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.

Supervised learning with quantum-enhanced feature spaces


Machine learning and quantum computing are two technologies that each have the potential to alter how computation is performed to address previously untenable problems. Kernel methods for machine learning are ubiquitous in pattern recognition, with support vector machines (SVMs) being the best known method for classification problems. However, there are limitations to the successful solution to such classification problems when the feature space becomes large, and the kernel functions become computationally expensive to estimate. A core element in the computational speed-ups enabled by quantum algorithms is the exploitation of an exponentially large quantum state space through controllable entanglement and interference. Here we propose and experimentally implement two quantum algorithms on a superconducting processor. A key component in both methods is the use of the quantum state space as feature space. The use of a quantum-enhanced feature space that is only efficiently accessible on a quantum computer provides a possible path to quantum advantage. The algorithms solve a problem of supervised learning: the construction of a classifier. One method, the quantum variational classifier, uses a variational quantum circuit1,2 to classify the data in a way similar to the method of conventional SVMs. The other method, a quantum kernel estimator, estimates the kernel function on the quantum computer and optimizes a classical SVM. The two methods provide tools for exploring the applications of noisy intermediate-scale quantum computers3 to machine 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: Quantum kernel functions.
Fig. 2: Experimental implementations.
Fig. 3: Convergence of the method and classification results.
Fig. 4: Kernels for Set III.

Data availability

All data generated or analysed during this study are included in this Letter (and its Supplementary Information).


  1. Mitarai, K., Negoro, M., Kitagawa, M. & Fujii, K. Quantum circuit learning. Preprint at (2018).

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

  3. Preskill, J. Quantum computing in the NISQ era and beyond. Preprint at (2018).

  4. Arunachalam, S. & de Wolf, R. Guest column: a survey of quantum learning theory. SIGACT News 48, 41–67 (2017).

    Article  MathSciNet  Google Scholar 

  5. Ciliberto, C. et al. Quantum machine learning: a classical perspective. Proc. R. Soc. Lond. A 474, 20170551 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  6. Dunjko, V. & Briegel, H. J. Machine learning & artificial intelligence in the quantum domain: a review of recent progress. Rep. Prog. Phys. 81, 074001 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  7. Biamonte, J. et al. Quantum machine learning. Nature 549, 195–202 (2017).

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

  9. Wan, K. H., Dahlsten, O., Kristjánsson, H., Gardner, R. & Kim, M. Quantum generalisation of feedforward neural networks. Preprint at (2016).

  10. Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  11. Li, Y. & Benjamin, S. C. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X 7, 021050 (2017).

    Google Scholar 

  12. Terhal, B. M. & DiVincenzo, D. P. Adaptive quantum computation, constant depth quantum circuits and Arthur-Merlin games. Quantum Inf. Comput. 4, 134–145 (2004).

    MathSciNet  MATH  Google Scholar 

  13. Bremner, M. J., Montanaro, A. & Shepherd, D. J. Achieving quantum supremacy with sparse and noisy commuting quantum computations. Quantum 1, 8 (2017).

    Article  Google Scholar 

  14. Vapnik, V. The Nature of Statistical Learning Theory (Springer Science & Business Media, 2013).

  15. Rebentrost, P., Mohseni, M. & Lloyd, S. Quantum support vector machine for big data classification. Phys. Rev. Lett. 113, 130503 (2014).

    Article  ADS  Google Scholar 

  16. Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

    Article  ADS  CAS  Google Scholar 

  17. Farhi, E., Goldstone, J., Gutmann, S. & Neven, H. Quantum algorithms for fixed qubit architectures. Preprint at (2017).

  18. Burges, C. J. A tutorial on support vector machines for pattern recognition. Data Min. Knowl. Discov. 2, 121–167 (1998).

    Article  Google Scholar 

  19. Boser, B. E., Guyon, I. M. & Vapnik, V. N. A training algorithm for optimal margin classifiers. In Proc. 5th Annual Workshop on Computational Learning Theory 144–152 (ACM, 1992).

  20. Goldberg, L. A. & Guo, H. The complexity of approximating complex-valued Ising and Tutte partition functions. Computat. Complex. 26, 765–833 (2017).

    Article  MathSciNet  Google Scholar 

  21. Demarie, T. F., Ouyang, Y. & Fitzsimons, J. F. Classical verification of quantum circuits containing few basis changes. Phys. Rev. A 97, 042319 (2018).

    Article  ADS  Google Scholar 

  22. Spall, J. C. A one-measurement form of simultaneous perturbation stochastic approximation. Automatica 33, 109–112 (1997).

    Article  MathSciNet  Google Scholar 

  23. Spall, J. C. Adaptive stochastic approximation by the simultaneous perturbation method. IEEE Trans. Automat. Contr. 45, 1839 (2000).

    Article  MathSciNet  Google Scholar 

  24. Kandala, A. et al. Extending the computational reach of a noisy superconducting quantum processor. Preprint at (2018).

  25. Buhrman, H., Cleve, R., Watrous, J. & De Wolf, R. Quantum fingerprinting. Phys. Rev. Lett. 87, 167902 (2001).

    Article  ADS  CAS  Google Scholar 

  26. Cincio, L., Subas, Y., Sornborger, A. T. & Coles, P. J. Learning the quantum algorithm for state overlap. Preprint at (2018).

  27. Smolin, J. A., Gambetta, J. M. & Smith, G. Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise. Phys. Rev. Lett. 108, 070502 (2012).

    Article  ADS  Google Scholar 

  28. Schuld, M. & Killoran, N. Quantum machine learning in feature Hilbert spaces. Preprint at (2018).

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

Download references


We thank S. Bravyi for discussions. A.W.H. acknowledges funding from the MIT-IBM Watson AI Lab under the project ‘Machine Learning in Hilbert Space’. The research was supported by the IBM Research Frontiers Institute. We acknowledge support from IARPA under contract W911NF-10-1-0324 for device fabrication.

Reviewer information

Nature thanks Christopher Eichler, Seth Lloyd, Maria Schuld and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Author information

Authors and Affiliations



The work on the classifier theory was led by V.H. and K.T. The experiment was designed by A.D.C., J.M.G. and K.T. and implemented by A.D.C. All authors contributed to the manuscript.

Corresponding authors

Correspondence to Antonio D. Córcoles or Kristan Temme.

Ethics declarations

Competing interests

The authors declare competing interests: Elements of this work are included in a patent filed by the International Business Machines Corporation with the US Patent and Trademark office.

Additional information

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

Supplementary information

Supplementary Information

This file contains supplementary text I–IX and references and includes Figs.1–9 and Tables I–II.

Supplementary Data

This file contains Supplementary Data tables.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Havlíček, V., Córcoles, A.D., Temme, K. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


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