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A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits

Author

Listed:
  • Michael Ragone

    (University of California Davis)

  • Bojko N. Bakalov

    (North Carolina State University)

  • Frédéric Sauvage

    (Los Alamos National Laboratory)

  • Alexander F. Kemper

    (North Carolina State University)

  • Carlos Ortiz Marrero

    (Pacific Northwest National Laboratory
    North Carolina State University)

  • Martín Larocca

    (Los Alamos National Laboratory
    Los Alamos National Laboratory)

  • M. Cerezo

    (Los Alamos National Laboratory)

Abstract

Variational quantum computing schemes train a loss function by sending an initial state through a parametrized quantum circuit, and measuring the expectation value of some operator. Despite their promise, the trainability of these algorithms is hindered by barren plateaus (BPs) induced by the expressiveness of the circuit, the entanglement of the input data, the locality of the observable, or the presence of noise. Up to this point, these sources of BPs have been regarded as independent. In this work, we present a general Lie algebraic theory that provides an exact expression for the variance of the loss function of sufficiently deep parametrized quantum circuits, even in the presence of certain noise models. Our results allow us to understand under one framework all aforementioned sources of BPs. This theoretical leap resolves a standing conjecture about a connection between loss concentration and the dimension of the Lie algebra of the circuit’s generators.

Suggested Citation

  • Michael Ragone & Bojko N. Bakalov & Frédéric Sauvage & Alexander F. Kemper & Carlos Ortiz Marrero & Martín Larocca & M. Cerezo, 2024. "A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits," Nature Communications, Nature, vol. 15(1), pages 1-10, December.
    • Handle: RePEc:nat:natcom:v:15:y:2024:i:1:d:10.1038_s41467-024-49909-3
    DOI: 10.1038/s41467-024-49909-3
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    References listed on IDEAS

    as
    1. Jacob Biamonte & Peter Wittek & Nicola Pancotti & Patrick Rebentrost & Nathan Wiebe & Seth Lloyd, 2017. "Quantum machine learning," Nature, Nature, vol. 549(7671), pages 195-202, September.
    2. M. Cerezo & Akira Sone & Tyler Volkoff & Lukasz Cincio & Patrick J. Coles, 2021. "Cost function dependent barren plateaus in shallow parametrized quantum circuits," Nature Communications, Nature, vol. 12(1), pages 1-12, December.
    3. Alberto Peruzzo & Jarrod McClean & Peter Shadbolt & Man-Hong Yung & Xiao-Qi Zhou & Peter J. Love & Alán Aspuru-Guzik & Jeremy L. O’Brien, 2014. "A variational eigenvalue solver on a photonic quantum processor," Nature Communications, Nature, vol. 5(1), pages 1-7, September.
    4. Samson Wang & Enrico Fontana & M. Cerezo & Kunal Sharma & Akira Sone & Lukasz Cincio & Patrick J. Coles, 2021. "Noise-induced barren plateaus in variational quantum algorithms," Nature Communications, Nature, vol. 12(1), pages 1-11, December.
    5. Jarrod R. McClean & Sergio Boixo & Vadim N. Smelyanskiy & Ryan Babbush & Hartmut Neven, 2018. "Barren plateaus in quantum neural network training landscapes," Nature Communications, Nature, vol. 9(1), pages 1-6, December.
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