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Efficient Quantum-Enhanced Classical Simulation for Patches of Quantum Landscapes

Sacha Lerch1,2,*, Ricard Puig1,2,*, Manuel S. Rudolph1,2,*, Armando Angrisani1,2, Tyson Jones1,2, M. Cerezo3,4, Supanut Thanasilp1,2,5,6,†, and Zoë Holmes1,2

  • *The first three authors contributed equally to this work.
  • †Contact author: supanut.thanasilp@gmail.com
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PRX Quantum 7, 020359 – Published 12 June, 2026

DOI: https://doi.org/10.1103/fhc5-8sm6

Abstract

Understanding the capabilities of classical simulation methods is key to identifying where quantum computers are advantageous. Not only does this ensure that quantum computers are used only where necessary, but also one can potentially identify subroutines that can be offloaded onto a classical device. In this work, we show that it is always possible to generate a classical surrogate of a subregion (dubbed a “patch”) of an expectation landscape produced by a parameterized quantum circuit. That is, we provide a quantum-enhanced classical algorithm which, after simple measurements on a quantum device, allows one to classically simulate approximate expectation values of a subregion of a landscape. We provide time and sample complexity guarantees for a range of families of circuits of interest, and further numerically demonstrate our simulation algorithms on an exactly verifiable simulation of a Hamiltonian variational Ansatz and long-time dynamics simulation on a 127-qubit heavy-hex topology.

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Popular Summary

Finding problems in science which are worthwhile to solve with quantum computers is a formidable challenge. It is yet unclear whether sufficiently accurate simulation of systems of interest, such as molecular dynamics and exotic materials, will even necessitate a quantum computer. This is because as experimentalists and algorithm designers devise techniques to simulate a widening pool of systems, theorists work closely behind them to classically simulate and dequantize their methods. The battle is well-intentioned; reducing or removing the quantum components of these algorithms expedites their real-world utility.

To this end, we present a “quantum-enhanced” classical method for surrogating subregions of the expectation landscape generated by a generic parametrized quantum circuit. That is, after simple measurements on a quantum device, the algorithm allows one to classically simulate approximate expectation values of patches of a landscape. We provide time and sample complexity guarantees for a range of families of circuits of interest, and further numerically demonstrate our simulation algorithms on an exactly verifiable simulation of a Hamiltonian variational Ansatz and long-time dynamics simulation on a 127-qubit heavy-hex topology. Our results can be applied to a wide range of quantum domains from variational quantum algorithms to dynamical simulation and quantum metrology.

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References (115)

  1. Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. Van Den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme et al., Evidence for the utility of quantum computing before fault tolerance, Nature 618, 500 (2023).
  2. M. S. Rudolph, E. Fontana, Z. Holmes, and L. Cincio, Classical surrogate simulation of quantum systems with LOWESA, arXiv:2308.09109.
  3. J. Tindall, M. Fishman, E. M. Stoudenmire, and D. Sels, Efficient tensor network simulation of IBM’s Eagle kicked Ising experiment, PRX Quantum 5, 010308 (2024).
  4. T. Begušić, J. Gray, and G. K.-L. Chan, Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance, Sci. Adv. 10, eadk4321 (2024).
  5. E. G. D. Torre and M. M. Roses, Dissipative mean-field theory of IBM utility experiment, arXiv:2308.01339.
  6. H.-J. Liao, K. Wang, Z.-S. Zhou, P. Zhang, and T. Xiang, Simulation of IBM’s kicked Ising experiment with projected entangled pair operator, arXiv:2308.03082.
  7. S. J. Thomson and J. Eisert, Unravelling quantum dynamics using flow equations, Nat. Phys. 20, 1401 (2024).
  8. M. Cerezo, M. Larocca, D. García-Martín, N. L. Diaz, P. Braccia, E. Fontana, M. S. Rudolph, P. Bermejo, A. Ijaz, S. Thanasilp et al., Does provable absence of barren plateaus imply classical simulability? Nat. Commun. 16, 7907 (2025).
  9. N. Dowling, P. Kos, and X. Turkeshi, Magic resources of the Heisenberg picture, Phys. Rev. Lett. 135, 050401 (2025).
  10. F. J. Schreiber, J. Eisert, and J. J. Meyer, Classical surrogates for quantum learning models, Phys. Rev. Lett. 131, 100803 (2023).
  11. S. Jerbi, C. Gyurik, S. C. Marshall, R. Molteni, and V. Dunjko, Shadows of quantum machine learning, Nat. Commun. 15, 5676 (2024).
  12. J. Landman, S. Thabet, C. Dalyac, H. Mhiri, and E. Kashefi, Classically approximating variational quantum machine learning with random Fourier features, arXiv:2210.13200.
  13. A. Angrisani, Learning unitaries with quantum statistical queries, Quantum 9, 1817 (2025).
  14. E. Fontana, M. S. Rudolph, R. Duncan, I. Rungger, and C. Cîrstoiu, Classical simulations of noisy variational quantum circuits, npj Quantum Inf. 11, 1 (2025).
  15. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  16. K. Bharti, T. Haug, V. Vedral, and L.-C. Kwek, Noisy intermediate-scale quantum algorithm for semidefinite programming, Phys. Rev. A 105, 052445 (2022).
  17. A. Basheer, Y. Feng, C. Ferrie, and S. Li, Alternating layered variational quantum circuits can be classically optimized efficiently using classical shadows, arXiv:2208.11623.
  18. S. Jerbi, J. Gibbs, M. S. Rudolph, M. C. Caro, P. J. Coles, H.-Y. Huang, and Z. Holmes, The power and limitations of learning quantum dynamics incoherently, arXiv:2303.12834.
  19. A. Basheer, Y. Feng, C. Ferrie, and S. Li, Ansatz-agnostic exponential resource saving in variational quantum algorithms using shallow shadows, arXiv:2309.04754.
  20. R. Shaffer, L. Kocia, and M. Sarovar, Surrogate-based optimization for variational quantum algorithms, Phys. Rev. A 107, 032415 (2023).
  21. R. D. Somma, Quantum computation, complexity, and many-body physics, arXiv:quant-ph/0512209.
  22. R. Somma, H. Barnum, G. Ortiz, and E. Knill, Efficient solvability of Hamiltonians and limits on the power of some quantum computational models, Phys. Rev. Lett. 97, 190501 (2006).
  23. V. Galitski, Quantum-to-classical correspondence and Hubbard-Stratonovich dynamical systems: A Lie-algebraic approach, Phys. Rev. A 84, 012118 (2011).
  24. M. L. Goh, M. Larocca, L. Cincio, M. Cerezo, and F. Sauvage, Lie-algebraic classical simulations for quantum computing, Phys. Rev. Res. 7, 033266 (2025).
  25. E. R. Anschuetz, A. Bauer, B. T. Kiani, and S. Lloyd, Efficient classical algorithms for simulating symmetric quantum systems, Quantum 7, 1189 (2023).
  26. A. Angrisani, A. Schmidhuber, M. S. Rudolph, M. Cerezo, Z. Holmes, and H.-Y. Huang, Classically estimating observables of noiseless quantum circuits, Phys. Rev. Lett. 135, 170602 (2025).
  27. P. Bermejo, P. Braccia, M. S. Rudolph, Z. Holmes, L. Cincio, and M. Cerezo, Quantum convolutional neural networks are (effectively) classically simulable, PRX Quantum 7, 020304 (2026).
  28. A. W. R. Smith, A. J. Paige, and M. S. Kim, Faster variational quantum algorithms with quantum kernel-based surrogate models, Quantum Sci. Technol. 8, 045016 (2023).
  29. A. A. Mele, A. Angrisani, S. Ghosh, S. Khatri, J. Eisert, D. Stilck França, and Y. Quek, Noise-induced shallow circuits and absence of barren plateaus, arXiv:2403.13927.
  30. R. Sweke, E. Recio, S. Jerbi, E. Gil-Fuster, B. Fuller, J. Eisert, and J. J. Meyer, Potential and limitations of random Fourier features for dequantizing quantum machine learning, Quantum 9, 1640 (2025).
  31. B. Koczor and S. C. Benjamin, Quantum analytic descent, Phys. Rev. Res. 4, 023017 (2022).
  32. E. J. Gustafson, J. Tiihonen, D. Chamaki, F. Sorourifar, J. W. Mullinax, A. C. Li, F. B. Maciejewski, N. P. Sawaya, J. T. Krogel, D. E. B. Neira et al., Surrogate optimization of variational quantum circuits, arXiv:2404.02951.
  33. K. Zhang, L. Liu, M.-H. Hsieh, and D. Tao, Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits, in Advances in Neural Information Processing Systems (2022), Vol. 35, pp. 18612–18627.
  34. C.-Y. Park and N. Killoran, Hamiltonian variational ansatz without barren plateaus, Quantum 8, 1239 (2024).
  35. Y. Wang, B. Qi, C. Ferrie, and D. Dong, Trainability enhancement of parameterized quantum circuits via reduced-domain parameter initialization, Phys. Rev. Appl. 22, 054005 (2024).
  36. C.-Y. Park, M. Kang, and J. Huh, Hardware-efficient ansatz without barren plateaus in any depth, arXiv:2403.04844.
  37. X. Shi and Y. Shang, Avoiding barren plateaus via Gaussian mixture model, arXiv:2402.13501.
  38. R. Puig, M. Drudis, S. Thanasilp, and Z. Holmes, Variational quantum simulation: A case study for understanding warm starts, PRX Quantum 6, 010317 (2025).
  39. H. Mhiri, L. Monbroussou, M. Herrero-Gonzalez, S. Thabet, E. Kashefi, and J. Landman, Constrained and vanishing expressivity of quantum Fourier models, Quantum 9, 1847 (2025).
  40. R. Puig, B. Casas, A. Cervera-Lierta, Z. Holmes, and A. Pérez-Salinas, Warm starts, cold states: Exploiting adiabaticity for variational ground-states, arXiv:2602.06137.
  41. H. Mhiri, R. Puig, S. Lerch, M. S. Rudolph, T. Chotibut, S. Thanasilp, and Z. Holmes, A general framework for analyzing warm starts in variational quantum computing (to be published).
  42. S. Lerch, J. Bowles, R. Puig, E. Armengol, Z. Holmes, and S. Thanasilp, IQP born machines under data-dependent and agnostic initialization strategies, arXiv:2603.14576.
  43. C. Huerta Alderete, M. H. Gordon, F. Sauvage, A. Sone, A. T. Sornborger, P. J. Coles, and M. Cerezo, Inference-based quantum sensing, Phys. Rev. Lett. 129, 190501 (2022).
  44. A. Ijaz, C. H. Alderete, F. Sauvage, L. Cincio, M. Cerezo, and M. L. Goh, More buck-per-shot: Why learning trumps mitigation in noisy quantum sensing, Mater. Today Quantum 6, 100042 (2025).
  45. M. Cerezo, K. Sharma, A. Arrasmith, and P. J. Coles, Variational quantum state eigensolver, npj Quantum Inf. 8, 1 (2022).
  46. C. Cirstoiu, Z. Holmes, J. Iosue, L. Cincio, P. J. Coles, and A. Sornborger, Variational fast forwarding for quantum simulation beyond the coherence time, npj Quantum Inf. 6, 1 (2020).
  47. P. Rall, D. Liang, J. Cook, and W. Kretschmer, Simulation of qubit quantum circuits via pauli propagation, Phys. Rev. A 99, 062337 (2019).
  48. D. Aharonov, X. Gao, Z. Landau, Y. Liu, and U. Vazirani, A polynomial-time classical algorithm for noisy random circuit sampling, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing (2023), p. 945.
  49. N. A. Nemkov, E. O. Kiktenko, and A. K. Fedorov, Fourier expansion in variational quantum algorithms, Phys. Rev. A 108, 032406 (2023).
  50. T. Begušić, K. Hejazi, and G. K. Chan, Simulating quantum circuit expectation values by Clifford perturbation theory, J. Chem. Phys. 162, 154110 (2025).
  51. Y. Shao, F. Wei, S. Cheng, and Z. Liu, Simulating noisy variational quantum algorithms: A polynomial approach, Phys. Rev. Lett. 133, 120603 (2024).
  52. T. Schuster, C. Yin, X. Gao, and N. Y. Yao, A polynomial-time classical algorithm for noisy quantum circuits, Phys. Rev. X 15, 041018 (2025).
  53. G. González-García, J. I. Cirac, and R. Trivedi, Pauli path simulations of noisy quantum circuits beyond average case, Quantum 9, 1730 (2025).
  54. M. S. Rudolph, A. Angrisani, A. Wright, I. Sanderski, R. Puig, and Z. Holmes, Thermal state simulation with Pauli and Majorana propagation, arXiv:2602.04878.
  55. M. S. Rudolph, T. Jones, Y. Teng, A. Angrisani, and Z. Holmes, Pauli propagation: A computational framework for simulating quantum systems, arXiv:2505.21606.
  56. M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2025).
  57. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 1 (2018).
  58. A. Arrasmith, Z. Holmes, M. Cerezo, and P. J. Coles, Equivalence of quantum barren plateaus to cost concentration and narrow gorges, Quantum Sci. Technol. 7, 045015 (2022).
  59. M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. Ortiz Marrero, M. Larocca, and M. Cerezo, A lie algebraic theory of barren plateaus for deep parameterized quantum circuits, Nat. Commun. 15, 7172 (2024).
  60. E. Fontana, D. Herman, S. Chakrabarti, N. Kumar, R. Yalovetzky, J. Heredge, S. H. Sureshbabu, and M. Pistoia, Characterizing barren plateaus in quantum ansätze with the adjoint representation, Nat. Commun. 15, 7171 (2024).
  61. C. Ortiz Marrero, M. Kieferová, and N. Wiebe, Entanglement-induced barren plateaus, PRX Quantum 2, 040316 (2021).
  62. K. Sharma, M. Cerezo, L. Cincio, and P. J. Coles, Trainability of dissipative perceptron-based quantum neural networks, Phys. Rev. Lett. 128, 180505 (2022).
  63. T. L. Patti, K. Najafi, X. Gao, and S. F. Yelin, Entanglement devised barren plateau mitigation, Phys. Rev. Res. 3, 033090 (2021).
  64. S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Noise-induced barren plateaus in variational quantum algorithms, Nat. Commun. 12, 1 (2021).
  65. M. Larocca, P. Czarnik, K. Sharma, G. Muraleedharan, P. J. Coles, and M. Cerezo, Diagnosing barren plateaus with tools from quantum optimal control, Quantum 6, 824 (2022).
  66. Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum 3, 010313 (2022).
  67. M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nat. Commun. 12, 1 (2021b).
  68. Z. Holmes, A. Arrasmith, B. Yan, P. J. Coles, A. Albrecht, and A. T. Sornborger, Barren plateaus preclude learning scramblers, Phys. Rev. Lett. 126, 190501 (2021).
  69. E. C. Martín, K. Plekhanov, and M. Lubasch, Barren plateaus in quantum tensor network optimization, Quantum 7, 974 (2023).
  70. G. Crognaletti, M. Grossi, and A. Bassi, Estimates of loss function concentration in noisy parametrized quantum circuits, arXiv:2410.01893.
  71. E. R. Anschuetz, A Unified Theory of Quantum Neural Network Loss Landscapes, in International Conference on Learning Representations, edited by Y. Yue, A. Garg, N. Peng, F. Sha and R. Yu (2025), pp. 97859–97918, https://proceedings.iclr.cc/paper_files/paper/2025/file/f3680ca31afcb9076846950dbc72c7e3-Paper-Conference.pdf.
  72. K. Srimahajariyapong, S. Thanasilp, and T. Chotibut, Connecting phases of matter to the flatness of the loss landscape in analog variational quantum algorithms, arXiv:2506.13865.
  73. M. S. Rudolph, S. Lerch, S. Thanasilp, O. Kiss, O. Shaya, S. Vallecorsa, M. Grossi, and Z. Holmes, Trainability barriers and opportunities in quantum generative modeling, npj Quantum Inf. 10, 116 (2024).
  74. M. Kieferova, O. M. Carlos, and N. Wiebe, Quantum generative training using Rényi divergences, arXiv:2106.09567.
  75. J. Tangpanitanon, S. Thanasilp, N. Dangniam, M.-A. Lemonde, and D. G. Angelakis, Expressibility and trainability of parametrized analog quantum systems for machine learning applications, Phys. Rev. Res. 2, 043364 (2020).
  76. S. Thanaslip, S. Wang, N. A. Nghiem, P. J. Coles, and M. Cerezo, Subtleties in the trainability of quantum machine learning models, Quantum Mach. Intell. 5, 21 (2023).
  77. S. Thanasilp, S. Wang, M. Cerezo, and Z. Holmes, Exponential concentration in quantum kernel methods, Nat. Commun. 15, 5200 (2024).
  78. A. Letcher, S. Woerner, and C. Zoufal, Tight and efficient gradient bounds for parameterized quantum circuits, Quantum 8, 1484 (2024)
  79. S. Y. Chang, S. Thanasilp, B. L. Saux, S. Vallecorsa, and M. Grossi, Latent style-based quantum GAN for high-quality image generation, arXiv:2406.02668.
  80. W. Xiong, G. Facelli, M. Sahebi, O. Agnel, T. Chotibut, S. Thanasilp, and Z. Holmes, On fundamental aspects of quantum extreme learning machines, Quantum Mach. Intell. 7, 20 (2025).
  81. W. Xiong, Z. Holmes, A. Angrisani, Y. Suzuki, T. Chotibut, and S. Thanasilp, Role of scrambling and noise in temporal information processing with quantum systems, arXiv:2505.10080.
  82. R. Aghaei Saem, B. Tafreshi, Z. Holmes, and S. Thanasilp, Pitfalls when tackling the exponential concentration of parameterized quantum models, Quantum Sci. Technol. 11, 015049 (2026).
  83. C. Cao, Y. Zhou, S. Tannu, N. Shannon, and R. Joynt, Exploiting many-body localization for scalable variational quantum simulation, Quantum 9, 1942 (2025).
  84. L. Xin and Z.-Q. Yin, Improve variational quantum eigensolver by many-body localization, Front. Phys. 20, 023202 (2025).
  85. A. A. Mele, G. B. Mbeng, G. E. Santoro, M. Collura, and P. Torta, Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz, Phys. Rev. A 106, L060401 (2022).
  86. A. Miller, J. Favre, Z. Holmes, Ö. Salehi, R. Chakraborty, A. Nykänen, Z. Zimboras, A. Glos, and G. García-Pérez, Simulation of fermionic circuits using majorana propagation, arXiv:2503.18939.
  87. A. Angrisani, A. A. Mele, M. S. Rudolph, M. Cerezo, and Z. Holmes, Simulating quantum circuits with arbitrary local noise using pauli propagation, arXiv:2501.13101.
  88. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  89. We note that surrogation beyond r∈O(1/m) is also achievable with low error, by extending the truncation order κ to be polynomial. This, in turn, increases the computational resources required to be super-polynomial or even exponential with m. Detailed derivations for various truncations are provided in Appendix 15g. Furthermore, these techniques could be adapted to other geometries. Although a rigorous generalization requires more technical work, Fig. 3 offers evidence that the results of Theorem 1 remain valid.
  90. J. M. Arrazola, O. Di Matteo, N. Quesada, S. Jahangiri, A. Delgado, and N. Killoran, Universal quantum circuits for quantum chemistry, Quantum 6, 742 (2022).
  91. D. Wierichs, J. Izaac, C. Wang, and C. Y.-Y. Lin, General parameter-shift rules for quantum gradients, Quantum 6, 677 (2022).
  92. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
  93. A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The randomized measurement toolbox, Nat. Rev. Phys. 5, 9 (2023).
  94. M. West, A. A. Mele, M. Larocca, and M. Cerezo, Real classical shadows, arXiv:2410.23481.
  95. D. Wecker, M. B. Hastings, and M. Troyer, Progress towards practical quantum variational algorithms, Phys. Rev. A 92, 042303 (2015).
  96. A. Miessen, D. J. Egger, I. Tavernelli, and G. Mazzola, Benchmarking digital quantum simulations above hundreds of qubits using quantum critical dynamics, PRX Quantum 5, 040320 (2024).
  97. W. H. Zurek, U. Dorner, and P. Zoller, Dynamics of a quantum phase transition, Phys. Rev. Lett. 95, 105701 (2005).
  98. T. Kibble, Some implications of a cosmological phase transition, Phys. Rep. 67, 183 (1980).
  99. W. H. Zurek, Cosmological experiments in superfluid helium? Nature 317, 505 (1985).
  100. A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Nature 607, 667 (2022).
  101. N. Herrmann, D. Arya, M. W. Doherty, A. Mingare, J. C. Pillay, F. Preis, and S. Prestel, Quantum utility–definition and assessment of a practical quantum advantage, in 2023 IEEE International Conference on Quantum Software (QSW) (IEEE, 2023), p. 162.
  102. R. Horst, P. M. Pardalos, and N. Van Thoai, Introduction to Global Optimization (Springer Science & Business Media, Springer New York, NY, 2000).
  103. B. Koczor and S. C. Benjamin, Quantum analytic descent, Phys. Rev. Res. 4, 023017 (2022).
  104. D. P. Dubhashi and A. Panconesi, Concentration of Measure for the Analysis of Randomized Algorithms (Cambridge University Press, Cambridge, UK, 2009).
  105. W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Am. Stat. Assoc. 58, 13 (1963).
  106. P. Billingsley, Probability and Measure, Wiley Series in Probability and Statistics (Wiley, 1995).
  107. A. Papoulis and S. Pillai, Probability, Random Variables, and Stochastic Processes, McGraw-Hill Series in Electrical and Computer Engineering (McGraw-Hill, 2002).
  108. M. Protter and C. Morrey, Intermediate Calculus, Undergraduate Texts in Mathematics (Springer, New York, 2012).
  109. J. J. Duistermaat and J. A. C. Kolk, in Distributions: Theory and Applications (Birkhäuser Boston, Boston, 2010), p. 59.
  110. A. Arrasmith, L. Cincio, R. D. Somma, and P. J. Coles, Operator sampling for shot-frugal optimization in variational algorithms, arXiv:2004.06252.
  111. N. C. Rubin, R. Babbush, and J. McClean, Application of fermionic marginal constraints to hybrid quantum algorithms, New J. Phys. 20, 053020 (2018).
  112. M. Cerezo and P. J. Coles, Higher order derivatives of quantum neural networks with barren plateaus, Quantum Sci. Technol. 6, 035006 (2021).
  113. We do not rule out that others using a different proof strategy could not prove guarantees in a larger region [112], although we think this unlikely.
  114. D. Aharonov, X. Gao, Z. Landau, Y. Liu, and U. Vazirani, A polynomial-time classical algorithm for noisy random circuit sampling, in Proceedings of the 55th Annual ACM Symposium on Theory of Computing (2023), pp. 945–957.
  115. D. Gottesman, Group22: Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, edited by S. P. Corney, R. Delbourgo, and P. D. Jarvis (Cambridge, MA, International Press, 1999), pp. 32–43.

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PRX Quantum 7, 020359– Published 12 June, 2026
  • Received 24 December 2024
  • Revised 22 December 2025
  • Accepted 13 April 2026
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