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  • Technical Review
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Tensor networks for quantum computing

Nature Reviews Physics volume 7, pages 581–593 (2025)Cite this article

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Abstract

Tensor networks have become a useful tool in many areas of physics, especially in quantum information science and quantum computing, where they are used to represent and manipulate quantum states and processes. The original use of tensor networks is the simulation of quantum systems, where tensor networks provide compressed representations of the structured systems. As research into quantum computing and tensor networks progresses, a plethora of new applications are becoming increasingly relevant. This Technical Review discusses the diverse applications of tensor networks to demonstrate that they are an important instrument for quantum computing. Specifically, we summarize the application of tensor networks in various domains of quantum computing, including simulation of quantum computation, quantum circuit synthesis, quantum error correction and mitigation, and quantum machine learning. Finally, we provide an outlook on the opportunities that tensor-network techniques provide and the challenges they may face in the future.

Key points

  • Tensor networks often provide efficient representations of mathematical objects encountered in quantum physics and quantum computing.

  • Tensor networks are powerful tools for classical simulations of quantum computing, and they are crucial to the understanding of the potential of quantum computational advantage.

  • Tensor networks are useful for synthesizing quantum circuits that prepare potentially interesting quantum states.

  • Tensor networks are a useful mathematical tool for understanding quantum error correction and provide a powerful computational tool to implement quantum error correction and error mitigation.

  • It is an open question whether tensor networks can provide advantage for machine learning using quantum computing.

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Fig. 1: Graphical representations of different mathematical objects and operations.
Fig. 2: Different types of tensor networks.
Fig. 3: Graphical representations of the mapping from tensors and tensor networks to quantum circuits.

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Acknowledgements

A.B., R.E., V.K., C.M., Al.M., Ar.M., V.M., D.M., F.N., M.Pe., M.Pf. and V.V. were supported by Terra Quantum AG. A.N. was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Alliance Quantum Program (Grant ALLRP-578555), and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program. S.M. is partially supported by the Prime Contract no. 80ARC020D0010 with the NASA Ames Research Center and acknowledges funding from DARPA under IAA 8839. A.B. thanks I. Luchnikov, M. Litvinov and S. Kourtis for discussions. A.A., Z.H., A.K., M.L., M.A.P., M.Pi., M.S. and R.S. thank their colleagues at the Global Technology Applied Research center of JPMorganChase for support and discussions.

Author information

Author notes

  1. These authors contributed equally: Aleksandr Berezutskii, Minzhao Liu.

Authors and Affiliations

  1. Terra Quantum AG, St Gallen, Switzerland

    Aleksandr Berezutskii, Roman Ellerbrock, Viacheslav Kuzmin, Christopher Mansell, Alexey Melnikov, Artem Melnikov, Vladimir Mironov, Dmitry Morozov, Florian Neukart, Michael Perelshtein, Markus Pflitsch & Valerii Vinokur

  2. Institut Quantique, Université de Sherbrooke, Sherbrooke, Quebec, Canada

    Aleksandr Berezutskii

  3. Département de Physique, Université de Sherbrooke, Sherbrooke, Quebec, Canada

    Aleksandr Berezutskii

  4. Global Technology Applied Research, JPMorganChase, New York, NY, USA

    Minzhao Liu, Atithi Acharya, Zichang He, Abid Khan, Michael A. Perlin, Matthew Steinberg, Ruslan Shaydulin & Marco Pistoia

  5. Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA, USA

    Johnnie Gray

  6. Quantinuum, Broomfield, CO, USA

    Reza Haghshenas

  7. NVIDIA Corporation, Santa Clara, CA, USA

    Dmitry Lyakh, Danylo Lykov & Yuri Alexeev

  8. Google Quantum AI, Venice, CA, USA

    Salvatore Mandrà & Benjamin Villalonga

  9. Quantum Artificial Intelligence Laboratory, NASA Ames Research Center, Moffett Field, CA, USA

    Salvatore Mandrà

  10. KBR Inc., Houston, TX, USA

    Salvatore Mandrà

  11. Nanoscience Center, University of Jyväskylä, Jyväskylä, Finland

    Dmitry Morozov

  12. Department of Chemistry, University of Jyväskylä, Jyväskylä, Finland

    Dmitry Morozov

  13. Department of Physics and Astronomy, The University of British Columbia, Vancouver, British Columbia, Canada

    Alberto Nocera

  14. Quantum Matter Institute, The University of British Columbia, Vancouver, British Columbia, Canada

    Alberto Nocera

Authors
  1. Aleksandr Berezutskii
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  2. Minzhao Liu
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Contributions

Yu.A., A.B., M.L. and V.V. conceptualized the work. A.B., R.E., J.G., R.H., A.K., M.L., A.N., F.N. and V.V. contributed to the ‘Tensor-network methods’ section. M.L., A.N. and F.N. contributed to the ‘Simulation of quantum computation’ section. A.A., R.H., Z.H., A.K., M.L., M.Pe. and V.V. contributed to the ‘Quantum circuit synthesis’ section. A.B., R.H., M.A.P., M.S. and B.V. contributed to the ‘Quantum error correction and mitigation’ section. A.A. and V.K. contributed to the ‘Tensor networks for quantum machine-learning’ section. All authors contributed to the ‘Introduction’ and the ‘Discussion and outlook’ sections. All authors were involved in shaping the direction of the manuscript, as well as in its discussion, review and editing.

Corresponding author

Correspondence to Yuri Alexeev.

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Nature Reviews Physics thanks Joonas Malmi, Matteo Rossi and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Berezutskii, A., Liu, M., Acharya, A. et al. Tensor networks for quantum computing. Nat Rev Phys 7, 581–593 (2025). https://doi.org/10.1038/s42254-025-00853-1

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