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Local minima in quantum systems

Nature Physics volume 21, pages 654–660 (2025)Cite this article

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Abstract

Finding ground states of quantum many-body systems is known to be hard for both classical and quantum computers. Consequently, when a quantum system is cooled in a low-temperature thermal bath, the ground state cannot always be found efficiently. Instead, the system may become trapped in a local minimum of the energy. In this work, we study the problem of finding local minima in quantum systems under thermal perturbations. Although local minima are much easier to find than ground states, we show that finding a local minimum is hard on classical computers, even when the task is merely to output a single-qubit observable at any local minimum. By contrast, we prove that a quantum computer can always find a local minimum efficiently using a thermal gradient descent algorithm that mimics natural cooling processes. To establish the classical hardness of finding local minima, we construct a family of two-dimensional Hamiltonians such that any problem solvable by polynomial-time quantum algorithms can be reduced to finding local minima of these Hamiltonians. Therefore, cooling systems to local minima is universal for quantum computation and, assuming that quantum computation is more powerful than classical computation, finding local minima is classically hard but quantumly easy.

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Fig. 1: Illustration of energy landscapes.

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We did not analyse or generate any datasets because our work proceeds within a theoretical and mathematical approach.

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We do not have any computer code because our work proceeds within a theoretical and mathematical approach. All algorithms are analysed mathematically in the Supplementary Information.

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Acknowledgements

We thank A. Anshu, R. Babbush, S. Bravyi, F. Brandao, G. Chan, S. Chen, S. Choi, J. Cotler, D. Gosset, J. R. McClean and M. Soleimanifar for their valuable input. H.H. thanks P. Coles, G. Crooks and F. Sbahi for the inspiring discussions and for sharing their recent works on classical thermodynamics for AI applications88,89. Part of the manuscript was previously published as a STOC abstract90. C.C. is supported by the AWS Center for Quantum Computing internship. H.H. is supported by a Google PhD fellowship and a MediaTek Research Young Scholarship. H.H. acknowledges the visiting associate position at the Massachusetts Institute of Technology. L.Z. acknowledges funding from the Walter Burke Institute for Theoretical Physics at Caltech. J.P. acknowledges support from the United States Department of Energy Office of Science, Office of Advanced Scientific Computing Research (Grant Nos. DE-NA0003525 and DE-SC0020290), the United States Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator, and the National Science Foundation (Grant No. PHY-1733907). The Institute for Quantum Information and Matter is an NSF Physics Frontiers Center.

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Authors and Affiliations

  1. California Institute of Technology, Pasadena, CA, USA

    Chi-Fang Chen, Hsin-Yuan Huang, John Preskill & Leo Zhou

  2. AWS Center for Quantum Computing, Pasadena, CA, USA

    Chi-Fang Chen & John Preskill

  3. Google Quantum AI, Venice, CA, USA

    Hsin-Yuan Huang

  4. Massachusetts Institute of Technology, Cambridge, MA, USA

    Hsin-Yuan Huang

  5. University of California Los Angeles, Los Angeles, CA, USA

    Leo Zhou

Authors
  1. Chi-Fang Chen
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  2. Hsin-Yuan Huang
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Contributions

H.H., J.P. and L.Z. conceived the study of local minima of quantum systems. H.H. and L.Z. developed the computational complexity framework and the analysis of local minima under local unitary perturbations. C.C. introduced the quantum thermodynamics aspect. C.C. and H.H. established the theoretical framework for quantum thermal gradient descent. C.C. and L.Z. proved the BQP-hardness of finding local minima under thermal perturbation. All authors contributed to the writing of the manuscript.

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Correspondence to Chi-Fang Chen, Hsin-Yuan Huang, John Preskill or Leo Zhou.

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Chen, CF., Huang, HY., Preskill, J. et al. Local minima in quantum systems. Nat. Phys. 21, 654–660 (2025). https://doi.org/10.1038/s41567-025-02781-4

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