IDEAS home Printed from https://ideas.repec.org/a/nat/natcom/v15y2024i1d10.1038_s41467-024-49909-3.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.nature.com/articles/s41467-024-49909-3
    File Function: Abstract
    Download Restriction: no
    File URL: https://libkey.io/10.1038/s41467-024-49909-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    4. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. 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.
    4. Eric R. Anschuetz & Bobak T. Kiani, 2022. "Quantum variational algorithms are swamped with traps," Nature Communications, Nature, vol. 13(1), pages 1-10, December.
    5. 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.
    6. 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.
    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. 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.
    9. Junyu Liu & Minzhao Liu & Jin-Peng Liu & Ziyu Ye & Yunfei Wang & Yuri Alexeev & Jens Eisert & Liang Jiang, 2024. "Towards provably efficient quantum algorithms for large-scale machine-learning models," Nature Communications, Nature, vol. 15(1), pages 1-6, December.
    10. Alen Senanian & Sridhar Prabhu & Vladimir Kremenetski & Saswata Roy & Yingkang Cao & Jeremy Kline & Tatsuhiro Onodera & Logan G. Wright & Xiaodi Wu & Valla Fatemi & Peter L. McMahon, 2024. "Microwave signal processing using an analog quantum reservoir computer," Nature Communications, Nature, vol. 15(1), pages 1-9, December.
    11. Wang, Shaoxuan & Shen, Yingtong & Liu, Xinjian & Zhang, Haoying & Wang, Yukun, 2024. "Variational quantum entanglement classification discrimination," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 637(C).
    12. 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.
    13. Ajagekar, Akshay & You, Fengqi, 2022. "Quantum computing and quantum artificial intelligence for renewable and sustainable energy: A emerging prospect towards climate neutrality," Renewable and Sustainable Energy Reviews, Elsevier, vol. 165(C).
    14. Xinbiao Wang & Yuxuan Du & Zhuozhuo Tu & Yong Luo & Xiao Yuan & Dacheng Tao, 2024. "Transition role of entangled data in quantum machine learning," Nature Communications, Nature, vol. 15(1), pages 1-8, December.
    15. Fangjun Hu & Saeed A. Khan & Nicholas T. Bronn & Gerasimos Angelatos & Graham E. Rowlands & Guilhem J. Ribeill & Hakan E. Türeci, 2024. "Overcoming the coherence time barrier in quantum machine learning on temporal data," Nature Communications, Nature, vol. 15(1), pages 1-12, December.
    16. Wei-Ming Li & Shi-Ju Ran, 2022. "Non-Parametric Semi-Supervised Learning in Many-Body Hilbert Space with Rescaled Logarithmic Fidelity," Mathematics, MDPI, vol. 10(6), pages 1-15, March.
    17. Camille Grange & Michael Poss & Eric Bourreau, 2023. "An introduction to variational quantum algorithms for combinatorial optimization problems," 4OR, Springer, vol. 21(3), pages 363-403, September.
    18. Daniel J. Egger & Claudio Gambella & Jakub Marecek & Scott McFaddin & Martin Mevissen & Rudy Raymond & Andrea Simonetto & Stefan Woerner & Elena Yndurain, 2020. "Quantum Computing for Finance: State of the Art and Future Prospects," Papers 2006.14510, arXiv.org, revised Jan 2021.
    19. El Amine Cherrat & Snehal Raj & Iordanis Kerenidis & Abhishek Shekhar & Ben Wood & Jon Dee & Shouvanik Chakrabarti & Richard Chen & Dylan Herman & Shaohan Hu & Pierre Minssen & Ruslan Shaydulin & Yue , 2023. "Quantum Deep Hedging," Papers 2303.16585, arXiv.org, revised Nov 2023.
    20. Sofiene Jerbi & Casper Gyurik & Simon C. Marshall & Riccardo Molteni & Vedran Dunjko, 2024. "Shadows of quantum machine learning," Nature Communications, Nature, vol. 15(1), pages 1-7, December.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:nat:natcom:v:15:y:2024:i:1:d:10.1038_s41467-024-49909-3. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.nature.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.