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Barren plateaus in variational quantum computing

Nature Reviews Physics volume 7, pages 174–189 (2025)Cite this article

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Abstract

Variational quantum computing offers a flexible computational approach with a broad range of applications. However, a key obstacle to realizing their potential is the barren plateau (BP) phenomenon. When a model exhibits a BP, its parameter optimization landscape becomes exponentially flat and featureless as the problem size increases. Importantly, all the moving pieces of an algorithm — choices of ansatz, initial state, observable, loss function and hardware noise — can lead to BPs if they are ill-suited. As BPs strongly impact on trainability, researchers have dedicated considerable effort to develop theoretical and heuristic methods to understand and mitigate their effects. As a result, the study of BPs has become a thriving area of research, influencing and exchanging ideas with other fields such as quantum optimal control, tensor networks and learning theory. This article provides a review of the current understanding of the BP phenomenon.

Key points

  • Variational quantum algorithms (VQAs) — this hybrid computational approach aims at training a quantum learning model (usually a parametrized quantum circuit) to solve a given task. The parameters in the model are trained by minimizing a loss function that encodes the degree to which the problem has been solved.

  • Barren plateaus — a phenomenon in which the gradients of the loss landscape of VQAs get exponentially suppressed. Currently, this issue is understood as a form of curse of dimensionality arising from operating in an unstructured manner in an exponentially large Hilbert space.

  • Trainability — in the context of VQAs, trainability refers to the ability to optimize parameters of a model and minimize the loss function. Barren plateaus are one of the main barriers to the trainability of VQAs.

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Fig. 1: Variational quantum computing.

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Change history

  • 09 April 2025

    In the version of this article initially published, the second affiliation for Jacob Biamonte (now reading ETS Montreal, University of Quebec, Montreal, Quebec, Canada) was incomplete and is now amended in the HTML and PDF versions of the article.

References

  1. Cerezo, M. et al. Variational quantum algorithms. Nat. Rev. Phys. 3, 625–644 (2021). Review on variational quantum algorithms presents details for newcomers to the field.

    Article  Google Scholar 

  2. Bharti, K. et al. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys. 94, 015004 (2022). Review on variational quantum algorithms presents details for newcomers to the field.

    Article  ADS  MathSciNet  Google Scholar 

  3. Endo, S., Cai, Z., Benjamin, S. C. & Yuan, X. Hybrid quantum-classical algorithms and quantum error mitigation. J. Phys. Soc. Jpn. 90, 032001 (2021).

    Article  ADS  Google Scholar 

  4. Schuld, M., Sinayskiy, I. & Petruccione, F. An introduction to quantum machine learning. Contemp. Phys. 56, 172–185 (2015). Pedagogical introduction to quantum machine learning.

    Article  ADS  Google Scholar 

  5. Biamonte, J. et al. Quantum machine learning. Nature 549, 195–202 (2017).

    Article  ADS  Google Scholar 

  6. Cerezo, M., Verdon, G., Huang, H.-Y., Cincio, L. & Coles, P. J. Challenges and opportunities in quantum machine learning. Nat. Comput. Sci. https://www.nature.com/articles/s43588-022-00311-3 (2022).

  7. Di Meglio, A. et al. Quantum computing for high-energy physics: state of the art and challenges. PRX Quantum 5, 037001 (2024).

    Article  Google Scholar 

  8. McClean, J. R., Boixo, S., Smelyanskiy, V. N., Babbush, R. & Neven, H. Barren plateaus in quantum neural network training landscapes. Nat. Commun. 9, 4812 (2018). Seminal work in which the barren plateaus phenomenon is discovered.

    Article  Google Scholar 

  9. Qi, H., Wang, L., Zhu, H., Gani, A. & Gong, C. The barren plateaus of quantum neural networks: review, taxonomy and trends. Quantum Inf. Process. 22, 435 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  10. Arrasmith, A., Holmes, Z., Cerezo, M. & Coles, P. J. Equivalence of quantum barren plateaus to cost concentration and narrow gorges. Quantum Sci. Technol. 7, 045015 (2022).

    Article  ADS  Google Scholar 

  11. Cerezo, M. et al. Does provable absence of barren plateaus imply classical simulability? Or, why we need to rethink variational quantum computing. Preprint at https://arxiv.org/abs/2312.09121 (2023). This work presents a deep connection between absence of barren plateaus and classical simulability.

  12. Bittel, L. & Kliesch, M. Training variational quantum algorithms is NP-hard. Phys. Rev. Lett. 127, 120502 (2021). In this work, it is shown that training variational quantum algorithms can be, in the worst case, NP-hard.

    Article  ADS  MathSciNet  Google Scholar 

  13. Fontana, E., Cerezo, M., Arrasmith, A., Rungger, I. & Coles, P. J. Non-trivial symmetries in quantum landscapes and their resilience to quantum noise. Quantum 6, 804 (2022).

    Article  Google Scholar 

  14. Anschuetz, E. R. & Kiani, B. T. Beyond barren plateaus: quantum variational algorithms are swamped with traps. Nat. Commun. 13, 7760 (2022). This work showcases the problems of local mínima in variational quantum computing.

    Article  ADS  Google Scholar 

  15. Anschuetz, E. R. Critical points in quantum generative models. In International Conference on Learning Representations https://openreview.net/forum?id=2f1z55GVQN (2022).

  16. Larocca, M. et al. Diagnosing barren plateaus with tools from quantum optimal control. Quantum 6, 824 (2022).

    Article  Google Scholar 

  17. Bermejo, P., Aizpurua, B. & Orús, R. Improving gradient methods via coordinate transformations: applications to quantum machine learning. Phys. Rev. Res. 6, 023069 (2024).

    Article  Google Scholar 

  18. Nádori, J., Morse, G., Majnay-Takács, Z., Zimborás, Z. & Rakyta, P. Line search strategy for navigating through barren plateaus in quantum circuit training. Preprint at https://arxiv.org/abs/2402.05227 (2024).

  19. Sim, S., Johnson, P. D. & Aspuru-Guzik, A. Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms. Adv. Quantum Technol. 2, 1900070 (2019).

    Article  Google Scholar 

  20. Akshay, V., Philathong, H., Morales, M. E. & Biamonte, J. D. Reachability deficits in quantum approximate optimization. Phys. Rev. Lett. 124, 090504 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  21. Miao, Q. & Barthel, T. Equivalence of cost concentration and gradient vanishing for quantum circuits: an elementary proof in the Riemannian formulation. Quantum Sci. Technol. 9, 045039 (2024).

    Article  Google Scholar 

  22. Cerezo, M. & Coles, P. J. Higher order derivatives of quantum neural networks with barren plateaus. Quantum Sci. Technol. 6, 035006 (2021).

    Article  ADS  Google Scholar 

  23. Puig, R., Drudis, M., Thanasilp, S. & Holmes, Z. Variational quantum simulation: a case study for understanding warm starts. PRX Quantum 6, 010317 (2025).

    Article  Google Scholar 

  24. Wang, S. et al. Noise-induced barren plateaus in variational quantum algorithms. Nat. Commun. 12, 6961 (2021). Here, it is shown that noise can induce barren plateaus and deterministic exponential concentration.

    Article  ADS  Google Scholar 

  25. Diaz, N. L., García-Martín, D., Kazi, S., Larocca, M. & Cerezo, M. Showcasing a barren plateau theory beyond the dynamical lie algebra. Preprint at https://arxiv.org/abs/2310.11505 (2023).

  26. Leone, L., Oliviero, S. F., Cincio, L. & Cerezo, M. On the practical usefulness of the hardware efficient ansatz. Quantum 8, 1395 (2024).

    Article  Google Scholar 

  27. Thanasilp, S., Wang, S., Nghiem, N. A., Coles, P. & Cerezo, M. Subtleties in the trainability of quantum machine learning models. Quant. Mach. Intell. 5, 21 (2023).

    Article  Google Scholar 

  28. Cybulski, J. L. & Nguyen, T. Impact of barren plateaus countermeasures on the quantum neural network capacity to learn. Quantum Inf. Process. 22, 442 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  29. Pérez-Salinas, A., Wang, H. & Bonet-Monroig, X. Analyzing variational quantum landscapes with information content. npj Quantum Inf. 10, 27 (2024).

    Article  ADS  Google Scholar 

  30. Zhang, H.-K., Zhu, C. & Wang, X. Predicting quantum learnability from landscape fluctuation. Preprint at https://arxiv.org/abs/2406.11805 (2024).

  31. Sack, S. H., Medina, R. A., Michailidis, A. A., Kueng, R. & Serbyn, M. Avoiding barren plateaus using classical shadows. PRX Quantum 3, 020365 (2022).

    Article  ADS  Google Scholar 

  32. Okumura, S. & Ohzeki, M. Fourier coefficient of parameterized quantum circuits and barren plateau problem. Preprint at https://arxiv.org/abs/2309.06740 (2023).

  33. Nemkov, N. A., Kiktenko, E. O. & Fedorov, A. K. Fourier expansion in variational quantum algorithms. Phys. Rev. A 108, 032406 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  34. Stęchły, M., Gao, L., Yogendran, B., Fontana, E. & Rudolph, M. Connecting the Hamiltonian structure to the QAOA energy and Fourier landscape structure. Preprint at https://arxiv.org/abs/2305.13594 (2023).

  35. Dankert, C., Cleve, R., Emerson, J. & Livine, E. Exact and approximate unitary 2-designs and their application to fidelity estimation. Phys. Rev. A 80, 012304 (2009).

    Article  ADS  Google Scholar 

  36. Sharma, K., Cerezo, M., Cincio, L. & Coles, P. J. Trainability of dissipative perceptron-based quantum neural networks. Phys. Rev. Lett. 128, 180505 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  37. Patti, T. L., Najafi, K., Gao, X. & Yelin, S. F. Entanglement devised barren plateau mitigation. Phys. Rev. Res. 3, 033090 (2021).

    Article  Google Scholar 

  38. Marrero, C. O., Kieferová, M. & Wiebe, N. Entanglement-induced barren plateaus. PRX Quantum 2, 040316 (2021).

    Article  Google Scholar 

  39. Holmes, Z. et al. Barren plateaus preclude learning scramblers. Phys. Rev. Lett. 126, 190501 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  40. Ragone, M. et al. A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits. Nat. Commun. 15, 7172 (2024). This is one of the two concurrent works that present a general theory for characterizing barren plateaus.

    Article  Google Scholar 

  41. Fontana, E. et al. Characterizing barren plateaus in quantum ansätze with the adjoint representation. Nat. Commun. 15, 7171 (2024).

    Article  Google Scholar 

  42. Schatzki, L., Larocca, M., Nguyen, Q. T., Sauvage, F. & Cerezo, M. Theoretical guarantees for permutation-equivariant quantum neural networks. npj Quantum Inf. 10, 12 (2024).

    Article  ADS  Google Scholar 

  43. Liu, J., Lin, Z. & Jiang, L. Laziness, barren plateau, and noise in machine learning. Mach. Learn. Sci. Technol. 5, 015058 (2024).

    Article  ADS  Google Scholar 

  44. West, M. T., Heredge, J., Sevior, M. & Usman, M. Provably trainable rotationally equivariant quantum machine learning. PRX Quantum 5, 030320 (2024).

    Article  Google Scholar 

  45. García-Martín, D., Larocca, M. & Cerezo, M. Deep quantum neural networks form Gaussian processes. Preprint at https://arxiv.org/abs/2305.09957 (2023).

  46. Mele, A. A. Introduction to Haar measure tools in quantum information: a beginner’s tutorial. Quantum 8, 1340 (2024).

    Article  Google Scholar 

  47. Ragone, M. et al. Representation theory for geometric quantum machine learning. Preprint at https://arxiv.org/abs/2210.07980 (2022).

  48. Cerezo, M., Sone, A., Volkoff, T., Cincio, L. & Coles, P. J. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nat. Commun. 12, 1791 (2021).

    Article  ADS  Google Scholar 

  49. Uvarov, A. & Biamonte, J. D. On barren plateaus and cost function locality in variational quantum algorithms. J. Phys. A Math. Theor. 54, 245301 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  50. Pesah, A. et al. Absence of barren plateaus in quantum convolutional neural networks. Phys. Rev. X 11, 041011 (2021).

    Google Scholar 

  51. Heyraud, V., Li, Z., Donatella, K., Boité, A. L. & Ciuti, C. Efficient estimation of trainability for variational quantum circuits. PRX Quantum 4, 040335 (2023).

    Article  ADS  Google Scholar 

  52. Liu, Z., Yu, L.-W., Duan, L.-M. & Deng, D.-L. The presence and absence of barren plateaus in tensor-network based machine learning. Phys. Rev. Lett. 129, 270501 (2022).

    Article  MathSciNet  Google Scholar 

  53. Barthel, T. & Miao, Q. Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states. Preprint at https://arxiv.org/abs/2304.00161 (2023).

  54. Miao, Q. & Barthel, T. Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus. Phys. Rev. A 109, L050402 (2024).

    Article  ADS  MathSciNet  Google Scholar 

  55. Garcia, R. J., Zhao, C., Bu, K. & Jaffe, A. Barren plateaus from learning scramblers with local cost functions. J. High Energy Phys. 2023, 1–79 (2023).

    Article  MathSciNet  Google Scholar 

  56. Letcher, A., Woerner, S. & Zoufal, C. Tight and efficient gradient bounds for parameterized quantum circuits. Quantum 8, 1484 (2024).

    Article  Google Scholar 

  57. Arrazola, J. M. et al. Universal quantum circuits for quantum chemistry. Quantum 6, 742 (2022).

    Article  Google Scholar 

  58. Zhang, H.-K., Liu, S. & Zhang, S.-X. Absence of barren plateaus in finite local-depth circuits with long-range entanglement. Phys. Rev. Lett. 132, 150603 (2024).

    Article  ADS  MathSciNet  Google Scholar 

  59. García-Martín, D., Braccia, P. & Cerezo, M. Architectures and random properties of symplectic quantum circuits. Preprint at https://arxiv.org/abs/2405.10264 (2024).

  60. Deneris, A. E., Bermejo, P., Braccia, P., Cincio, L. & Cerezo, M. Exact spectral gaps of random one-dimensional quantum circuits. Preprint at https://arxiv.org/abs/2408.11201 (2024).

  61. Napp, J. Quantifying the barren plateau phenomenon for a model of unstructured variational ansätze. Preprint at https://arxiv.org/abs/2203.06174 (2022).

  62. Braccia, P., Bermejo, P., Cincio, L. & Cerezo, M. Computing exact moments of local random quantum circuits via tensor networks. Quant. Mach. Intell. 6, 54 (2024).

    Article  Google Scholar 

  63. Hu, H.-Y. et al. Demonstration of robust and efficient quantum property learning with shallow shadows. Preprint at https://arxiv.org/abs/2402.17911 (2024).

  64. Zhao, C. & Gao, X.-S. Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus. Quantum 5, 466 (2021).

    Article  Google Scholar 

  65. Zhang, K., Liu, L., Hsieh, M.-H. & Tao, D. Escaping from the barren plateau via Gaussian initializations in deep variational quantum circuits. Adv. Neural Inf. Process. Syst. https://openreview.net/forum?id=jXgbJdQ2YIy (2022).

  66. Wang, Y., Qi, B., Ferrie, C. & Dong, D. Trainability enhancement of parameterized quantum circuits via reduced-domain parameter initialization. Phys. Rev. Appl. 22, 054005 (2024).

    Article  Google Scholar 

  67. Park, C.-Y. & Killoran, N. Hamiltonian variational ansatz without barren plateaus. Quantum 8, 1239 (2024).

    Article  Google Scholar 

  68. Park, C.-Y., Kang, M. & Huh, J. Hardware-efficient ansatz without barren plateaus in any depth. Preprint at https://arxiv.org/abs/2403.04844 (2024).

  69. Sannia, A., Tacchino, F., Tavernelli, I., Giorgi, G. L. & Zambrini, R. Engineered dissipation to mitigate barren plateaus. npj Quantum Inf. 10, 81 (2024).

    Article  Google Scholar 

  70. Liu, J., Wilde, F., Mele, A. A., Jiang, L. & Eisert, J. Stochastic noise can be helpful for variational quantum algorithms. Preprint at https://arxiv.org/abs/2210.06723 (2022).

  71. Schumann, M., Wilhelm, F. K. & Ciani, A. Emergence of noise-induced barren plateaus in arbitrary layered noise models. Quantum Sci. Technol. 9, 045019 (2024).

    Article  Google Scholar 

  72. Singkanipa, P. & Lidar, D. A. Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and fixed points. Quantum 9, 1617 (2025).

    Article  Google Scholar 

  73. Bremner, M. J., Mora, C. & Winter, A. Are random pure states useful for quantum computation? Phys. Rev. Lett. 102, 190502 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  74. Gross, D., Flammia, S. T. & Eisert, J. Most quantum states are too entangled to be useful as computational resources. Phys. Rev. Lett. 102, 190501 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  75. Yang, G. et al. Tensor programs v: tuning large neural networks via zero-shot hyperparameter transfer. Adv. Neural Inf. Process. Syst. 34, 17084–17097 (2021).

    Google Scholar 

  76. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).

  77. Horgan, J. Scott Aaronson answers every ridiculously big question throw at him. Sci. Am. https://blogs.scientificamerican.com/cross-check/scott-aaronson-answers-every-ridiculously-big-question-i-throw-at-him (2016).

  78. Liu, Y., Arunachalam, S. & Temme, K. A rigorous and robust quantum speed-up in supervised machine learning. Nat. Phys. https://www.nature.com/articles/s41567-021-01287-z (2021).

  79. Jäger, J. & Krems, R. V. Universal expressiveness of variational quantum classifiers and quantum kernels for support vector machines. Nat. Commun. 14, 576 (2023).

    Article  ADS  Google Scholar 

  80. Holmes, Z., Sharma, K., Cerezo, M. & Coles, P. J. Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3, 010313 (2022).

    Article  ADS  Google Scholar 

  81. Friedrich, L. & Maziero, J. Quantum neural network cost function concentration dependency on the parametrization expressivity. Sci. Rep. 13, 9978 (2023).

    Article  ADS  Google Scholar 

  82. Abbas, A. et al. The power of quantum neural networks. Nat. Comput. Sci. 1, 403–409 (2021).

    Article  Google Scholar 

  83. Haug, T., Bharti, K. & Kim, M. S. Capacity and quantum geometry of parametrized quantum circuits. PRX Quantum 2, 040309 (2021).

    Article  ADS  Google Scholar 

  84. Larocca, M., Ju, N., García-Martín, D., Coles, P. J. & Cerezo, M. Theory of overparametrization in quantum neural networks. Nat. Comput. Sci. 3, 542–551 (2023).

    Article  Google Scholar 

  85. Somma, R., Ortiz, G., Barnum, H., Knill, E. & Viola, L. Nature and measure of entanglement in quantum phase transitions. Phys. Rev. A 70, 042311 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  86. Somma, R. D. Quantum computation, complexity, and many-body physics. Preprint at https://arxiv.org/abs/quant-ph/0512209 (2005).

  87. Khatri, S. et al. Quantum-assisted quantum compiling. Quantum 3, 140 (2019).

    Article  Google Scholar 

  88. Uvarov, A., Biamonte, J. D. & Yudin, D. Variational quantum eigensolver for frustrated quantum systems. Phys. Rev. B 102, 075104 (2020).

    Article  ADS  Google Scholar 

  89. Kashif, M. & Al-Kuwari, S. The impact of cost function globality and locality in hybrid quantum neural networks on NISQ devices. Mach. Learn. Sci. Technol. 4, 015004 (2023).

    Article  ADS  Google Scholar 

  90. Leadbeater, C., Sharrock, L., Coyle, B. & Benedetti, M. F-divergences and cost function locality in generative modelling with quantum circuits. Entropy 23, 1281 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  91. Cerezo, M., Sharma, K., Arrasmith, A. & Coles, P. J. Variational quantum state eigensolver. npj Quantum Inf. 8, 1–11 (2022).

    Article  Google Scholar 

  92. Kiani, B. T., De Palma, G., Marvian, M., Liu, Z.-W. & Lloyd, S. Learning quantum data with the quantum earth mover’s distance. Quantum Sci. Technol. 7, 045002 (2022).

    Article  ADS  Google Scholar 

  93. Zambrano, L., Muñoz-Moller, A. D., Muñoz, M., Pereira, L. & Delgado, A. Avoiding barren plateaus in the variational determination of geometric entanglement. Quantum Sci. Technol. 9, 025016 (2024).

    Article  ADS  Google Scholar 

  94. Ogunkoya, O., Morris, K. & Kürkçüoglu, D. M. Investigating parameter trainability in the snap-displacement protocol of a qudit system. Preprint at https://arxiv.org/abs/2309.14942 (2023).

  95. Zhang, B. & Zhuang, Q. Energy-dependent barren plateau in bosonic variational quantum circuits. Quantum Sci. Technol. 10, 015009 (2024).

    Article  Google Scholar 

  96. Wiersema, R., Zhou, C., Carrasquilla, J. F. & Kim, Y. B. Measurement-induced entanglement phase transitions in variational quantum circuits. SciPost Phys. 14, 147 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  97. Shaydulin, R. & Wild, S. M. Importance of kernel bandwidth in quantum machine learning. Phys. Rev. A 106, 042407 (2022).

    Article  ADS  Google Scholar 

  98. Kashif, M. & Al-Kuwari, S. The unified effect of data encoding, ansatz expressibility and entanglement on the trainability of HQNNs. Int. J. Parallel Emergent Distrib. Syst. 38, 362–400 (2023).

    Article  Google Scholar 

  99. Das, S., Martina, S. & Caruso, F. The role of data embedding in equivariant quantum convolutional neural networks. Quant. Mach. Intell. 6, 82 (2024).

    Article  Google Scholar 

  100. Mhiri, H. et al. Constrained and vanishing expressivity of quantum Fourier models. Preprint at https://arxiv.org/abs/2403.09417 (2024).

  101. Stilck França, D. & Garcia-Patron, R. Limitations of optimization algorithms on noisy quantum devices. Nat. Phys. 17, 1221–1227 (2021).

    Article  Google Scholar 

  102. De Palma, G., Marvian, M., Rouzé, C. & França, D. S. Limitations of variational quantum algorithms: a quantum optimal transport approach. PRX Quantum 4, 010309 (2023).

    Article  Google Scholar 

  103. Crognaletti, G., Grossi, M. & Bassi, A. Estimates of loss function concentration in noisy parametrized quantum circuits. Preprint at https://arxiv.org/abs/2410.01893 (2024).

  104. Mele, A. A. et al. Noise-induced shallow circuits and absence of barren plateaus. Preprint at https://arxiv.org/abs/2403.13927 (2024).

  105. Fefferman, B., Ghosh, S., Gullans, M., Kuroiwa, K. & Sharma, K. Effect of non-unital noise on random circuit sampling. PRX Quantum 5, 030317 (2024).

    Article  Google Scholar 

  106. Deshpande, A. et al. Dynamic parameterized quantum circuits: expressive and barren-plateau free. Preprint at https://arxiv.org/abs/2411.05760 (2024).

  107. Wilde, M. M. Quantum Information Theory (Cambridge Univ. Press, 2013).

  108. Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).

    Article  ADS  Google Scholar 

  109. He, Z., Li, L., Zheng, S., Li, Y. & Situ, H. Variational quantum compiling with double q-learning. N. J. Phys. 23, 033002 (2021).

    Article  Google Scholar 

  110. Moro, L., Paris, M. G. A., Restelli, M. & Prati, E. Quantum compiling by deep reinforcement learning. Commun. Phys. 4, 178 (2021).

    Article  Google Scholar 

  111. Beer, K. et al. Training deep quantum neural networks. Nat. Commun. 11, 808 (2020).

    Article  ADS  Google Scholar 

  112. Buonaiuto, G., Gargiulo, F., De Pietro, G., Esposito, M. & Pota, M. The effects of quantum hardware properties on the performances of variational quantum learning algorithms. Quant. Mach. Intell. 6, 9 (2024).

    Article  Google Scholar 

  113. Liu, X., Liu, G., Zhang, H.-K., Huang, J. & Wang, X. Mitigating barren plateaus of variational quantum eigensolvers. IEEE Trans. Quant. Eng. 1–19 https://ieeexplore.ieee.org/abstract/document/10485449 (2024).

  114. Zhang, H.-K., Zhu, C., Liu, G. & Wang, X. Exponential hardness of optimization from the locality in quantum neural networks. In Proc. AAAI Conference on Artificial Intelligence Vol. 38, 16741–16749 https://ojs.aaai.org/index.php/AAAI/article/view/29614 (2024).

  115. Wecker, D., Hastings, M. B. & Troyer, M. Progress towards practical quantum variational algorithms. Phys. Rev. A 92, 042303 (2015).

    Article  ADS  Google Scholar 

  116. Wiersema, R. et al. Exploring entanglement and optimization within the Hamiltonian variational ansatz. PRX Quant. 1, 020319 (2020).

    Article  Google Scholar 

  117. Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).

    Article  ADS  Google Scholar 

  118. Wiersema, R., Kökcü, E., Kemper, A. F. & Bakalov, B. N. Classification of dynamical Lie algebras of 2-local spin systems on linear, circular and fully connected topologies. npj Quantum Inf. 10, 110 (2024).

    Article  Google Scholar 

  119. Farhi, E., Goldstone, J. & Gutmann, S. A quantum approximate optimization algorithm. Preprint at https://arxiv.org/abs/1411.4028 (2014).

  120. Hadfield, S. et al. From the quantum approximate optimization algorithm to a quantum alternating operator ansatz. Algorithms 12, 34 (2019).

    Article  MathSciNet  Google Scholar 

  121. Bärtschi, A. & Eidenbenz, S. Grover mixers for QAOA: shifting complexity from mixer design to state preparation. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) 72–82 https://www.computer.org/csdl/proceedings-article/qce/2020/896900a072/1p2VnUCmpYA (IEEE, 2020).

  122. Fuchs, F. G., Lye, K. O., Møll Nilsen, H., Stasik, A. J. & Sartor, G. Constraint preserving mixers for the quantum approximate optimization algorithm. Algorithms 15, 202 (2022).

    Article  Google Scholar 

  123. Tate, R., Moondra, J., Gard, B., Mohler, G. & Gupta, S. Warm-started QAOA with custom mixers provably converges and computationally beats Goemans–Williamson’s max-cut at low circuit depths. Quantum 7, 1121 (2023).

    Article  Google Scholar 

  124. Zhang, B., Sone, A. & Zhuang, Q. Quantum computational phase transition in combinatorial problems. npj Quantum Inf. 8, 87 (2022).

    Article  ADS  Google Scholar 

  125. Kazi, S. et al. Analyzing the quantum approximate optimization algorithm: ansätze, symmetries, and Lie algebras. Preprint at https://arxiv.org/abs/2410.05187 (2024).

  126. Crooks, G. E. Performance of the quantum approximate optimization algorithm on the maximum cut problem. Preprint at https://arxiv.org/abs/1811.08419 (2018).

  127. Boulebnane, S. & Montanaro, A. Solving Boolean satisfiability problems with the quantum approximate optimization algorithm. PRX Quantum 5, 030348 (2024).

    Article  Google Scholar 

  128. Montanaro, A. & Zhou, L. Quantum speedups in solving near-symmetric optimization problems by low-depth QAOA. Preprint at https://arxiv.org/abs/2411.04979 (2024).

  129. Herrman, R., Lotshaw, P. C., Ostrowski, J., Humble, T. S. & Siopsis, G. Multi-angle quantum approximate optimization algorithm. Sci. Rep. 12, 1–10 (2022).

    Article  Google Scholar 

  130. Aguilar, G., Cichy, S., Eisert, J. & Bittel, L. Full classification of Pauli Lie algebras Preprint at https://arxiv.org/abs/2408.00081 (2024).

  131. Chapman, A. & Flammia, S. T. Characterization of solvable spin models via graph invariants. Quantum 4, 278 (2020).

    Article  Google Scholar 

  132. Kökcü, E., Wiersema, R., Kemper, A. F. & Bakalov, B. N. Classification of dynamical Lie algebras generated by spin interactions on undirected graphs. Preprint at https://arxiv.org/abs/2409.19797 (2024).

  133. Haah, J., Liu, Y. & Tan, X. Efficient approximate unitary designs from random Pauli rotations. In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) 463–475 (IEEE, 2024).

  134. Zhou, N.-R., Liu, X.-X., Chen, Y.-L. & Du, N.-S. Quantum k-nearest-neighbor image classification algorithm based on K-L transform. Int. J. Theor. Phys. 60, 1209–1224 (2021).

    Article  Google Scholar 

  135. Chai, Y. et al. Shortcuts to the quantum approximate optimization algorithm. Phys. Rev. A 105, 042415 (2022).

    Article  ADS  Google Scholar 

  136. Chandarana, P. et al. Digitized-counterdiabatic quantum approximate optimization algorithm. Phys. Rev. Res. 4, 013141 (2022).

    Article  Google Scholar 

  137. Vizzuso, M., Passarelli, G., Cantele, G. & Lucignano, P. Convergence of digitized-counterdiabatic QAOA: circuit depth versus free parameters. N. J. Phys. 26, 013002 (2024).

    Article  MathSciNet  Google Scholar 

  138. Shaydulin, R., Lotshaw, P. C., Larson, J., Ostrowski, J. & Humble, T. S. Parameter transfer for quantum approximate optimization of weighted maxcut. ACM Trans. Quantum Comput. 4, 115 (2023).

    Article  MathSciNet  Google Scholar 

  139. Farhi, E., Goldstone, J., Gutmann, S. & Zhou, L. The quantum approximate optimization algorithm and the Sherrington–Kirkpatrick model at infinite size. Quantum 6, 759 (2022).

    Article  Google Scholar 

  140. Taube, A. G. & Bartlett, R. J. New perspectives on unitary coupled-cluster theory. Int. J. Quantum Chem. 106, 3393–3401 (2006).

    Article  ADS  Google Scholar 

  141. Lee, J., Huggins, W. J., Head-Gordon, M. & Whaley, K. B. Generalized unitary coupled cluster wave functions for quantum computation. J. Chem. Theory Comput. 15, 311–324 (2018).

    Article  Google Scholar 

  142. Mao, R., Tian, G. & Sun, X. Towards determining the presence of barren plateaus in some chemically inspired variational quantum algorithms. Commun. Phys. 7, 342 (2024).

    Article  Google Scholar 

  143. Bermejo, P. et al. Quantum convolutional neural networks are (effectively) classically simulable. Preprint at https://arxiv.org/abs/2408.12739 (2024).

  144. Cichy, S., Faehrmann, P. K., Khatri, S. & Eisert, J. Perturbative gadgets for gate-based quantum computing: nonrecursive constructions without subspace restrictions. Phys. Rev. A 109, 052624 (2024).

    Article  ADS  Google Scholar 

  145. Kiani, B. T., Lloyd, S. & Maity, R. Learning unitaries by gradient descent. Preprint at https://arxiv.org/abs/2001.11897 (2020).

  146. Rajakumar, J., Golden, J., Bärtschi, A. & Eidenbenz, S. Trainability barriers in low-depth QAOA landscapes. In Proc. 21st ACM International Conference on Computing Frontiers, CF ’24, 199–206 https://doi.org/10.1145/3649153.3649204 (Association for Computing Machinery, 2024).

  147. Marshall, S. C., Gyurik, C. & Dunjko, V. High dimensional quantum machine learning with small quantum computers. Quantum 7, 1078 (2023).

    Article  Google Scholar 

  148. Zhang, K., Hsieh, M.-H., Liu, L. & Tao, D. Toward trainability of quantum neural networks. Preprint at https://arxiv.org/abs/2011.06258 (2020).

  149. Larocca, M. et al. Group-invariant quantum machine learning. PRX Quantum 3, 030341 (2022).

    Article  ADS  Google Scholar 

  150. Nguyen, Q. T. et al. Theory for equivariant quantum neural networks. PRX Quantum 5, 020328 (2024).

    Article  ADS  Google Scholar 

  151. Meyer, J. J. et al. Exploiting symmetry in variational quantum machine learning. PRX Quantum 4, 010328 (2023).

    Article  ADS  Google Scholar 

  152. Skolik, A., Cattelan, M., Yarkoni, S., Bäck, T. & Dunjko, V. Equivariant quantum circuits for learning on weighted graphs. npj Quantum Inf. 9, 47 (2023).

    Article  ADS  Google Scholar 

  153. Volkoff, T. & Coles, P. J. Large gradients via correlation in random parameterized quantum circuits. Quantum Sci. Technol. 6, 025008 (2021).

    Article  ADS  Google Scholar 

  154. Sauvage, F., Larocca, M., Coles, P. J. & Cerezo, M. Building spatial symmetries into parameterized quantum circuits for faster training. Quantum Sci. Technol. 9, 015029 (2024).

    Article  ADS  Google Scholar 

  155. Kazi, S., Larocca, M. & Cerezo, M. On the universality of sn-equivariant k-body gates. N. J. Phys. 26, 053030 (2024).

    Article  Google Scholar 

  156. Monbroussou, L., Landman, J., Grilo, A. B., Kukla, R. & Kashefi, E. Trainability and expressivity of Hamming-weight preserving quantum circuits for machine learning. Preprint at https://arxiv.org/abs/2309.15547 (2023).

  157. Raj, S. et al. Quantum deep hedging. Quantum 7, 1191 (2023).

    Article  Google Scholar 

  158. Lee, J., Magann, A. B., Rabitz, H. A. & Arenz, C. Progress toward favorable landscapes in quantum combinatorial optimization. Phys. Rev. A 104, 032401 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  159. Jozsa, R. & Miyake, A. Matchgates and classical simulation of quantum circuits. Proc. R. Soc. A Math. Phys. Eng. Sci. 464, 3089–3106 (2008).

    ADS  MathSciNet  Google Scholar 

  160. Wan, K., Huggins, W. J., Lee, J. & Babbush, R. Matchgate shadows for fermionic quantum simulation. Commun. Math. Phys. 404, 629 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  161. De Melo, F., Ćwikliński, P. & Terhal, B. M. The power of noisy fermionic quantum computation. N. J. Phys. 15, 013015 (2013).

    Article  MathSciNet  Google Scholar 

  162. Oszmaniec, M., Dangniam, N., Morales, M. E. & Zimborás, Z. Fermion sampling: a robust quantum computational advantage scheme using fermionic linear optics and magic input states. PRX Quantum 3, 020328 (2022).

    Article  ADS  Google Scholar 

  163. Matos, G., Self, C. N., Papić, Z., Meichanetzidis, K. & Dreyer, H. Characterization of variational quantum algorithms using free fermions. Quantum 7, 966 (2023).

    Article  Google Scholar 

  164. Diaz, N. L. et al. Parallel-in-time quantum simulation via Page and Wootters quantum time. Preprint at https://arxiv.org/abs/2308.12944 (2023).

  165. Kökcü, E. et al. Fixed depth Hamiltonian simulation via Cartan decomposition. Phys. Rev. Lett. 129, 070501 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  166. Volkoff, T. J. Efficient trainability of linear optical modules in quantum optical neural networks. J. Russ. Laser Res. 42, 250–260 (2021).

    Article  Google Scholar 

  167. Volkoff, T., Holmes, Z. & Sornborger, A. Universal compiling and (no-)free-lunch theorems for continuous-variable quantum learning. PRX Quantum 2, 040327 (2021).

    Article  ADS  Google Scholar 

  168. Arjovsky, M., Shah, A. & Bengio, Y. Unitary evolution recurrent neural networks. In International Conference on Machine Learning 1120–1128. https://proceedings.mlr.press/v48/arjovsky16.html (PMLR, 2016).

  169. Jing, L. et al. Tunable efficient unitary neural networks (EUNN) and their application to RNNs. In International Conference on Machine Learning 1733–1741. https://proceedings.mlr.press/v70/jing17a.html (PMLR, 2017).

  170. Bilkis, M., Cerezo, M., Verdon, G., Coles, P. J. & Cincio, L. A semi-agnostic ansatz with variable structure for variational quantum algorithms. Quant. Mach. Intell. 5, 43 (2023).

    Article  Google Scholar 

  171. Du, Y., Huang, T., You, S., Hsieh, M.-H. & Tao, D. Quantum circuit architecture search: error mitigation and trainability enhancement for variational quantum solvers. npj Quantum Inf. 8, 62 (2022).

    Article  ADS  Google Scholar 

  172. Grimsley, H. R., Economou, S. E., Barnes, E. & Mayhall, N. J. An adaptive variational algorithm for exact molecular simulations on a quantum computer. Nat. Commun. 10, 3007 (2019).

    Article  ADS  Google Scholar 

  173. Tang, H. L. et al. qubit-ADAPT-VQE: an adaptive algorithm for constructing hardware-efficient ansätze on a quantum processor. PRX Quantum 2, 020310 (2021).

    Article  ADS  Google Scholar 

  174. Sim, S., Romero, J., Gonthier, J. F. & Kunitsa, A. A. Adaptive pruning-based optimization of parameterized quantum circuits. Quantum Sci. Technol. 6, 025019 (2021).

    Article  ADS  Google Scholar 

  175. Zhang, Z.-J., Kyaw, T. H., Kottmann, J., Degroote, M. & Aspuru-Guzik, A. Mutual information-assisted adaptive variational quantum eigensolver. Quantum Sci. Technol. https://iopscience.iop.org/article/10.1088/2058-9565/abdca4 (2021).

  176. Tkachenko, N. V. et al. Correlation-informed permutation of qubits for reducing ansatz depth in VQE. PRX Quantum 2, 020337 (2021).

    Article  ADS  Google Scholar 

  177. Claudino, D., Wright, J., McCaskey, A. J. & Humble, T. S. Benchmarking adaptive variational quantum eigensolvers. Front. Chem. 8, 1152 (2020).

    Article  ADS  Google Scholar 

  178. Rattew, A. G., Hu, S., Pistoia, M., Chen, R. & Wood, S. A domain-agnostic, noise-resistant, hardware-efficient evolutionary variational quantum eigensolver. Preprint at https://arxiv.org/abs/1910.09694 (2019).

  179. Chivilikhin, D. et al. MoG-VQE: multiobjective genetic variational quantum eigensolver. Preprint at https://arxiv.org/abs/2007.04424 (2020).

  180. Cincio, L., Subaşı, Y., Sornborger, A. T. & Coles, P. J. Learning the quantum algorithm for state overlap. N. J. Phys. 20, 113022 (2018).

    Article  Google Scholar 

  181. Zhang, S.-X., Hsieh, C.-Y., Zhang, S. & Yao, H. Differentiable quantum architecture search. Quantum Sci. Technol. 7, 045023 (2022).

    Article  ADS  Google Scholar 

  182. Wada, K., Raymond, R., Sato, Y. & Watanabe, H. C. Sequential optimal selections of single-qubit gates in parameterized quantum circuits. Quantum Sci. Technol. 9, 035030 (2024).

    Article  ADS  Google Scholar 

  183. Grimsley, H. R., Mayhall, N. J., Barron, G. S., Barnes, E. & Economou, S. E. Adaptive, problem-tailored variational quantum eigensolver mitigates rough parameter landscapes and barren plateaus. npj Quantum Inf. 9, 19 (2023).

    Article  ADS  Google Scholar 

  184. Anastasiou, P. G., Chen, Y., Mayhall, N. J., Barnes, E. & Economou, S. E. TETRIS-ADAPT-VQE: an adaptive algorithm that yields shallower, denser circuit ansätze. Phys. Rev. Res. 6, 013254 (2024).

    Article  Google Scholar 

  185. Van Dyke, J. S. et al. Scaling adaptive quantum simulation algorithms via operator pool tiling. Phys. Rev. Res. 6, L012030 (2024).

    Article  Google Scholar 

  186. Romero, A., Engel, J., Tang, H. L. & Economou, S. E. Solving nuclear structure problems with the adaptive variational quantum algorithm. Phys. Rev. C 105, 064317 (2022).

    Article  ADS  Google Scholar 

  187. Zhu, L. et al. Adaptive quantum approximate optimization algorithm for solving combinatorial problems on a quantum computer. Phys. Rev. Res. 4, 033029 (2022).

    Article  Google Scholar 

  188. Grant, E., Wossnig, L., Ostaszewski, M. & Benedetti, M. An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019).

    Article  Google Scholar 

  189. Jain, N., Coyle, B., Kashefi, E. & Kumar, N. Graph neural network initialisation of quantum approximate optimisation. Quantum 6, 861 (2022).

    Article  Google Scholar 

  190. Skolik, A., McClean, J. R., Mohseni, M., van der Smagt, P. & Leib, M. Layerwise learning for quantum neural networks. Quant. Mach. Intell. 3, 1–11 (2021).

    Article  Google Scholar 

  191. Kulshrestha, A. & Safro, I. BEINIT: avoiding barren plateaus in variational quantum algorithms. In 2022 IEEE International Conference on Quantum Computing and Engineering (QCE) 197–203 https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=9951195 (IEEE, 2022).

  192. Rad, A., Seif, A. & Linke, N. M. Surviving the barren plateau in variational quantum circuits with Bayesian learning initialization. Preprint at https://arxiv.org/abs/2203.02464 (2022).

  193. Astrakhantsev, N., Mazzola, G., Tavernelli, I. & Carleo, G. Phenomenological theory of variational quantum ground-state preparation. Phys. Rev. Res. 5, 033225 (2023).

    Article  Google Scholar 

  194. Haug, T. & Kim, M. Optimal training of variational quantum algorithms without barren plateaus. Preprint at https://arxiv.org/abs/2104.14543 (2021).

  195. Kashif, M., Rashid, M., Al-Kuwari, S. & Shafique, M. Alleviating barren plateaus in parameterized quantum machine learning circuits: investigating advanced parameter initialization strategies. In 2024 Design, Automation & Test in Europe Conference & Exhibition (DATE) https://ieeexplore.ieee.org/document/10546644 (IEEE 2024).

  196. Shi, X. & Shang, Y. Avoiding barren plateaus via Gaussian mixture model. Preprint at https://arxiv.org/abs/2402.13501 (2024).

  197. Friedrich, L. & Maziero, J. Avoiding barren plateaus with classical deep neural networks. Phys. Rev. A 106, 042433 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  198. Rudolph, M. S. et al. Synergistic pretraining of parametrized quantum circuits via tensor networks. Nat. Commun. 14, 8367 (2023).

    Article  ADS  Google Scholar 

  199. Marin-Sanchez, G., Gonzalez-Conde, J. & Sanz, M. Quantum algorithms for approximate function loading. Phys. Rev. Res. 5, 033114 (2023).

    Article  Google Scholar 

  200. Cervera-Lierta, A., Kottmann, J. S. & Aspuru-Guzik, A. The meta-variational quantum eigensolver (meta-VQE): learning energy profiles of parameterized Hamiltonians for quantum simulation. PRX Quantum 2, 020329 (2021).

    Article  ADS  Google Scholar 

  201. Goh, M. L., Larocca, M., Cincio, L., Cerezo, M. & Sauvage, F. Lie-algebraic classical simulations for variational quantum computing. Preprint at https://arxiv.org/abs/2308.01432 (2023).

  202. Brandao, F. G., Broughton, M., Farhi, E., Gutmann, S. & Neven, H. For fixed control parameters the quantum approximate optimization algorithm’s objective function value concentrates for typical instances. Preprint at https://arxiv.org/abs/1812.04170 (2018).

  203. Zhou, L., Wang, S.-T., Choi, S., Pichler, H. & Lukin, M. D. Quantum approximate optimization algorithm: performance, mechanism, and implementation on near-term devices. Phys. Rev. X 10, 021067 (2020).

    Google Scholar 

  204. Wurtz, J. & Lykov, D. Fixed-angle conjectures for the quantum approximate optimization algorithm on regular maxcut graphs. Phys. Rev. A 104, 052419 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  205. Boulebnane, S. & Montanaro, A. Predicting parameters for the quantum approximate optimization algorithm for max-cut from the infinite-size limit. Preprint at https://arxiv.org/abs/2110.10685 (2021).

  206. Galda, A., Liu, X., Lykov, D., Alexeev, Y. & Safro, I. Transferability of optimal QAOA parameters between random graphs. In 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 171–180 https://www.computer.org/csdl/proceedings-article/qce/2021/169100a171/1yEZ9MWWjSg (IEEE, 2021).

  207. Mele, A. A., Mbeng, G. B., Santoro, G. E., Collura, M. & Torta, P. Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz. Phys. Rev. A 106, L060401 (2022).

    Article  ADS  Google Scholar 

  208. Liu, H.-Y., Sun, T.-P., Wu, Y.-C., Han, Y.-J. & Guo, G.-P. Mitigating barren plateaus with transfer-learning-inspired parameter initializations. N. J. Phys. 25, 013039 (2023).

    Article  MathSciNet  Google Scholar 

  209. Zhang, Z.-J., Sun, J., Yuan, X. & Yung, M.-H. Low-depth Hamiltonian simulation by an adaptive product formula. Phys. Rev. Lett. 130, 040601 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  210. Gacon, J., Nys, J., Rossi, R., Woerner, S. & Carleo, G. Variational quantum time evolution without the quantum geometric tensor. Phys. Rev. Res. 6, 013143 (2024).

    Article  Google Scholar 

  211. Campos, E., Rabinovich, D., Akshay, V. & Biamonte, J. Training saturation in layerwise quantum approximate optimization. Phys. Rev. A 104, L030401 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  212. Campos, E., Nasrallah, A. & Biamonte, J. Abrupt transitions in variational quantum circuit training. Phys. Rev. A 103, 032607 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  213. Fitzek, D., Jonsson, R. S., Dobrautz, W. & Schäfer, C. Optimizing variational quantum algorithms with qBang: efficiently interweaving metric and momentum to navigate flat energy landscapes. Quantum 8, 1313 (2024).

    Article  Google Scholar 

  214. Acampora, G., Chiatto, A. & Vitiello, A. A comparison of evolutionary algorithms for training variational quantum classifiers. In 2023 IEEE Congress on Evolutionary Computation (CEC) 1–8 https://ieeexplore.ieee.org/abstract/document/10254076 (IEEE, 2023).

  215. Stokes, J., Izaac, J., Killoran, N. & Carleo, G. Quantum natural gradient. Quantum 4, 269 (2020).

    Article  Google Scholar 

  216. Koczor, B. & Benjamin, S. C. Quantum natural gradient generalized to noisy and nonunitary circuits. Phys. Rev. A 106, 062416 (2022).

    Article  ADS  MathSciNet  Google Scholar 

  217. Arrasmith, A., Cerezo, M., Czarnik, P., Cincio, L. & Coles, P. J. Effect of barren plateaus on gradient-free optimization. Quantum 5, 558 (2021).

    Article  Google Scholar 

  218. Temme, K., Bravyi, S. & Gambetta, J. M. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett. 119, 180509 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  219. Li, Y. & Benjamin, S. C. Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X 7, 021050 (2017).

    Google Scholar 

  220. Cai, Z. et al. Quantum error mitigation. Rev. Mod. Phys. 95, 045005 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  221. Wang, K., Chen, Y.-A. & Wang, X. Mitigating quantum errors via truncated Neumann series. Sci. China Inf. Sci. 66, 180508 (2023).

    Article  MathSciNet  Google Scholar 

  222. Takagi, R., Endo, S., Minagawa, S. & Gu, M. Fundamental limits of quantum error mitigation. npj Quantum Inf. 8, 114 (2022).

    Article  ADS  Google Scholar 

  223. Takagi, R., Tajima, H. & Gu, M. Universal sampling lower bounds for quantum error mitigation. Phys. Rev. Lett. 131, 210602 (2023).

    Article  ADS  MathSciNet  Google Scholar 

  224. Quek, Y., França, D. S., Khatri, S., Meyer, J. J. & Eisert, J. Exponentially tighter bounds on limitations of quantum error mitigation. Nat. Phys. 20, 1648–1658 (2024).

    Article  Google Scholar 

  225. Perdomo-Ortiz, A., Benedetti, M., Realpe-Gómez, J. & Biswas, R. Opportunities and challenges for quantum-assisted machine learning in near-term quantum computers. Quantum Sci. Technol. 3, 030502 (2018).

    Article  ADS  Google Scholar 

  226. Gao, X., Anschuetz, E. R., Wang, S.-T., Cirac, J. I. & Lukin, M. D. Enhancing generative models via quantum correlations. Phys. Rev. X 12, 021037 (2022).

    Google Scholar 

  227. Benedetti, M. et al. A generative modeling approach for benchmarking and training shallow quantum circuits. npj Quantum Inf. 5, 45 (2019).

    Article  ADS  Google Scholar 

  228. Liu, J.-G. & Wang, L. Differentiable learning of quantum circuit born machines. Phys. Rev. A 98, 062324 (2018).

    Article  ADS  Google Scholar 

  229. Coyle, B., Mills, D., Danos, V. & Kashefi, E. The Born supremacy: quantum advantage and training of an Ising Born Machine. npj Quantum Inf. 6, 60 (2020).

    Article  ADS  Google Scholar 

  230. Rudolph, M. S. et al. Trainability barriers and opportunities in quantum generative modeling. npj Quantum Inf. 10, 116 (2024).

    Article  Google Scholar 

  231. Zoufal, C. Generative quantum machine learning. Preprint at https://arxiv.org/abs/2111.12738 (2021).

  232. Lloyd, S. & Weedbrook, C. Quantum generative adversarial learning. Phys. Rev. Lett. 121, 040502 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  233. Amin, M. H., Andriyash, E., Rolfe, J., Kulchytskyy, B. & Melko, R. Quantum Boltzmann machine. Phys. Rev. X 8, 021050 (2018).

    Google Scholar 

  234. Coopmans, L. & Benedetti, M. On the sample complexity of quantum Boltzmann machine learning. Commun. Phys. 7, 274 (2024).

    Article  Google Scholar 

  235. Chang, S. Y., Thanasilp, S., Saux, B. L., Vallecorsa, S. & Grossi, M. Latent style-based quantum GAN for high-quality image generation. Preprint at https://arxiv.org/abs/2406.02668 (2024).

  236. Kieferova, M., Carlos, O. M. & Wiebe, N. Quantum generative training using Rényi divergences. Preprint at https://arxiv.org/abs/2106.09567 (2021).

  237. Huang, H.-Y. et al. Power of data in quantum machine learning. Nat. Commun. 12, 2631 (2021).

    Article  ADS  Google Scholar 

  238. Thanasilp, S., Wang, S., Cerezo, M. & Holmes, Z. Exponential concentration in quantum kernel methods. Nat. Commun. 15, 5200 (2024).

    Article  ADS  Google Scholar 

  239. Kübler, J., Buchholz, S. & Schölkopf, B. The inductive bias of quantum kernels. Adv. Neural Inf. Process. Syst. 34, 12661–12673 (2021).

    Google Scholar 

  240. Canatar, A., Peters, E., Pehlevan, C., Wild, S. M. & Shaydulin, R. Bandwidth enables generalization in quantum kernel models. Trans. Mach. Learn. Res. https://openreview.net/forum?id=A1N2qp4yAq (2023).

  241. Suzuki, Y. & Li, M. Effect of alternating layered ansatzes on trainability of projected quantum kernel. Phys. Rev. A 110, 012409 (2024).

    Article  MathSciNet  Google Scholar 

  242. Suzuki, Y., Kawaguchi, H. & Yamamoto, N. Quantum Fisher kernel for mitigating the vanishing similarity issue. Quantum Sci. Technol. 9, 035050 (2024).

    Article  Google Scholar 

  243. Xiong, W. et al. On fundamental aspects of quantum extreme learning machines. Quant. Mach. Intell. 7, 20 (2025).

    Article  Google Scholar 

  244. Yu, L.-W. et al. Expressibility-induced concentration of quantum neural tangent kernels. Rep. Prog. Phys. 87, 110501 (2024).

    Article  MathSciNet  Google Scholar 

  245. Schuld, M. & Petruccione, F. Supervised Learning with Quantum Computers Vol. 17 (Springer, 2018).

  246. Havlíček, V. et al. Supervised learning with quantum-enhanced feature spaces. Nature 567, 209–212 (2019).

    Article  ADS  Google Scholar 

  247. Schuld, M. Supervised quantum machine learning models are kernel methods. Preprint at https://arxiv.org/abs/2101.11020 (2021).

  248. Gan, B. Y., Leykam, D. & Thanasilp, S. A unified framework for trace-induced quantum kernels. Preprint at https://arxiv.org/abs/2311.13552 (2023).

  249. Magann, A. B. et al. From pulses to circuits and back again: a quantum optimal control perspective on variational quantum algorithms. PRX Quantum 2, 010101 (2021).

    Article  Google Scholar 

  250. Ge, X., Wu, R.-B. & Rabitz, H. The optimization landscape of hybrid quantum–classical algorithms: from quantum control to NISQ applications. Annu. Rev. Control https://www.sciencedirect.com/science/article/pii/S1367578822000840 (2022).

  251. Broers, L. & Mathey, L. Mitigated barren plateaus in the time-nonlocal optimization of analog quantum-algorithm protocols. Phys. Rev. Res. 6, 013076 (2024).

    Article  Google Scholar 

  252. Tao, H.-X., Hu, J. & Wu, R.-B. Unleashing the expressive power of pulse-based quantum neural networks. Preprint at https://arxiv.org/abs/2402.02880 (2024).

  253. de Keijzer, R., Tse, O. & Kokkelmans, S. Pulse based variational quantum optimal control for hybrid quantum computing. Quantum 7, 908 (2023).

    Article  Google Scholar 

  254. Pecci, G., Wang, R., Torta, P., Mbeng, G. B. & Santoro, G. Beyond quantum annealing: optimal control solutions to maxcut problems. Quantum Sci. Technol. 9, 045013 (2024).

    Article  Google Scholar 

  255. Martín, E. C., Plekhanov, K. & Lubasch, M. Barren plateaus in quantum tensor network optimization. Quantum 7, 974 (2023).

    Article  Google Scholar 

  256. Feldman, V. Statistical Query Learning, 2090–2095. https://doi.org/10.1007/978-1-4939-2864-4_401 (Springer New York, 2016).

  257. Arunachalam, S., Grilo, A. B. & Yuen, H. Quantum statistical query learning. Preprint at https://arxiv.org/abs/2002.08240 (2020).

  258. Anshu, A. & Arunachalam, S. A survey on the complexity of learning quantum states. Nat. Rev. Phys. 6, 59–69 (2024).

    Article  Google Scholar 

  259. Angrisani, A. Learning unitaries with quantum statistical queries. Preprint at https://arxiv.org/abs/2310.02254 (2023).

  260. Wadhwa, C. & Doosti, M. Learning quantum processes with quantum statistical queries. Preprint at https://arxiv.org/abs/2310.02075 (2023).

  261. Nietner, A. Unifying (quantum) statistical and parametrized (quantum) algorithms. Preprint at https://arxiv.org/abs/2310.17716 (2023).

  262. Nietner, A. et al. On the average-case complexity of learning output distributions of quantum circuits. Preprint at https://arxiv.org/abs/2305.05765 (2023).

  263. Anschuetz, E. R. & Kiani, B. T. Quantum variational algorithms are swamped with traps. Nat. Commun. 13, 7760 (2022).

    Article  ADS  Google Scholar 

  264. You, X. & Wu, X. Exponentially many local minima in quantum neural networks. In International Conference on Machine Learning 12144–12155. https://proceedings.mlr.press/v139/you21c.html (PMLR, 2021).

  265. Huang, H.-Y. et al. Learning shallow quantum circuits. In Proc. 56th Annual ACM Symposium on Theory of Computing 1343–1351. https://dl.acm.org/doi/10.1145/3618260.3649722 (2024).

  266. Nemkov, N. A., Kiktenko, E. O. & Fedorov, A. K. Barren plateaus are swamped with traps. Phys. Rev. A 111, 012441 (2025).

    Article  Google Scholar 

  267. Anschuetz, E. R. A unified theory of quantum neural network loss landscapes. Preprint at https://arxiv.org/abs/2408.11901 (2024).

  268. Tikku, A. & Kim, I. H. Circuit depth versus energy in topologically ordered systems. Preprint at https://arxiv.org/abs/2210.06796 (2022).

  269. Basheer, A., Feng, Y., Ferrie, C. & Li, S. Alternating layered variational quantum circuits can be classically optimized efficiently using classical shadows. Proceedings of the AAAI Conference on Artificial Intelligence 37, 6 (AAAI, 2023).

  270. Jerbi, S. et al. The power and limitations of learning quantum dynamics incoherently. Preprint at https://arxiv.org/abs/2303.12834 (2023).

  271. Somma, R., Barnum, H., Ortiz, G. & Knill, E. Efficient solvability of Hamiltonians and limits on the power of some quantum computational models. Phys. Rev. Lett. 97, 190501 (2006).

    Article  ADS  Google Scholar 

  272. Galitski, V. Quantum-to-classical correspondence and Hubbard–Stratonovich dynamical systems: a Lie-algebraic approach. Phys. Rev. A 84, 012118 (2011).

    Article  ADS  Google Scholar 

  273. Anschuetz, E. R., Bauer, A., Kiani, B. T. & Lloyd, S. Efficient classical algorithms for simulating symmetric quantum systems. Quantum 7, 1189 (2023).

    Article  Google Scholar 

  274. Angrisani, A. et al. Classically estimating observables of noiseless quantum circuits. Preprint at https://arxiv.org/abs/2409.01706 (2024).

  275. Rudolph, M. S., Fontana, E., Holmes, Z. & Cincio, L. Classical surrogate simulation of quantum systems with lowesa. Preprint at https://arxiv.org/abs/2308.09109 (2023).

  276. Fontana, E., Rudolph, M. S., Duncan, R., Rungger, I. & Cîrstoiu, C. Classical simulations of noisy variational quantum circuits. Preprint at https://arxiv.org/abs/2306.05400 (2023).

  277. Lerch, S. et al. Efficient quantum-enhanced classical simulation for patches of quantum landscapes. Preprint at https://arxiv.org/abs/2411.19896 (2024).

  278. Bharti, K. & Haug, T. Iterative quantum-assisted eigensolver. Phys. Rev. A 104, L050401 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  279. Bharti, K. & Haug, T. Quantum-assisted simulator. Phys. Rev. A 104, 042418 (2021).

    Article  ADS  Google Scholar 

  280. Gil-Fuster, E., Gyurik, C., Pérez-Salinas, A. & Dunjko, V. On the relation between trainability and dequantization of variational quantum learning models. Preprint at https://arxiv.org/abs/2406.07072 (2024).

  281. Parrish, R. M., Hohenstein, E. G., McMahon, P. L. & Martínez, T. J. Quantum computation of electronic transitions using a variational quantum eigensolver. Phys. Rev. Lett. 122, 230401 (2019).

    Article  ADS  Google Scholar 

  282. Jerbi, S., Gyurik, C., Marshall, S. C., Molteni, R. & Dunjko, V. Shadows of quantum machine learning. Nat. Commun. 15, 5676 (2024).

    Article  Google Scholar 

  283. Gyurik, C., Molteni, R. & Dunjko, V. Limitations of measure-first protocols in quantum machine learning. Preprint at https://arxiv.org/abs/2311.12618 (2023).

  284. Elben, A. et al. The randomized measurement toolbox. Nat. Rev. Phys. 5, 1 (2023).

    Google Scholar 

  285. White, S. R. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69, 2863 (1992).

    Article  ADS  Google Scholar 

  286. Lee, S. et al. Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry. Nat. Commun. 14, 1952 (2023).

    Article  ADS  Google Scholar 

  287. Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science 355, 602–606 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  288. Sharir, O., Levine, Y., Wies, N., Carleo, G. & Shashua, A. Deep autoregressive models for the efficient variational simulation of many-body quantum systems. Phys. Rev. Lett. 124, 020503 (2020).

    Article  ADS  Google Scholar 

  289. Carrasquilla, J. & Melko, R. G. Machine learning phases of matter. Nat. Phys. 13, 431 (2017).

    Article  Google Scholar 

  290. Torlai, G. et al. Neural-network quantum state tomography. Nat. Phys. 14, 447–450 (2018).

    Article  Google Scholar 

  291. Glorot, X. & Bengio, Y. Understanding the difficulty of training deep feedforward neural networks. In Proc. Thirteenth International Conference on Artificial Intelligence and Statistics 249–256. https://proceedings.mlr.press/v9/glorot10a (JMLR Workshop and Conference Proceedings, 2010).

  292. Ioffe, S. & Szegedy, C. Batch normalization: accelerating deep network training by reducing internal covariate shift. In International Conference on Machine Learning 448–456. https://proceedings.mlr.press/v37/ioffe15.html (PMLR, 2015).

  293. Ba, J. L., Kiros, J. R. & Hinton, G. E. Layer normalization. Preprint at https://arxiv.org/abs/1607.06450 (2016).

  294. Nair, V. & Hinton, G. E. Rectified linear units improve restricted Boltzmann machines. In Proc. 27th International Conference on Machine Learning (ICML-10) 807–814. https://www.cs.toronto.edu/~hinton/absps/reluICML.pdf (2010).

  295. Hochreiter, S. The vanishing gradient problem during learning recurrent neural nets and problem solutions. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 6, 107–116 (1998).

    Article  Google Scholar 

  296. Hochreiter, S. & Schmidhuber, J. Long short-term memory. Neural Comput. 9, 1735–1780 (1997).

    Article  Google Scholar 

  297. Narkhede, M. V., Bartakke, P. P. & Sutaone, M. S. A review on weight initialization strategies for neural networks. Artif. Intell. Rev. 55, 291–322 (2022).

    Article  Google Scholar 

  298. Chen, M., Pennington, J. & Schoenholz, S. Dynamical isometry and a mean field theory of RNNs: gating enables signal propagation in recurrent neural networks. In Proc. 35th International Conference on Machine Learning (eds Dy, J. & Krause, A.) Vol. 80 of Proceedings of Machine Learning Research 873–882. https://proceedings.mlr.press/v80/chen18i.html (PMLR, 2018).

  299. Xiao, L., Bahri, Y., Sohl-Dickstein, J., Schoenholz, S. & Pennington, J. Dynamical isometry and a mean field theory of CNNs: how to train 10,000-layer vanilla convolutional neural networks. In Proc. 35th International Conference on Machine Learning (eds Dy, J. & Krause, A.) Vol. 80 of Proceedings of Machine Learning Research 5393–5402. https://proceedings.mlr.press/v80/xiao18a.html (PMLR, 2018).

  300. Saxe, A. M., McClelland, J. L. & Ganguli, S. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. Preprint at https://doi.org/10.48550/arXiv.1312.6120 (2013).

  301. Baird, L. & Moore, A. Gradient descent for general reinforcement learning. Adv. Neural Inform. Process. Syst. 11 https://proceedings.neurips.cc/paper/1998/hash/af5afd7f7c807171981d443ad4f4f648-Abstract.html (1998).

  302. Salimans, T., Ho, J., Chen, X., Sidor, S. & Sutskever, I. Evolution strategies as a scalable alternative to reinforcement learning. Preprint at https://arxiv.org/abs/1703.03864 (2017).

  303. Standard Deviation. https://en.wikipedia.org/wiki/Standard_deviation#Sample_standard_deviation.

  304. Zeier, R. & Schulte-Herbrüggen, T. Symmetry principles in quantum systems theory. J. Math. Phys. 52, 113510 (2011).

    Article  ADS  MathSciNet  Google Scholar 

  305. Wierichs, D., East, R. D., Larocca, M., Cerezo, M. & Killoran, N. Symmetric derivatives of parametrized quantum circuits. Preprint at https://arxiv.org/abs/2312.06752 (2023).

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Acknowledgements

The authors thank M. Kieferova, P. Bermejo and T. O’Leary for their feedback on our manuscript. M.L. was supported by the Center for Nonlinear Studies at Los Alamos National Laboratory (LANL). M.L. acknowledges support by the Laboratory Directed Research and Development (LDRD) program of LANL under project number 20230049DR. S.T. and Z.H. acknowledge support from the Sandoz Family Foundation-Monique de Meuron program for Academic Promotion. This research was partly supported (L.C.) by the Quantum Science Center, a National Quantum Science Initiative of the Department of Energy, managed by Oak Ridge National Laboratory. L.C. was also supported by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under Computational Partnerships program. M.C. acknowledges support by the LANL ASC Beyond Moore’s Law project and by LDRD program of LANL under project number 20230527ECR.

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Authors and Affiliations

  1. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    Martín Larocca & Lukasz Cincio

  2. Center for Non-Linear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA

    Martín Larocca

  3. Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Supanut Thanasilp & Zoë Holmes

  4. Chula Intelligent and Complex Systems, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok, Thailand

    Supanut Thanasilp

  5. Institute for Quantum Information and Matter, Caltech, Pasadena, CA, USA

    Samson Wang

  6. IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

    Kunal Sharma

  7. Tensor Institute, Montreal, Quebec, Canada

    Jacob Biamonte

  8. ETS Montreal, University of Quebec, Montreal, Quebec, Canada

    Jacob Biamonte

  9. Normal Computing Corporation, New York, NY, USA

    Patrick J. Coles

  10. Quantum Science Center, Oak Ridge, TN, USA

    Lukasz Cincio & M. Cerezo

  11. Google Quantum AI, Venice, CA, USA

    Jarrod R. McClean

  12. Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA

    M. Cerezo

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All authors contributed to writing and reviewing of the manuscript. M.L., S.T., Z.H. and M.C. compiled and edited the main text.

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Larocca, M., Thanasilp, S., Wang, S. et al. Barren plateaus in variational quantum computing. Nat Rev Phys 7, 174–189 (2025). https://doi.org/10.1038/s42254-025-00813-9

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