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A rigorous and robust quantum speed-up in supervised machine learning


Recently, several quantum machine learning algorithms have been proposed that may offer quantum speed-ups over their classical counterparts. Most of these algorithms are either heuristic or assume that data can be accessed quantum-mechanically, making it unclear whether a quantum advantage can be proven without resorting to strong assumptions. Here we construct a classification problem with which we can rigorously show that heuristic quantum kernel methods can provide an end-to-end quantum speed-up with only classical access to data. To prove the quantum speed-up, we construct a family of datasets and show that no classical learner can classify the data inverse-polynomially better than random guessing, assuming the widely believed hardness of the discrete logarithm problem. Furthermore, we construct a family of parameterized unitary circuits, which can be efficiently implemented on a fault-tolerant quantum computer, and use them to map the data samples to a quantum feature space and estimate the kernel entries. The resulting quantum classifier achieves high accuracy and is robust against additive errors in the kernel entries that arise from finite sampling statistics.

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Fig. 1: Quantum kernel estimation.
Fig. 2: Learning concept class \({\mathcal{C}}\) by a quantum feature map.

Data availability

Data sharing is not applicable to this paper as no datasets were generated or analysed during the current study.


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We thank S. Bravyi and R. Kothari for helpful comments and discussions. Y.L. was supported by DOE NQISRC QSA grant number FP00010905, Vannevar Bush faculty fellowship N00014-17-1-3025 and NSF award DMR-1747426. Part of this work was done when Y.L. was a research intern at IBM. S.A. and K.T. acknowledge support from the MIT-IBM Watson AI Lab under the project ‘Machine Learning in Hilbert Space’, the IBM Research Frontiers Institute and ARO grant W911NF-20-1-0014.

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All authors contributed important ideas during initial discussions and contributed equally to deriving the technical proofs and writing the manuscript.

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Correspondence to Kristan Temme.

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The authors declare no competing interests.

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Peer review informationNature Physics thanks Ewin Tang and Zhikuan Zhao for their contribution to the peer review of this work.

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Liu, Y., Arunachalam, S. & Temme, K. A rigorous and robust quantum speed-up in supervised machine learning. Nat. Phys. (2021).

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