Supervised learning with quantum-enhanced feature spaces

Abstract

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.

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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).

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Acknowledgements

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.

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Contributions

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.

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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.

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Supplementary Information

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

Supplementary Data

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Havlíček, V., Córcoles, A.D., Temme, K. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212 (2019). https://doi.org/10.1038/s41586-019-0980-2

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