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Scalability challenges in variational quantum optimization under stochastic noise

Adelina Bärligea1,2,*, Benedikt Poggel1, and Jeanette Miriam Lorenz1,3

  • *Contact author: adelina.baerligea@tum.de
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Phys. Rev. A 112, 032407 – Published 3 September, 2025

DOI: https://doi.org/10.1103/rgyh-8xw8

Abstract

With rapid advances in quantum hardware, a central question is whether quantum devices with or without full error correction can outperform classical computers on practically relevant problems. Variational quantum algorithms (VQAs) have gained significant attention as promising candidates in this pursuit, particularly for combinatorial optimization problems. While reports of their challenges and limitations continue to accumulate, many studies remain optimistic based on small-scale, idealized testing setups, leaving doubt about the scalability of VQAs for large-scale problems. We systematically investigate this scaling behavior by analyzing how classical optimizers minimize variational quantum loss functions for random Quadratic Unconstrained Binary Optimization instances in the presence of uncertainty, modeled as effective Gaussian noise. We find that the critical noise threshold for successful optimization decreases rapidly as system size grows. This decline exceeds what can be explained solely by shrinking loss variance, confirming deeper, fundamental limitations in the loss landscapes of VQAs beyond barren plateaus. Translating these thresholds into required measurement shots reveals that achieving sufficient precision in the evaluated loss values quickly becomes impractical, even for moderatelysized problems. Our findings demonstrate serious scalability challenges for VQAs in optimization stemming from mere uncertainty, indicating potential barriers to achieving practical quantum advantage with current hybrid approaches.

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References (136)

  1. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, and D. A. Buell et al., Quantum supremacy using a programmable superconducting processor, Nature (London) 574, 505 (2019).
  2. C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Appl. Phys. Rev. 6, 021314 (2019).
  3. H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, and Y. Hu et al., Quantum computational advantage using photons, Science 370, 1460 (2020).
  4. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  5. J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New J. Phys. 18, 023023 (2016).
  6. 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).
  7. K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, and T. Menke et al., Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
  8. A. Abbas, A. Ambainis, B. Augustino, A. Bärtschi, H. Buhrman, C. Coffrin, G. Cortiana, V. Dunjko, D. J. Egger, and B. G. Elmegreen et al., Challenges and opportunities in quantum optimization, Nat. Rev. Phys. 6, 718 (2024).
  9. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
  10. M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Biamonte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, A review of barren plateaus in variational quantum computing, Nat. Rev. Phys. 7, 174 (2025).
  11. M. Cerezo, M. Larocca, D. García-Martín, N. L. Diaz, P. Braccia, E. Fontana, M. S. Rudolph, P. Bermejo, A. Ijaz, S. Thanasilp, E. R. Anschuetz, and Z. Holmes, Does provable absence of barren plateaus imply classical simulability? Or, why we need to rethink variational quantum computing, arXiv:2312.09121.
  12. M. L. Goh, M. Larocca, L. Cincio, M. Cerezo, and F. Sauvage, Lie-algebraic classical simulations for quantum computing, arXiv:2308.01432.
  13. L. Bittel and M. Kliesch, Training variational quantum algorithms is NP-hard, Phys. Rev. Lett. 127, 120502 (2021).
  14. E. R. Anschuetz and B. T. Kiani, Quantum variational algorithms are swamped with traps, Nat. Commun. 13, 7760 (2022).
  15. F. Leymann and J. Barzen, The bitter truth about gate-based quantum algorithms in the NISQ era, Quantum Sci. Technol. 5, 044007 (2020).
  16. 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, 6961 (2021).
  17. G. González-García, R. Trivedi, and J. I. Cirac, Error propagation in NISQ devices for solving classical optimization problems, PRX Quantum 3, 040326 (2022).
  18. G. Scriva, N. Astrakhantsev, S. Pilati, and G. Mazzola, Challenges of variational quantum optimization with measurement shot noise, Phys. Rev. A 109, 032408 (2024).
  19. D. S. França and R. García-Patrón, Limitations of optimization algorithms on noisy quantum devices, Nat. Phys. 17, 1221 (2021).
  20. Y. Shao, F. Wei, S. Cheng, and Z. Liu, Simulating noisy variational quantum algorithms: A polynomial approach, Phys. Rev. Lett. 133, 120603 (2024).
  21. T. Schuster, C. Yin, X. Gao, and N. Y. Yao, A polynomial-time classical algorithm for noisy quantum circuits, arXiv:2407.12768.
  22. V. Akshay, H. Philathong, M. E. S. Morales, and J. D. Biamonte, Reachability deficits in quantum approximate optimization, Phys. Rev. Lett. 124, 090504 (2020).
  23. 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, 1791 (2021).
  24. 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).
  25. 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).
  26. M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. O. Marrero, M. Larocca, and M. Cerezo, A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits, Nat. Commun. 15, 7172 (2024).
  27. S. Bravyi, D. Gosset, and R. Movassagh, Classical algorithms for quantum mean values, Nat. Phys. 17, 337 (2021).
  28. A. Deshpande, P. Niroula, O. Shtanko, A. V. Gorshkov, B. Fefferman, and M. J. Gullans, Tight bounds on the convergence of noisy random circuits to the uniform distribution, PRX Quantum 3, 040329 (2022).
  29. G. De Palma, M. Marvian, C. Rouzé, and D. S. França, Limitations of variational quantum algorithms: A quantum optimal transport approach, PRX Quantum 4, 010309 (2023).
  30. K. M. Nakanishi, K. Fujii, and S. Todo, Sequential minimal optimization for quantum-classical hybrid algorithms, Phys. Rev. Res. 2, 043158 (2020).
  31. K. J. Sung, J. Yao, M. P. Harrigan, N. C. Rubin, Z. Jiang, L. Lin, R. Babbush, and J. R. McClean, Using models to improve optimizers for variational quantum algorithms, Quantum Sci. Technol. 5, 044008 (2020).
  32. W. Lavrijsen, A. Tudor, J. Muller, C. Iancu, and W. de Jong, Classical optimizers for noisy intermediate-scale quantum devices, in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Denver, CO, 2020), pp. 267–277.
  33. A. Pellow-Jarman, I. Sinayskiy, A. Pillay, and F. Petruccione, A comparison of various classical optimizers for a variational quantum linear solver, Quantum Info. Proc. 20, 202 (2021).
  34. V. P. Soloviev, P. Larrañaga, and C. Bielza, Quantum parametric circuit optimization with estimation of distribution algorithms, in Proceedings of the Genetic and Evolutionary Computation Conference Companion, GECCO'22 (ACM, Boston, Massachusetts, 2022), pp. 2247–2250.
  35. H. Singh, S. Majumder, and S. Mishra, Benchmarking of different optimizers in the variational quantum algorithms for applications in quantum chemistry, J. Chem. Phys. 159, 044117 (2023).
  36. L. Palackal, B. Poggel, M. Wulff, H. Ehm, J. M. Lorenz, and C. B. Mendl, Quantum-assisted solution paths for the capacitated vehicle routing problem, in 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Bellevue, WA, 2023), pp. 648–658.
  37. X. Bonet-Monroig, H. Wang, D. Vermetten, B. Senjean, C. Moussa, T. Bäck, V. Dunjko, and T. E. O'Brien, Performance comparison of optimization methods on variational quantum algorithms, Phys. Rev. A 107, 032407 (2023).
  38. A. Pellow-Jarman, S. McFarthing, I. Sinayskiy, D. K. Park, A. Pillay, and F. Petruccione, The effect of classical optimizers and ansatz depth on QAOA performance in noisy devices, Sci. Rep. 14, 16011 (2024).
  39. A. Bärligea, Scalability of variational quantum optimization in the presence of noise, Master's thesis, Technical University of Munich, Garching, Germany, 2024.
  40. V. Kungurtsev, G. Korpas, J. Marecek, and E. Y. Zhu, Iteration complexity of variational quantum algorithms, Quantum 8, 1495 (2024).
  41. B. Korte and J. Vygen, Combinatorial Optimization: Theory and Algorithms (Springer, Berlin, 2018).
  42. R. M. Karp, Reducibility among combinatorial problems, Complexity of Computer Computations, edited by R. E. Miller and J. W. Thatcher, The IBM Research Symposia Series (Springer, New York, 1972), pp. 85–103.
  43. G. Ausiello, A. Marchetti-Spaccamela, P. Crescenzi, G. Gambosi, M. Protasi, and V. Kann, Complexity and Approximation (Springer, Berlin, 1999).
  44. L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC '96 (Association for Computing Machinery, New York, 1996), pp. 212–219.
  45. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2016).
  46. E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Quantum computation by adiabatic evolution, arXiv:quant-ph/0001106.
  47. A. Bochkarev, R. Heese, S. Jäger, P. Schiewe, and A. Schöbel, Quantum computing for discrete optimization: A highlight of three technologies, arXiv:2409.01373.
  48. S. Heng, D. Kim, T. Kim, and Y. Han, How to solve combinatorial optimization problems using real quantum machines: A recent survey, IEEE Access 10, 120106 (2022).
  49. E. Zahedinejad and A. Zaribafiyan, Combinatorial optimization on gate model quantum computers: A survey, arXiv:1708.05294.
  50. K. Sankar, A. Scherer, S. Kako, S. Reifenstein, N. Ghadermarzy, W. B. Krayenhoff, Y. Inui, E. Ng, T. Onodera, P. Ronagh, and Y. Yamamoto, A benchmarking study of quantum algorithms for combinatorial optimization, npj Quantum Inf. 10, 64 (2024).
  51. M. Dupont, B. Sundar, B. Evert, D. E. B. Neira, Z. Peng, S. Jeffrey, and M. J. Hodson, Benchmarking quantum optimization for the maximum-cut problem on a superconducting quantum computer, Phys. Rev. Appl. 23, 014045 (2025).
  52. Y. Fu and P. W. Anderson, Application of statistical mechanics to NP-complete problems in combinatorial optimisation, J. Phys. A: Math. Gen. 19, 1605 (1986).
  53. G. Kochenberger, J.-K. Hao, F. Glover, M. Lewis, Z. Lü, H. Wang, and Y. Wang, The unconstrained binary quadratic programming problem: A survey, J. Comb. Optim. 28, 58 (2014).
  54. F. Glover, G. Kochenberger, R. Hennig, and Y. Du, Quantum bridge analytics I: A tutorial on formulating and using QUBO models, Ann. Oper. Res. 314, 141 (2022).
  55. F. Glover, G. Kochenberger, and Y. Du, Applications and computational advances for solving the QUBO model, in The Quadratic Unconstrained Binary Optimization Problem: Theory, Algorithms, and Applications, edited by A. P. Punnen (Springer International Publishing, Cham, 2022), pp. 39–56.
  56. F. Barahona, On the computational complexity of Ising spin glass models, J. Phys. A: Math. Gen. 15, 3241 (1982).
  57. A. Lucas, Ising formulations of many NP problems, Front. Phys. 2, 5 (2014).
  58. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O'Brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 4213 (2014).
  59. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  60. N. Moll, P. Barkoutsos, L. S. Bishop, J. M. Chow, A. Cross, D. J. Egger, S. Filipp, A. Fuhrer, J. M. Gambetta, M. Ganzhorn, A. Kandala, A. Mezzacapo, P. Müller, W. Riess, G. Salis, J. Smolin, I. Tavernelli, and K. Temme, Quantum optimization using variational algorithms on near-term quantum devices, Quantum Sci. Technol. 3, 030503 (2018).
  61. G. Nannicini, Performance of hybrid quantum-classical variational heuristics for combinatorial optimization, Phys. Rev. E 99, 013304 (2019).
  62. J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, and J. Tennyson, The variational quantum eigensolver: A review of methods and best practices, Phys. Rep. 986, 1 (2022).
  63. K. Blekos, D. Brand, A. Ceschini, C.-H. Chou, R.-H. Li, K. Pandya, and A. Summer, A review on quantum approximate optimization algorithm and its variants, Phys. Rep. 1068, 1 (2024).
  64. N. Kuete Meli, F. Mannel, and J. Lellmann, A universal quantum algorithm for weighted maximum cut and Ising problems, Quantum Info. Proc. 22, 279 (2023).
  65. A. Bärligea, B. Poggel, and J. M. Lorenz, Solving combinatorial optimization problems with a block encoding quantum optimizer, arXiv:2404.14054.
  66. J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, Grand unification of quantum algorithms, PRX Quantum 2, 040203 (2021).
  67. D. Aharonov, V. Jones, and Z. Landau, A polynomial quantum algorithm for approximating the Jones polynomial, Algorithmica 55, 395 (2009).
  68. J. Rajakumar, J. Golden, A. Bärtschi, and S. Eidenbenz, Trainability barriers in low-depth QAOA landscapes, in Proceedings of the 21st ACM International Conference on Computing Frontiers, CF'24 (ACM, Ischia, Italy, 2024), pp. 199–206.
  69. This is a conservative assumption on parameter scaling, as larger problem instances typically require deeper circuits with more parameters, leading to higher-dimensional loss landscapes and potentially better optimization conditions [136]. Fixing this relationship therefore ensures a fair scalability comparison without biases from varying parameter growth.
  70. R. Wiersema, E. Kökcü, A. F. Kemper, and B. N. Bakalov, Classification of dynamical Lie algebras of 2-local spin systems on linear, circular and fully connected topologies, npj Quantum Inf. 10, 110 (2024).
  71. E. Kökcü, R. Wiersema, A. F. Kemper, and B. N. Bakalov, Classification of dynamical Lie algebras generated by spin interactions on undirected graphs, arXiv:2409.19797.
  72. A. Javadi-Abhari, M. Treinish, K. Krsulich, C. J. Wood, J. Lishman, J. Gacon, S. Martiel, P. D. Nation, L. S. Bishop, A. W. Cross, B. R. Johnson, and J. M. Gambetta, Quantum computing with Qiskit, arXiv:2405.08810.
  73. P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, and J. Bright et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  74. R. Murray, B. Swenson, and S. Kar, Revisiting normalized gradient descent: Fast evasion of saddle points, IEEE Trans. Autom. Control 64, 4818 (2019).
  75. Y. Suzuki, H. Yano, R. Raymond, and N. Yamamoto, Normalized gradient descent for variational quantum algorithms, in 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Broomfield, CO, 2021), pp. 1–9.
  76. R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Comput. 16, 1190 (1995).
  77. R. Fletcher, Function minimization by conjugate gradients, Comput. J. 7, 149 (1964).
  78. D. Kraft, A software package for sequential quadratic programming (1988), https://degenerateconic.com/uploads/2018/03/DFVLR_FB_88_28.pdf.
  79. J. C. Spall, Multivariate stochastic approximation using a simultaneous perturbation gradient approximation, IEEE Trans. Autom. Control 37, 332 (1992).
  80. J. C. Spall, An overview of the simultaneous perturbation method for efficient optimization, Johns Hopkins APL Tech. Dig. 19, 482 (1998).
  81. M. J. D. Powell, Direct search algorithms for optimization calculations, Acta Numer. 7, 287 (1998).
  82. M. J. D. Powell, A direct search optimization method that models the objective and constraint functions by linear interpolation, in Advances in Optimization and Numerical Analysis (Springer Netherlands, Amsterdam, 1994), pp. 51–67.
  83. M. J. D. Powell, An efficient method for finding the minimum of a function of several variables without calculating derivatives, Comput. J. 7, 155 (1964).
  84. K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).
  85. M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Evaluating analytic gradients on quantum hardware, Phys. Rev. A 99, 032331 (2019).
  86. 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, edited by S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Curran Associates, New Orleans, LA, 2022), Vol. 35, pp. 18612–18627.
  87. J. Branke and C. Schmidt, Selection in the presence of noise, in Genetic and Evolutionary Computation—GECCO 2003 (Springer, Berlin, 2003), pp. 766–777.
  88. S. Stich and H. Harshvardhan, Escaping local minima with stochastic noise, in OPT2021: 13th Annual Workshop on Optimization for Machine Learning (NeurIPS, San Diego, CA, 2021), pp. 18612–18627.
  89. J. Liu, F. Wilde, A. A. Mele, L. Jiang, and J. Eisert, Stochastic noise can be helpful for variational quantum algorithms, Phys. Rev. A 111, 052441 (2025).
  90. E. Kaminishi, T. Mori, M. Sugawara, and N. Yamamoto, Impact of measurement noise on escaping saddles in variational quantum algorithms, arXiv:2406.09780.
  91. E. Fontana, M. Cerezo, A. Arrasmith, I. Rungger, and P. J. Coles, Non-trivial symmetries in quantum landscapes and their resilience to quantum noise, Quantum 6, 804 (2022).
  92. J. Kim and Y. Oz, Quantum energy landscape and circuit optimization, Phys. Rev. A 106, 052424 (2022).
  93. E. Fontana, N. Fitzpatrick, D. M. Ramo, R. Duncan, and I. Rungger, Evaluating the noise resilience of variational quantum algorithms, Phys. Rev. A 104, 022403 (2021).
  94. O. Ezratty, Where are we heading with NISQ? arXiv:2305.09518.
  95. I. Quantum, Development & innovation roadmap (2024), https://www.ibm.com/quantum/assets/IBM_Quantum_Development_&_Innovation_Roadmap.pdf.
  96. Quantinuum, Quantinuum Unveils Accelerated Roadmap to Achieve Universal, Fully Fault-Tolerant Quantum Computing by 2030 (2024), https://www.quantinuum.com/press-releases/quantinuum-unveils-accelerated-roadmap-to-achieve-universal-fault-tolerant-quantum-computing-by-2030.
  97. Using an exponential fit for the loss variance—consistent with a barren plateau assumption—would imply an increase in the noise resilience for system sizes n≥61 in the power-law case (31), and n≥27 for logarithmic decay (32). While this would not change the overall implications of Sec. 6b, this scenario is unlikely.
  98. If σ* were to decay logarithmically, ε* would eventually start increasing again due to the faster power-law decay of the loss variance. However, since this effect only becomes relevant for n>e8≈2981, it remains negligible within the scope of our findings.
  99. G. Mazzola and G. Carleo, Exponential challenges in unbiasing quantum monte carlo algorithms with quantum computers, arXiv:2205.09203.
  100. S. Wang, P. Czarnik, A. Arrasmith, M. Cerezo, L. Cincio, and P. J. Coles, Can error mitigation improve trainability of noisy variational quantum algorithms? Quantum 8, 1287 (2024).
  101. D. Applegate, R. E. Bixby, V. Chvátal, and W. J. Cook, Concorde TSP solver (2003), https://www.math.uwaterloo.ca/tsp/concorde/index.html.
  102. D. L. Applegate, R. E. Bixby, V. Chvatal, and W. J. Cook, The Traveling Salesman Problem: A Computational Study, Princeton Series in Applied Mathematics (Princeton University Press, Princeton, 2007).
  103. J. Pawlowski, J. Tuziemski, P. Tarasiuk, A. Przybysz, R. Adamski, K. Hendzel, L. Pawela, and B. Gardas, VeloxQ: A fast and efficient QUBO solver, arXiv:2501.19221.
  104. O. Salehi, A. Glos, and J. A. Miszczak, Unconstrained binary models of the travelling salesman problem variants for quantum optimization, Quantum Info. Proc. 21, 67 (2022).
  105. M. Schnaus, L. Palackal, B. Poggel, X. Runge, H. Ehm, J. M. Lorenz, and C. B. Mendl, Efficient encodings of the travelling salesperson problem for variational quantum algorithms, in 2024 IEEE International Conference on Quantum Software (QSW) (IEEE, Shenzhen, China, 2024), pp. 81–87.
  106. D. J. Egger, J. Mareček, and S. Woerner, Warm-starting quantum optimization, Quantum 5, 479 (2021).
  107. S. H. Sack and M. Serbyn, Quantum annealing initialization of the quantum approximate optimization algorithm, Quantum 5, 491 (2021).
  108. S. Boulebnane and A. Montanaro, Predicting parameters for the quantum approximate optimization algorithm for MAX-CUT from the infinite-size limit, arXiv:2110.10685.
  109. N. Jain, B. Coyle, E. Kashefi, and N. Kumar, Graph neural network initialisation of quantum approximate optimisation, Quantum 6, 861 (2022).
  110. A. N. Cáliz, J. Riu, J. Bosch, P. Torrente, J. Miralles, and A. Riera, A coherent approach to quantum-classical optimization, Commun. Phys. 8, 197 (2025).
  111. E. Grant, L. Wossnig, M. Ostaszewski, and M. Benedetti, An initialization strategy for addressing barren plateaus in parametrized quantum circuits, Quantum 3, 214 (2019).
  112. 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).
  113. T. Volkoff and P. J. Coles, Large gradients via correlation in random parameterized quantum circuits, Quantum Sci. Technol. 6, 025008 (2021).
  114. C.-Y. Park, M. Kang, and J. Huh, Hardware-efficient ansatz without barren plateaus in any depth, arXiv:2403.04844.
  115. 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, Chicago, IL, 2023), pp. 162–174.
  116. M. Sciorilli, L. Borges, T. L. Patti, D. García-Martín, G. Camilo, A. Anandkumar, and L. Aolita, Towards large-scale quantum optimization solvers with few qubits, Nat. Commun. 16, 476 (2025).
  117. P. Bermejo and R. Orús, Variational quantum non-orthogonal optimization, Sci. Rep. 13, 9840 (2023).
  118. F. B. Maciejewski, B. G. Bach, M. Dupont, P. A. Lott, B. Sundar, D. E. B. Neira, I. Safro, and D. Venturelli, A multilevel approach for solving large-scale QUBO problems with noisy hybrid quantum approximate optimization, arXiv:2408.07793.
  119. J. B. Larsen, M. D. Grace, A. D. Baczewski, and A. B. Magann, Feedback-based quantum algorithms for ground state preparation, Phys. Rev. Res. 6, 033336 (2024).
  120. J. Jiang, P. Huang, Z. Wu, X. Sun, Z. Guo, W. Huang, L. Zhang, Y. Zhou, J. Zhang, W. Guo, X. Linpeng, S. Liu, W. Ren, Z. Tao, J. Chu, J. Niu, Y. Zhong, and D. Yu, Improving quantum optimization to achieve quadratic time complexity, arXiv:2501.13469.
  121. T. Bennett, L. Noakes, and J. Wang, Non-variational quantum combinatorial optimisation, arXiv:2404.03167.
  122. A. B. Catli, S. Simon, and N. Wiebe, Exponentially better bounds for quantum optimization via dynamical simulation, arXiv:2502.04285.
  123. S. P. Jordan, N. Shutty, M. Wootters, A. Zalcman, A. Schmidhuber, R. King, S. V. Isakov, T. Khattar, and R. Babbush, Optimization by decoded quantum interferometry, arXiv:2408.08292.
  124. P.-W. Huang and P. Rebentrost, Post-variational quantum neural networks, arXiv:2307.10560 [quant-ph].
  125. C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silberhorn, and I. Jex, Gaussian boson sampling, Phys. Rev. Lett. 119, 170501 (2017).
  126. M. Slysz, K. Kurowski, and G. Waligóra, Solving combinatorial optimization problems on a photonic quantum computer, arXiv:2409.13781.
  127. B. Go, C. Oh, and H. Jeong, Quantum computational advantage of noisy boson sampling with partially distinguishable photons, arXiv:2501.13433.
  128. A. Finnila, M. Gomez, C. Sebenik, C. Stenson, and J. Doll, Quantum annealing: A new method for minimizing multidimensional functions, Chem. Phys. Lett. 219, 343 (1994).
  129. S. Yarkoni, E. Raponi, T. Bäck, and S. Schmitt, Quantum annealing for industry applications: Introduction and review, Rep. Prog. Phys. 85, 104001 (2022).
  130. R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. 21, 467 (1982).
  131. S. Lloyd, Universal quantum simulators, Science 273, 1073 (1996).
  132. A. J. Daley, I. Bloch, C. Kokail, S. Flannigan, N. Pearson, M. Troyer, and P. Zoller, Practical quantum advantage in quantum simulation, Nature (London) 607, 667 (2022).
  133. B. Fauseweh, Quantum many-body simulations on digital quantum computers: State-of-the-art and future challenges, Nat. Commun. 15, 2123 (2024).
  134. F. Barahona, M. Jünger, and G. Reinelt, Experiments in quadratic 0-1 programming, Math. Program. 44, 127 (1989).
  135. H. Fischer, A History of the Central Limit Theorem: From Classical to Modern Probability Theory (Springer, New York, 2011).
  136. M. Larocca, N. Ju, D. García-Martín, P. J. Coles, and M. Cerezo, Theory of overparametrization in quantum neural networks, Nat. Comput. Sci. 3, 542 (2023).

Outline

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Phys. Rev. A 112, 032407– Published 3 September, 2025
  • Received 13 March 2025
  • Accepted 29 July 2025
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