Skip to main content

Thank you for visiting nature.com. 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.

  • Article
  • Published:

Theory of overparametrization in quantum neural networks

Nature Computational Science volume 3, pages 542–551 (2023)Cite this article

Subjects

A preprint version of the article is available at arXiv.

Abstract

The prospect of achieving quantum advantage with quantum neural networks (QNNs) is exciting. Understanding how QNN properties (for example, the number of parameters M) affect the loss landscape is crucial to designing scalable QNN architectures. Here we rigorously analyze the overparametrization phenomenon in QNNs, defining overparametrization as the regime where the QNN has more than a critical number of parameters Mc allowing it to explore all relevant directions in state space. Our main results show that the dimension of the Lie algebra obtained from the generators of the QNN is an upper bound for Mc, and for the maximal rank that the quantum Fisher information and Hessian matrices can reach. Underparametrized QNNs have spurious local minima in the loss landscape that start disappearing when M ≥ Mc. Thus, the overparametrization onset corresponds to a computational phase transition where the QNN trainability is greatly improved. We then connect the notion of overparametrization to the QNN capacity, so that when a QNN is overparametrized, its capacity achieves its maximum possible value.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Overparametrization in QNNs.
Fig. 2: The loss function as a composition of maps.
Fig. 3: Training curves for VQE implementation L.
Fig. 4: Overparametrization threshold for VQE implementation.
Fig. 5: QNN ansatz for the numerical simulations.

Similar content being viewed by others

Data availability

All data were generated by simulating quantum circuits with the open-source library Qibo35. The dataset utilized in this study was taken from the NTangled dataset48 and can be downloaded from GitHub at https://github.com/LSchatzki/NTangled_Datasets. Source data are available with this paper and in ref. 49.

Code availability

All code to generate the figures and analyses in this study is publicly available from ref. 49.

References

  1. Mohri, M., Rostamizadeh, A. & Talwalkar, A. Foundations of Machine Learning (MIT Press, 2018).

  2. Vamathevan, J. et al. Applications of machine learning in drug discovery and development. Nat. Rev. Drug Discov. 18, 463–477 (2019).

    Article  Google Scholar 

  3. Schmidt, J., Marques, M. R., Botti, S. & Marques, M. A. Recent advances and applications of machine learning in solid-state materials science. npj Comput. Mater. 5, 83 (2019).

    Article  Google Scholar 

  4. Blum, A. L. & Rivest, R. L. Training a 3-node neural network is NP-complete. Neural Netw. 5, 117–127 (1992).

    Article  Google Scholar 

  5. Neyshabur, B., Li, Z., Bhojanapalli, S., LeCun, Y., & Srebro, N. Towards understanding the role of over-parametrization in generalization of neural networks. Preprint at arXiv https://doi.org/10.48550/arXiv.1805.12076 (2018).

  6. Zhang, C., Bengio, S., Hardt, M., Recht, B. & Vinyals, O. Understanding deep learning (still) requires rethinking generalization. Commun. ACM 64, 107–115 (2021).

    Article  Google Scholar 

  7. Allen-Zhu, Z., Li, Y. & Song, Z. A convergence theory for deep learning via over-parameterization. In Proceedings of the 36th International Conference on Machine Learning 242–252 (PMLR, 2019).

  8. Du, S. S., Zhai, X., Poczos, B., & Singh, A. Gradient descent provably optimizes over-parameterized neural networks. Preprint at arXiv https://doi.org/10.48550/arXiv.1810.02054 (2018).

  9. Preskill, J. Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018).

    Article  Google Scholar 

  10. Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625–644 (2021).

    Article  Google Scholar 

  11. Bharti, K. et al. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys. 94, 015004 (2022).

    Article  MathSciNet  Google Scholar 

  12. Huang, H.-Y. et al. Power of data in quantum machine learning. Nat. Commun. 12, 2631 (2021).

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. Bittel, L. & Kliesch, M. Training variational quantum algorithms is NP-hard. Phys. Rev. Lett. 127, 120502 (2021).

    Article  MathSciNet  Google Scholar 

  15. Wierichs, D., Gogolin, C. & Kastoryano, M. Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer. Phys. Rev. Res. 2, 043246 (2020).

    Article  Google Scholar 

  16. Anschuetz, E. R. Critical points in quantum generative models. Preprint at arXiv https://doi.org/10.48550/arXiv.2109.06957 (2021).

  17. McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, 4812 (2018).

    Article  Google Scholar 

  18. Cerezo, M., Sone, A., Volkoff, T., Cincio, L. & Coles, P. J. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nat. Commun. 12, 1791 (2021).

    Article  Google Scholar 

  19. Wang, S. et al. Noise-induced barren plateaus in variational quantum algorithms. Nat. Commun. 12, 6961 (2021).

    Article  Google Scholar 

  20. Stilck França, D. & Garcia-Patron, R. Limitations of optimization algorithms on noisy quantum devices. Nat. Phys. 17, 1221–1227 (2021).

    Article  Google Scholar 

  21. Wiersema, R. et al. Exploring entanglement and optimization within the Hamiltonian variational ansatz. PRX Quantum 1, 020319 (2020).

    Article  Google Scholar 

  22. Kiani, B. T., Lloyd, S. & Maity, R. Learning unitaries by gradient descent. Preprint at arXiv https://doi.org/10.48550/arXiv.2001.11897 (2020).

  23. Funcke, L. et al. Best-approximation error for parametric quantum circuits. In 2021 IEEE International Conference on Web Services (ICWS) 693–702 (IEEE, 2021).

  24. Lee, J., Magann, A. B., Rabitz, H. A. & Arenz, C. Progress toward favorable landscapes in quantum combinatorial optimization. Phys. Rev. A 104, 032401 (2021).

    Article  MathSciNet  Google Scholar 

  25. Zeier, R. & Schulte-Herbrüggen, T. Symmetry principles in quantum systems theory. J. Math. Phys. 52, 113510 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  26. Meyer, J. J. Fisher information in noisy intermediate-scale quantum applications. Quantum 5, 539 (2021).

    Article  Google Scholar 

  27. Moore, K. W. & Rabitz, H. Exploring constrained quantum control landscapes. J. Chem. Phys. 137, 134113 (2012).

    Article  Google Scholar 

  28. Larocca, M., Calzetta, E. & Wisniacki, D. A. Exploiting landscape geometry to enhance quantum optimal control. Phys. Rev. A 101, 023410 (2020).

    Article  MathSciNet  Google Scholar 

  29. Wu, R.-B., Long, R., Dominy, J., Ho, T.-S. & Rabitz, H. Singularities of quantum control landscapes. Phys. Rev. A 86, 013405 (2012).

    Article  Google Scholar 

  30. Rach, N., Müller, M. M., Calarco, T. & Montangero, S. Dressing the chopped-random-basis optimization: a bandwidth-limited access to the trap-free landscape. Phys. Rev. A 92, 062343 (2015).

    Article  Google Scholar 

  31. Larocca, M., Poggi, P. M. & Wisniacki, D. A. Quantum control landscape for a two-level system near the quantum speed limit. J. Phys. A Math. Theor. 51, 385305 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  32. Larocca, M. et al. Diagnosing barren plateaus with tools from quantum optimal control. Quantum 6, 824 (2022).

    Article  Google Scholar 

  33. Haug, T., Bharti, K. & Kim, M. S. Capacity and quantum geometry of parametrized quantum circuits. PRX Quantum 2, 040309 (2021).

    Article  Google Scholar 

  34. Larocca, M., Calzetta, E. & Wisniacki, D. Fourier compression: a customization method for quantum control protocols. Phys. Rev. A 102, 033108 (2020).

    Article  Google Scholar 

  35. Efthymiou, S. et al. Qibo: a framework for quantum simulation with hardware acceleration. Quantum Sci. Technol. 7, 015018 (2021).

    Article  Google Scholar 

  36. Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).

    Article  Google Scholar 

  37. García-Martín, D., Larocca, M. & Cerezo, M. Effects of noise on the overparametrization of quantum neural networks. Preprint at arXiv https://doi.org/10.48550/arXiv.2302.05059 (2023).

  38. Fukumizu, K. A regularity condition of the information matrix of a multilayer perceptron network. Neural Netw. 9, 871–879 (1996).

    Article  Google Scholar 

  39. Chan, N. & Kwong, M. K. Hermitian matrix inequalities and a conjecture. Am. Math. Monthly 92, 533–541 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  40. Glaser, S. J. et al. Training Schrödinger’s cat: quantum optimal control. Eur. Phys. J. D 69, 279 (2015).

    Article  Google Scholar 

  41. Rembold, P. et al. Introduction to quantum optimal control for quantum sensing with nitrogen-vacancy centers in diamond. AVS Quantum Sci. 2, 024701 (2020).

    Article  Google Scholar 

  42. Ebadi, S. et al. Quantum phases of matter on a 256-atom programmable quantum simulator. Nature 595, 227–232 (2021).

    Article  Google Scholar 

  43. Magann, A. B., Rudinger, K. M., Grace, M. D. & Sarovar, M. Feedback-based quantum optimization. Phys. Rev. Lett. 129, 250502 (2022).

    Article  Google Scholar 

  44. Magann, A. B. et al. From pulses to circuits and back again: a quantum optimal control perspective on variational quantum algorithms. PRX Quantum 2, 010101 (2021).

    Article  Google Scholar 

  45. Hsieh, M., Wu, R. & Rabitz, H. Topology of the quantum control landscape for observables. J. Chem. Phys. 130, 104109 (2009).

    Article  Google Scholar 

  46. Ho, T.-S., Dominy, J. & Rabitz, H. Landscape of unitary transformations in controlled quantum dynamics. Phys. Rev. A 79, 013422 (2009).

    Article  MathSciNet  Google Scholar 

  47. Riviello, G. et al. Searching for quantum optimal control fields in the presence of singular critical points. Phys. Rev. A 90, 013404 (2014).

    Article  Google Scholar 

  48. Schatzki, L., Arrasmith, A., Coles, P. J. & Cerezo, M. Entangled datasets for quantum machine learning. Preprint at arXiv https://doi.org/10.48550/arXiv.2109.03400 (2021).

  49. García-Martín, D. DiegoGM91/theory-of-overparametrization: v0.0.1. Zenodo https://doi.org/10.5281/zenodo.7916659 (2023).

Download references

Acknowledgements

We thank P. de Niverville, J. Nakhleh, S. Efthymiou, L. Schatzki, M. Farinati and Z. Szabo for useful conversations. N.J. and D.G.-M. were supported by the US DOE through a quantum computing program sponsored by the Los Alamos National Laboratory (LANL) Information Science & Technology Institute. D.G.-M. acknowledges partial financial support from project QuantumCAT (ref. 001-P-001644), co-funded by the Generalitat de Catalunya and the European Union Regional Development Fund within the ERDF Operational Program of Catalunya, and from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 951911 (AI4Media). P.J.C. and M.C. were initially supported by the Laboratory Directed Research and Development (LDRD) program of LANL under project no. 20190065DR. P.J.C. also acknowledges support from the LANL ASC Beyond Moore’s Law project. M.C. also acknowledges support from the Center for Nonlinear Studies at LANL. This work was supported by the US DOE, Office of Science, Office of Advanced Scientific Computing Research, under the Accelerated Research in Quantum Computing (ARQC) program.

Author information

Author notes

  1. These authors contributed equally: Martín Larocca, Diego García-Martín.

Authors and Affiliations

  1. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    Martín Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles & Marco Cerezo

  2. Departamento de Física ‘J. J. Giambiagi’ and IFIBA, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina

    Martín Larocca

  3. Barcelona Supercomputing Center, Barcelona, Spain

    Diego García-Martín

  4. Instituto de Física Teórica, UAM-CSIC, Madrid, Spain

    Diego García-Martín

  5. Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA

    Marco Cerezo

  6. Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA

    Marco Cerezo

Authors
  1. Martín Larocca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nathan Ju
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Diego García-Martín
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Patrick J. Coles
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Marco Cerezo
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The project was conceived by M.L., P.J.C. and M.C. The manuscript was written by N.J., M.L., D.G.-M., P.J.C. and M.C. Theoretical results were proved by N.J., M.L., P.J.C. and M.C. Numerical implementations were performed by D.G.-M.

Corresponding author

Correspondence to Martín Larocca.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Computational Science thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available. Primary Handling Editor: Jie Pan, in collaboration with the Nature Computational Science team.

Additional information

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

Extended data

Extended Data Fig. 1 Noise and overparametrization.

Eigenvalues of the QFIM for the same ansatz used in the VQE implementation of Fig. 3, but where local depolarizing noise acts before and after each gate.

Source data

Supplementary information

Supplementary Information

Supplementary Sections 1–8 and Figs. 1–5.

Peer Review File

Supplementary Data 1

Supplementary Section 7 and Fig. 1. This folder contains data from the VQE simulations. Each file contains an array with the eigenvalues of the quantum Fisher information or Hessian matrices.

Supplementary Data 2

Supplementary Section 7 and Fig. 2. This folder contains data from the VQE simulations. Each file contains an array with the eigenvalues of the quantum Fisher information matrix.

Supplementary Data 3

Supplementary Section 7 and Fig. 4. This folder contains data from the unitary compilation simulations. Each file contains either an array of cost function values for every optimization step, or an array with the eigenvalues of the quantum Fisher information or Hessian matrices.

Supplementary Data 4

Supplementary Section 7 and Fig. 5. This folder contains data from the autoencoder simulations. Each file contains either an array of cost function values for every optimization step, or an array with the eigenvalues of the quantum Fisher information matrix.

Source data

Source Data Fig. 3

This folder contains data from the VQE simulations. Each file contains an array of cost function values for every optimization step.

Source Data Fig. 4

This folder contains data from the VQE simulations. Each file contains an array with the eigenvalues of the quantum Fisher information or Hessian matrices.

Source Data Extended Data Fig. 1

This folder contains data from the VQE simulations in the presence of noise. Each file contains an array with the eigenvalues of the quantum Fisher information matrix or with the matrix itself.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Larocca, M., Ju, N., García-Martín, D. et al. Theory of overparametrization in quantum neural networks. Nat Comput Sci 3, 542–551 (2023). https://doi.org/10.1038/s43588-023-00467-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s43588-023-00467-6

This article is cited by

Search

Advanced search

Quick links

Nature Briefing AI and Robotics

Sign up for the Nature Briefing: AI and Robotics newsletter — what matters in AI and robotics research, free to your inbox weekly.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing: AI and Robotics