549, 203–209 (2017).

    ADS  Google Scholar 

  9. Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at (2014).

  10. McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016).

    ADS  MATH  Google Scholar 

  11. Romero, J., Olson, J. P. & Aspuru-Guzik, A. Quantum autoencoders for efficient compression of quantum data. Quantum Sci. Technol. 2, 045001 (2017).

    ADS  Google Scholar 

  12. Farhi, E. & Neven, H. Classification with quantum neural networks on near term processors. Preprint at (2018).

  13. Schuld, M., Bocharov, A., Svore, K. & Wiebe, N. Circuit-centric quantum classifiers. Preprint at (2018).

  14. Chen, H., Wossnig, L., Severini, S., Neven, H. & Mohseni, M. Universal discriminative quantum neural networks. Preprint at (2018).

  15. Steinbrecher, G. R., Olson, J. P., Englund, D. & Carolan, J. Quantum optical neural networks. npj Quantum Inf. 5, 60 (2019).

    ADS  Google Scholar 

  16. Aaronson, S. & Arkhipov, A. The computational complexity of linear optics. In Proc. 43rd Annual ACM Symposium on Theory of Computing 333–342 (ACM, 2011).

  17. Bouland, A., Fefferman, B., Nirkhe, C. & Vazirani, U. On the complexity and verification of quantum random circuit sampling. Nat. Phys. 15, 159–163 (2019).

    Google Scholar 

  18. Boixo, S. et al. Characterizing quantum supremacy in near-term devices. Nat. Phys. 14, 595–600 (2018).

    Google Scholar 

  19. Neville, A. et al. Classical boson sampling algorithms with superior performance to near-term experiments. Nat. Phys. 13, 1153–1157 (2017).

    Google Scholar 

  20. Zhang, J. et al. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator. Nature 551, 601–604 (2017).

    ADS  Google Scholar 

  21. Bernien, H. et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature 551, 579–584 (2017).

    ADS  Google Scholar 

  22. Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).

    ADS  Google Scholar 

  23. Shor, P. W. Algorithms for quantum computation: discrete logarithms and factoring. In Proc. 35th Annual Symposium on Foundations of Computer Science 124–134 (Society for Industrial and Applied Mathematics, 1994).

  24. Hangleiter, D., Kliesch, M., Eisert, J. & Gogolin, C. Sample complexity of device-independently certified ‘quantum supremacy’. Phys. Rev. Lett. 122, 210502 (2019).

    ADS  Google Scholar 

  25. Carolan, J. et al. On the experimental verification of quantum complexity in linear optics. Nat. Photon. 8, 621–626 (2014).

    ADS  Google Scholar 

  26. Mohseni, M., Rezakhani, A. T. & Lidar, D. A. Quantum-process tomography: resource analysis of different strategies. Phys. Rev. A 77, 032322 (2008).

    ADS  Google Scholar 

  27. O’Donnell, R. & Wright, J. Efficient quantum tomography. In Proc. 48th Annual ACM Symposium on Theory of Computing 899–912 (ACM, 2016).

  28. Cramer, M. et al. Efficient quantum state tomography. Nat. Commun. 1, 149 (2010).

    ADS  Google Scholar 

  29. Cong, I., Choi, S. & Lukin, M. D. Quantum convolutional neural networks. Nat. Phys. (2019).

  30. Grant, E. et al. Hierarchical quantum classifiers. npj Quantum Inf. 4, 65 (2018).

    ADS  Google Scholar 

  31. Kokail, C. et al. Self-verifying variational quantum simulation of lattice models. Nature 569, 355–660 (2019).

    ADS  Google Scholar 

  32. McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, 4812 (2018).

    ADS  Google Scholar 

  33. Grant, E., Wossnig, L., Ostaszewski, M. & Benedetti, M. An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Preprint at (2019).

  34. Eldan, R. & Shamir, O. The power of depth for feedforward neural networks. In Proc. Conference on Learning Theory (eds Feldman, V., Rakhlin, A. & Shamir, O.) 907–940 (JMLR, 2016).

  35. Arora, S., Cohen, N. & Hazan, E. E. On the optimization of deep networks: implicit acceleration by overparameterization. In Proc. 35th International Conference on Machine Learning, ICML 2018 (eds Dy., J & Krause, A.) 372–389 (International Machine Learning Society, 2018).

  36. Bengio, Y., Lamblin, P., Popovic, D. & Larochelle, H. Greedy layer-wise training of deep networks. In Advances in Neural Information Processing 19 (eds Schölkopf, B., Platt, J. C. & Hoffman, T.) 153–160 (MIT Press, 2007).

  37. Hettinger, C. et al. Forward thinking: building and training neural networks one layer at a time. Preprint at (2017).

  38. Barenco, A. et al. Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995).

    ADS  Google Scholar 

  39. Mitarai, K., Negoro, M., Kitagawa, M. & Fujii, K. Quantum circuit learning. Phys. Rev. A 98, 032309 (2018).

    ADS  Google Scholar 

  40. Khatri, S. et al. Quantum-assisted quantum compiling. Quantum 3, 140 (2019).

    Google Scholar 

  41. Romero, J. et al. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Quantum Sci. Technol. 4, 014008 (2018).

    ADS  Google Scholar 

  42. Grimsley, H. R., Economou, S. E., Barnes, E. & Mayhall, N. J. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nat. Commun. 10, 3007 (2019).

    ADS  Google Scholar 

  43. Scheel, S. Permanents in linear optical networks. Preprint at (2004).

  44. Valiant, L. G. The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979).

    MathSciNet  MATH  Google Scholar 

  45. Reck, M., Zeilinger, A., Bernstein, H. J. & Bertani, H. Experimental realization of any discrete unitary operator. Phys. Rev. Lett. 73, 58–61 (1994).

    ADS  Google Scholar 

  46. Rahimi-Keshari, S. et al. Direct characterization of linear-optical networks. Opt. Exp. 21, 13450–13459 (2013).

    ADS  Google Scholar 

  47. Laing, A. & O’Brien, J. L. Super-stable tomography of any linear optical device. Preprint at (2012).

  48. Rubin, N. C., Babbush, R. & McClean, J. Application of fermionic marginal constraints to hybrid quantum algorithms. New J. Phys. 20, 053020 (2018).

    ADS  MathSciNet  Google Scholar 

  49. Russell, N. J., Chakhmakhchyan, L., O’Brien, J. L. & Laing, A. Direct dialling of Haar random unitary matrices. New J. Phys. 19, 033007 (2017).

    ADS  MathSciNet  Google Scholar 

  50. Notaros, J. et al. Ultra-efficient CMOS fiber-to-chip grating couplers. In Proc. Optical Fiber Communication Conference M2I–M25 (2016).

  51. Silverstone, J. W. et al. On-chip quantum interference between silicon photon-pair sources. Nat. Photon. 8, 104–108 (2014).

    ADS  Google Scholar 

  52. Carolan, J. et al. Scalable feedback control of single photon sources for photonic quantum technologies. Optica 6, 335–340 (2019).

    ADS  Google Scholar 

  53. Najafi, F. et al. On-chip detection of non-classical light by scalable integration of single-photon detectors. Nat. Commun. 6, 5873 (2015).

    ADS  Google Scholar 

  54. Brinks, D. et al. Visualizing and controlling vibrational wave packets of single molecules. Nature 465, 905–908 (2010).

    ADS  Google Scholar 

  55. Guha, S. et al. Quantum enigma machines and the locking capacity of a quantum channel. Phys. Rev. X 4, 011016 (2014).

    Google Scholar 

  56. Guha, S. Structured optical receivers to attain superadditive capacity and the holevo limit. Phys. Rev. Lett. 106, 240502 (2011).

    ADS  Google Scholar 

  57. Niu, M. Y., Chuang, I. L. & Shapiro, J. H. Hardware-efficient bosonic quantum error-correcting codes based on symmetry operators. Phys. Rev. A 97, 032323 (2018).

    ADS  Google Scholar 

  58. Moroder, T. et al. Certifying systematic errors in quantum experiments. Phys. Rev. Lett. 110, 180401 (2013).

    ADS  Google Scholar 

  59. Powell, M. J. The bobyqa algorithm for bound constrained optimization without derivatives (2009).

  60. Johnson, S. G. The NLopt nonlinear-optimization package (2011);

  61. Chen, C. et al. Efficient generation and characterization of spectrally factorable biphotons. Opt. Exp. 25, 7300–7313 (2017).

    ADS  Google Scholar 

  62. Carolan, J. et al. Universal linear optics. Science 349, 711–716 (2015).

    MathSciNet  MATH  Google Scholar 

  63. Harris, N. C. et al. Quantum transport simulations in a programmable nanophotonic processor. Nat. Photon. 11, 447–452 (2017).

    ADS  Google Scholar 

  64. Harris, N. C. et al. Linear programmable nanophotonic processors. Optica 5, 1623–1631 (2018).

    ADS  Google Scholar 

Download references


We thank E. Farhi, E. Grant, D. Hangleiter, I. Marvian, J. McClean, M. Pant, M. Schuld, P. Shadbolt, S. Sim and G. Steinbrecher for insightful discussions. This work was supported by the AFOSR MURI for Optimal Measurements for Scalable Quantum Technologies (FA9550-14-1-0052), the MITRE Quantum Moonshot Program and by the AFOSR programme FA9550-16-1-0391, supervised by G. Pomrenke. J.C. is supported by EU H2020 Marie Sklodowska-Curie grant number 751016.

Author information

Authors and Affiliations



J.C., M.M., S.L. and D.E. conceived the project. J.C., M.M., J.P.O., M.Y.N, S.L. and D.E. developed the theory. J.C., M.P., C.C., D.B., N.C.H., F.N.C.W., M.H. and D.E. contributed to the experimental set-up. J.C., M.P., C.C. and D.B. performed the experiment and analysed data. J.C. performed numerical experiments. All authors contributed to the discussion of the results and the writing of the manuscript.

Corresponding author

Correspondence to Jacques Carolan.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Ashley Montanaro 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. 1–4 and Sections I–V.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carolan, J., Mohseni, M., Olson, J.P. et al. Variational quantum unsampling on a quantum photonic processor. Nat. Phys. 16, 322–327 (2020).

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