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Cost function dependent barren plateaus in shallow parametrized quantum circuits

Author

Listed:
  • M. Cerezo

    (Los Alamos National Laboratory
    Los Alamos National Laboratory)

  • Akira Sone

    (Los Alamos National Laboratory
    Los Alamos National Laboratory)

  • Tyler Volkoff

    (Los Alamos National Laboratory)

  • Lukasz Cincio

    (Los Alamos National Laboratory)

  • Patrick J. Coles

    (Los Alamos National Laboratory)

Abstract

Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(θ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ) is $${\mathcal{O}}(\mathrm{log}\,n)$$ O ( log n ) . Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.

Suggested Citation

  • 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.
  • Handle: RePEc:nat:natcom:v:12:y:2021:i:1:d:10.1038_s41467-021-21728-w
    DOI: 10.1038/s41467-021-21728-w
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    Cited by:

    1. Manuel S. Rudolph & Jacob Miller & Danial Motlagh & Jing Chen & Atithi Acharya & Alejandro Perdomo-Ortiz, 2023. "Synergistic pretraining of parametrized quantum circuits via tensor networks," Nature Communications, Nature, vol. 14(1), pages 1-10, December.
    2. 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.
    3. Eric R. Anschuetz & Bobak T. Kiani, 2022. "Quantum variational algorithms are swamped with traps," Nature Communications, Nature, vol. 13(1), pages 1-10, December.
    4. Enrico Fontana & Dylan Herman & Shouvanik Chakrabarti & Niraj Kumar & Romina Yalovetzky & Jamie Heredge & Shree Hari Sureshbabu & Marco Pistoia, 2024. "Characterizing barren plateaus in quantum ansätze with the adjoint representation," Nature Communications, Nature, vol. 15(1), pages 1-12, December.
    5. Matthias C. Caro & Hsin-Yuan Huang & M. Cerezo & Kunal Sharma & Andrew Sornborger & Lukasz Cincio & Patrick J. Coles, 2022. "Generalization in quantum machine learning from few training data," Nature Communications, Nature, vol. 13(1), pages 1-11, December.
    6. Matthias C. Caro & Hsin-Yuan Huang & Nicholas Ezzell & Joe Gibbs & Andrew T. Sornborger & Lukasz Cincio & Patrick J. Coles & Zoë Holmes, 2023. "Out-of-distribution generalization for learning quantum dynamics," Nature Communications, Nature, vol. 14(1), pages 1-9, December.
    7. He, Zhimin & Deng, Maijie & Zheng, Shenggen & Li, Lvzhou & Situ, Haozhen, 2023. "GSQAS: Graph Self-supervised Quantum Architecture Search," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 630(C).
    8. 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.
    9. Elies Gil-Fuster & Jens Eisert & Carlos Bravo-Prieto, 2024. "Understanding quantum machine learning also requires rethinking generalization," Nature Communications, Nature, vol. 15(1), pages 1-12, December.
    10. Sofiene Jerbi & Lukas J. Fiderer & Hendrik Poulsen Nautrup & Jonas M. Kübler & Hans J. Briegel & Vedran Dunjko, 2023. "Quantum machine learning beyond kernel methods," Nature Communications, Nature, vol. 14(1), pages 1-8, December.

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