Abstract
The use of kernel functions is a common technique to extract important features from datasets. A quantum computer can be used to estimate kernel entries as transition amplitudes of unitary circuits. Quantum kernels exist that, subject to computational hardness assumptions, cannot be computed classically. The learning problems for these cases are constructed artificially and it is an important challenge to find quantum kernels that have the potential to be relevant for real-world data. Here we identify a suitable class of learning problems on data that have a group structure, which are amenable to kernel methods. We introduce a family of quantum kernels that can be applied to such data, generalizing from a kernel that is known to have a quantum–classical separation when solving a particular set of problems. We use 27 qubits of a superconducting processor to demonstrate our method with a learning problem that embodies the structure of many essential learning problems on groups.
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Data availability
Experiment data are available in the Supplementary Information. Source data are provided with this paper.
Code availability
Code for training the model using kernel alignment is available through the Qiskit machine learning module: https://github.com/qiskit-community/qiskit-machine-learning.
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Acknowledgements
We thank S. Arunachalam and V. Havlicek for helpful comments and discussions and D. T. McClure and N. Sundaresan for technical assistance with the device. Funding: this work was supported by the IBM Research Frontiers Institute and K.T. acknowledges support from the ARO Grant W911NF-20-1-0014.
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K.T., J.M.G., J.R.G., T.P.G. and A.D.C. designed the algorithms and experiments. A.D.C., A.K. and Y.K. conducted the experiments and analysis. All authors contributed to the data analysis and writing of the manuscript.
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Supplementary Text and Figs. 1–5.
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Source Data Fig. 3
Experimental data in a 26-sheet Excel file displayed in the subplots of Fig. 3.
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Glick, J.R., Gujarati, T.P., Córcoles, A.D. et al. Covariant quantum kernels for data with group structure. Nat. Phys. 20, 479–483 (2024). https://doi.org/10.1038/s41567-023-02340-9
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DOI: https://doi.org/10.1038/s41567-023-02340-9
