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The power of quantum neural networks

A preprint version of the article is available at arXiv.


It is unknown whether near-term quantum computers are advantageous for machine learning tasks. In this work we address this question by trying to understand how powerful and trainable quantum machine learning models are in relation to popular classical neural networks. We propose the effective dimension—a measure that captures these qualities—and prove that it can be used to assess any statistical model’s ability to generalize on new data. Crucially, the effective dimension is a data-dependent measure that depends on the Fisher information, which allows us to gauge the ability of a model to train. We demonstrate numerically that a class of quantum neural networks is able to achieve a considerably better effective dimension than comparable feedforward networks and train faster, suggesting an advantage for quantum machine learning, which we verify on real quantum hardware.

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Fig. 1: Overview of the quantum neural network used in this study.
Fig. 2: Average Fisher information spectrum distribution.
Fig. 3: Normalized effective dimension and training loss.

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Data availability

The data for the graphs and analyses in this study was generated using Python. Source data are provided with this paper. All other data can be accessed via the following Zenodo repository: (ref. 56).

Code availability

All code to generate the data, figures and analyses in this study is publicly available with detailed information on the implementation via the following Zenodo repository: (ref. 56).


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We thank M. Schuld for insightful discussions on data embedding in quantum models. We also thank T. L. Scholten for constructive feedback on the manuscript. C.Z. acknowledges support from the National Centre of Competence in Research Quantum Science and Technology (QSIT).

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Authors and Affiliations



The main ideas were developed by all of the authors. A.A. provided numerical simulations. D.S. and A.F. proved the technical claims. All authors contributed to the write-up.

Corresponding author

Correspondence to Stefan Woerner.

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The authors declare no competing interests.

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Peer review informationNature Computational Science thanks Patrick Coles and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Handling editor: Jie Pan, in collaboration with the Nature Computational Science team.

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

Supplementary information

Supplementary Information

Supplementary Figs. 1–11, Sections 1–5 and Table 1.

Source data

Source Data Fig. 2

Unprocessed raw text data are stored in text files, containing the numerical values used to generate the eigenvalue distributions for each model in the subplots of Fig. 2.

Source Data Fig. 3

In the zip folder there are text data files containing the numerical values for the effective dimension for each model and labeled accordingly, there is also a file containing the values of the axis. Then, the the raw numerical data for the loss values, their averages and the standard deviation around the average loss values are stored in text files for each model and labeled transparently.

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Abbas, A., Sutter, D., Zoufal, C. et al. The power of quantum neural networks. Nat Comput Sci 1, 403–409 (2021).

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