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Theory of overparametrization in quantum neural networks

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

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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.

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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.

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.

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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

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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.

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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.

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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.

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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

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