Effect of barren plateaus on gradient-free optimization

Andrew Arrasmith; M. Cerezo; Piotr Czarnik; Lukasz Cincio; Patrick J. Coles

doi:10.22331/q-2021-10-05-558

Effect of barren plateaus on gradient-free optimization

Andrew Arrasmith¹, M. Cerezo^1,2, Piotr Czarnik¹, Lukasz Cincio¹, and Patrick J. Coles¹

¹Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
²Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, USA

Published:	2021-10-05, volume 5, page 558
Eprint:	arXiv:2011.12245v2
Doi:	https://doi.org/10.22331/q-2021-10-05-558
Citation:	Quantum 5, 558 (2021).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizers will be significantly affected by barren plateaus. However, whether or not gradient-free optimizers are impacted is a topic of debate, with some arguing that gradient-free approaches are unaffected by barren plateaus. Here we show that, indeed, gradient-free optimizers do not solve the barren plateau problem. Our main result proves that cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau. Hence, without exponential precision, gradient-free optimizers will not make progress in the optimization. We numerically confirm this by training in a barren plateau with several gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show that the numbers of shots required in the optimization grows exponentially with the number of qubits.

Popular summary

Quantum machine learning and variational quantum algorithms provide the potential for near-term, practical applications of quantum computers. However, these techniques require the use of a classical optimization loop that can, in some cases, prove prohibitive. In particular, some of these optimization landscapes present barren plateaus, where the gradient of the cost function is exponentially suppressed except within an exponentially small region. As a result, optimization on a barren plateau landscape requires computational resources that scale exponentially with the number of qubits used.

Perhaps because of the mention of gradients in the definition of a barren plateau, a number of researchers have postulated that a gradient free optimizer might be able to beat the resource scaling. This thought turns out to be incorrect. Here we provide a proof that no gradient free strategy (including models that build surrogate gradients) can escape this resource scaling in barren plateau landscapes.

► BibTeX data

@article{Arrasmith2021effectofbarren,
  doi = {10.22331/q-2021-10-05-558},
  url = {https://doi.org/10.22331/q-2021-10-05-558},
  title = {Effect of barren plateaus on gradient-free optimization},
  author = {Arrasmith, Andrew and Cerezo, M. and Czarnik, Piotr and Cincio, Lukasz and Coles, Patrick J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {558},
  month = oct,
  year = {2021}
}

► References

[1] 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, Nature Reviews Physics 1, 19 (2021a).
https://doi.org/10.1038/s42254-021-00348-9

[2] K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum (nisq) algorithms, arXiv preprint arXiv:2101.08448 (2021).
arXiv:2101.08448

[3] 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, Nature communications 5, 1 (2014).
https://doi.org/10.1038/ncomms5213

[4] J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New Journal of Physics 18, 023023 (2016).
https://doi.org/10.1007/978-94-015-8330-5_4

[5] E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv preprint arXiv:1411.4028 (2014).
arXiv:1411.4028

[6] J. Romero, J. P. Olson, and A. Aspuru-Guzik, Quantum autoencoders for efficient compression of quantum data, Quantum Science and Technology 2, 045001 (2017).
https://doi.org/10.1088/2058-9565/aa8072

[7] S. Khatri, R. LaRose, A. Poremba, L. Cincio, A. T. Sornborger, and P. J. Coles, Quantum-assisted quantum compiling, Quantum 3, 140 (2019).
https://doi.org/10.22331/q-2019-05-13-140

[8] R. LaRose, A. Tikku, É. O'Neel-Judy, L. Cincio, and P. J. Coles, Variational quantum state diagonalization, npj Quantum Information 5, 1 (2019).
https://doi.org/10.1038/s41534-019-0167-6

[9] A. Arrasmith, L. Cincio, A. T. Sornborger, W. H. Zurek, and P. J. Coles, Variational consistent histories as a hybrid algorithm for quantum foundations, Nature communications 10, 1 (2019).
https://doi.org/10.1038/s41467-019-11417-0

[10] M. Cerezo, A. Poremba, L. Cincio, and P. J. Coles, Variational quantum fidelity estimation, Quantum 4, 248 (2020a).
https://doi.org/10.22331/q-2020-03-26-248

[11] 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 Information 6, 1 (2020).
https://doi.org/10.1038/s41534-020-00302-0

[12] K. Sharma, S. Khatri, M. Cerezo, and P. J. Coles, Noise resilience of variational quantum compiling, New Journal of Physics 22, 043006 (2020a).
https://doi.org/10.1088/1367-2630/ab784c

[13] C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P. Coles, Variational quantum linear solver, arXiv preprint arXiv:1909.05820 (2019).
arXiv:1909.05820

[14] M. Cerezo, K. Sharma, A. Arrasmith, and P. J. Coles, Variational quantum state eigensolver, arXiv preprint arXiv:2004.01372 (2020b).
arXiv:2004.01372

[15] M. Schuld, I. Sinayskiy, and F. Petruccione, The quest for a quantum neural network, Quantum Information Processing 13, 2567 (2014).
https://doi.org/10.1007/s11128-014-0809-8

[16] I. Cong, S. Choi, and M. D. Lukin, Quantum convolutional neural networks, Nature Physics 15, 1273 (2019).
https://doi.org/10.1038/s41567-019-0648-8

[17] K. Beer, D. Bondarenko, T. Farrelly, T. J. Osborne, R. Salzmann, D. Scheiermann, and R. Wolf, Training deep quantum neural networks, Nature Communications 11, 808 (2020).
https://doi.org/10.1038/s41467-020-14454-2

[18] G. Verdon, J. Pye, and M. Broughton, A universal training algorithm for quantum deep learning, arXiv preprint arXiv:1806.09729 (2018).
arXiv:1806.09729

[19] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nature communications 9, 1 (2018).
https://doi.org/10.1038/s41467-018-07090-4

[20] M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Cost function dependent barren plateaus in shallow parametrized quantum circuits, Nature Communications 12, 1791 (2021b).
https://doi.org/10.1038/s41467-021-21728-w

[21] K. Sharma, M. Cerezo, L. Cincio, and P. J. Coles, Trainability of dissipative perceptron-based quantum neural networks, arXiv preprint arXiv:2005.12458 (2020b).
arXiv:2005.12458

[22] S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Noise-induced barren plateaus in variational quantum algorithms, arXiv preprint arXiv:2007.14384 (2020).
arXiv:2007.14384

[23] M. Cerezo and P. J. Coles, Higher order derivatives of quantum neural networks with barren plateaus, Quantum Science and Technology 6, 035006 (2021).
https://doi.org/10.1088/2058-9565/abf51a

[24] Z. Holmes, A. Arrasmith, B. Yan, P. J. Coles, A. Albrecht, and A. T. Sornborger, Barren plateaus preclude learning scramblers, Physical Review Letters 126, 190501 (2021a).
https://doi.org/10.1103/PhysRevLett.126.190501

[25] A. Pesah, M. Cerezo, S. Wang, T. Volkoff, A. T. Sornborger, and P. J. Coles, Absence of barren plateaus in quantum convolutional neural networks, arXiv preprint arXiv:2011.02966 (2020).
arXiv:2011.02966

[26] K. Zhang, M.-H. Hsieh, L. Liu, and D. Tao, Toward trainability of quantum neural networks, arXiv preprint arXiv:2011.06258 (2020).
arXiv:2011.06258

[27] A. Abbas, D. Sutter, C. Zoufal, A. Lucchi, A. Figalli, and S. Woerner, The power of quantum neural networks, Nature Computational Science 1, 403 (2021).
https://doi.org/10.1038/s43588-021-00084-1

[28] C. O. Marrero, M. Kieferová, and N. Wiebe, Entanglement induced barren plateaus, arXiv preprint arXiv:2010.15968 (2020).
arXiv:2010.15968

[29] T. L. Patti, K. Najafi, X. Gao, and S. F. Yelin, Entanglement devised barren plateau mitigation, Physical Review Research 3, 033090 (2021).
https://doi.org/10.1103/PhysRevResearch.3.033090

[30] A. Uvarov and J. D. Biamonte, On barren plateaus and cost function locality in variational quantum algorithms, Journal of Physics A: Mathematical and Theoretical 54, 245301 (2021).
https://doi.org/10.1088/1751-8121/abfac7

[31] Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, arXiv preprint arXiv:2101.02138 (2021b).
arXiv:2101.02138

[32] Y. Du, M.-H. Hsieh, T. Liu, S. You, and D. Tao, On the learnability of quantum neural networks, arXiv preprint arXiv:2007.12369 (2020).
arXiv:2007.12369

[33] A. Arrasmith, Z. Holmes, M. Cerezo, and P. J. Coles, Equivalence of quantum barren plateaus to cost concentration and narrow gorges, arXiv preprint arXiv:2104.05868 (2021).
arXiv:2104.05868

[34] G. Verdon, M. Broughton, J. R. McClean, K. J. Sung, R. Babbush, Z. Jiang, H. Neven, and M. Mohseni, Learning to learn with quantum neural networks via classical neural networks, arXiv preprint arXiv:1907.05415 (2019).
arXiv:1907.05415

[35] T. Volkoff and P. J. Coles, Large gradients via correlation in random parameterized quantum circuits, Quantum Science and Technology 6, 025008 (2021).
https://doi.org/10.1088/2058-9565/abd89

[36] A. Skolik, J. R. McClean, M. Mohseni, P. van der Smagt, and M. Leib, Layerwise learning for quantum neural networks, Quantum Machine Intelligence 3, 1 (2021).
https://doi.org/10.1007/s42484-020-00036-4

[37] E. Grant, L. Wossnig, M. Ostaszewski, and M. Benedetti, An initialization strategy for addressing barren plateaus in parametrized quantum circuits, Quantum 3, 214 (2019).
https://doi.org/10.22331/q-2019-12-09-214

[38] E. Campos, A. Nasrallah, and J. Biamonte, Abrupt transitions in variational quantum circuit training, Physical Review A 103, 032607 (2021).
https://doi.org/10.1103/PhysRevA.103.032607

[39] E. Farhi and H. Neven, Classification with quantum neural networks on near term processors, arXiv preprint arXiv:1802.06002 (2018).
arXiv:1802.06002

[40] N. Killoran, T. R. Bromley, J. M. Arrazola, M. Schuld, N. Quesada, and S. Lloyd, Continuous-variable quantum neural networks, Physical Review Research 1, 033063 (2019).
https://doi.org/10.1103/PhysRevResearch.1.033063

[41] J. M. Kübler, A. Arrasmith, L. Cincio, and P. J. Coles, An adaptive optimizer for measurement-frugal variational algorithms, Quantum 4, 263 (2020).
https://doi.org/10.22331/q-2020-05-11-263

[42] R. Sweke, F. Wilde, J. J. Meyer, M. Schuld, P. K. Fährmann, B. Meynard-Piganeau, and J. Eisert, Stochastic gradient descent for hybrid quantum-classical optimization, Quantum 4, 314 (2020).
https://doi.org/10.22331/q-2020-08-31-314

[43] A. Arrasmith, L. Cincio, R. D. Somma, and P. J. Coles, Operator sampling for shot-frugal optimization in variational algorithms, arXiv preprint arXiv:2004.06252 (2020).
arXiv:2004.06252

[44] J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum natural gradient, Quantum 4, 269 (2020).
https://doi.org/10.22331/q-2020-05-25-269

[45] J. A. Nelder and R. Mead, A simplex method for function minimization, The computer journal 7, 308 (1965).
https://doi.org/10.1093/comjnl/7.4.308

[46] R. R. Barton and J. S. Ivey Jr, Modifications of the nelder-mead simplex method for stochastic simulation response optimization, 1991 Winter Simulation Conference Proceedings (1991), 10.1109/WSC.1991.185709.
https://doi.org/10.1109/WSC.1991.185709

[47] X. Huang, Robust simplex algorithm for online optimization, Physical Review Accelerators and Beams 21, 104601 (2018).
https://doi.org/10.1103/PhysRevAccelBeams.21.104601

[48] M. J. Powell, An efficient method for finding the minimum of a function of several variables without calculating derivatives, The computer journal 7, 155 (1964).
https://doi.org/10.1093/comjnl/7.2.155

[49] R. P. Brent, Algorithms for minimization without derivatives (Courier Corporation, 2013).

[50] M. J. Powell, in Advances in optimization and numerical analysis (Springer, 1994) pp. 51–67.
https://doi.org/10.1007/978-94-015-8330-5_4

[51] A. W. Harrow and R. A. Low, Random quantum circuits are approximate 2-designs, Communications in Mathematical Physics 291, 257 (2009).
https://doi.org/10.1007/s00220-009-0873-6

[52] F. G. Brandao, A. W. Harrow, and M. Horodecki, Local random quantum circuits are approximate polynomial-designs, Communications in Mathematical Physics 346, 397 (2016).
https://doi.org/10.1007/s00220-016-2706-8

[53] A. Harrow and S. Mehraban, Approximate unitary $ t $-designs by short random quantum circuits using nearest-neighbor and long-range gates, arXiv preprint arXiv:1809.06957 (2018).
arXiv:1809.06957

[54] J. Močkus, in Optimization techniques IFIP technical conference (Springer, 1975) pp. 400–404.
https://doi.org/10.1007/978-3-662-38527-2_55

[55] A. Anand, M. Degroote, and A. Aspuru-Guzik, Natural evolutionary strategies for variational quantum computation, Machine Learning: Science and Technology (2021), 10.1088/2632-2153/abf3ac.
https://doi.org/10.1088/2632-2153/abf3ac

Cited by

[1] M. Cerezo, Guillaume Verdon, Hsin-Yuan Huang, Lukasz Cincio, and Patrick J. Coles, "Challenges and opportunities in quantum machine learning", Nature Computational Science 2 9, 567 (2022).

[2] Saad Yalouz, Bruno Senjean, Filippo Miatto, and Vedran Dunjko, "Encoding strongly-correlated many-boson wavefunctions on a photonic quantum computer: application to the attractive Bose-Hubbard model", Quantum 5, 572 (2021).

[3] Zhehao Yi, Yanying Liang, and Haozhen Situ, "Enhancing variational quantum circuit training: an improved neural network approach for barren plateau mitigation", Physica Scripta 100 8, 086004 (2025).

[4] Andoni Agirre, Evert van Nieuwenburg, and Matteo M Wauters, "A Monte Carlo Tree Search approach to QAOA: finding a needle in the haystack", New Journal of Physics 27 4, 043014 (2025).

[5] Yuanyang Zhou, Huaxin He, Fengtao Pang, Hao Lyu, Yongping Zhang, and Xi Chen, "Variational quantum compiling for three-qubit-gate design in quantum dots", Physical Review A 111 4, 042616 (2025).

[6] Hiroshi Ohno, "Grover’s search with learning oracle for constrained binary optimization problems", Quantum Machine Intelligence 6 1, 12 (2024).

[7] Syed Masiur Rahman, Omar Hamad Alkhalaf, Md Shafiul Alam, Surya Prakash Tiwari, Md Shafiullah, Sarah Mohammed Al-Judaibi, and Fahad Saleh Al-Ismail, "Climate Change Through Quantum Lens: Computing and Machine Learning", Earth Systems and Environment 8 3, 705 (2024).

[8] Mandaar B. Pande, Lecture Notes in Electrical Engineering 1221, 3 (2025) ISBN:978-981-97-4710-8.

[9] Cedric Gaberle and Manpreet S Jattana, "Slice-wise initial state optimization to improve cost and accuracy of the VQE on lattice models", Physica Scripta 100 12, 125112 (2025).

[10] Lucas Friedrich, Tiago de Souza Farias, and Jonas Maziero, "Barren plateaus are amplified by the dimension of qudits", Quantum Machine Intelligence 7 1, 56 (2025).

[11] Jonathan Wei Zhong Lau, Kian Hwee Lim, Harshank Shrotriya, and Leong Chuan Kwek, "NISQ computing: where are we and where do we go?", AAPPS Bulletin 32 1, 27 (2022).

[12] Giovanni Acampora and Autilia Vitiello, "On the Stability of Local Interpretable Model-Agnostic Explanations for Quantum Classifiers", Machine Learning 115 6, 141 (2026).

[13] Cenk Tüysüz, Giuseppe Clemente, Arianna Crippa, Tobias Hartung, Stefan Kühn, and Karl Jansen, "Classical splitting of parametrized quantum circuits", Quantum Machine Intelligence 5 2, 34 (2023).

[14] Yuelin Li, Yangyang Li, Zhengya Qi, Haorui Yang, Ronghua Shang, and Licheng Jiao, "QD-MSA : A quantum distributed tensor network framework for multimodal sentiment analysis", Information Fusion 124, 103404 (2025).

[15] Steven Kordonowy and Hannes Leipold, "The Lie algebra of XY-mixer topologies and warm starting QAOA for constrained optimization", npj Quantum Information 12 1, 61 (2026).

[16] Supanut Thanasilp, Samson Wang, M. Cerezo, and Zoë Holmes, "Exponential concentration in quantum kernel methods", Nature Communications 15 1, 5200 (2024).

[17] Mrittunjoy Guha Majumdar, "Fault-tolerant quantum neural networks suppressing barren plateaus via localised cost functions and shallow parametrised architectures", Pramana 100 2, 93 (2026).

[18] Hannes Leipold, Federico M. Spedalieri, and Eleanor Rieffel, "Tailored Quantum Alternating Operator Ansätzes for Circuit Fault Diagnostics", Algorithms 15 10, 356 (2022).

[19] G. Paradezhenko, A. Pervishko, and D. Yudin, "Quantum-Assisted Open-Pit Optimization", JETP Letters 119 6, 470 (2024).

[20] Filippo Brozzi, Gloria Turati, Maurizio Ferrari Dacrema, Filippo Caruso, and Paolo Cremonesi, "Hamiltonian expressibility for ansatz selection in variational quantum algorithms", Quantum Machine Intelligence 8 2, 76 (2026).

[21] Rui Mao, Guojing Tian, and Xiaoming Sun, "Towards determining the presence of barren plateaus in some chemically inspired variational quantum algorithms", Communications Physics 7 1, 342 (2024).

[22] Hiroyoshi Kurogi, Katsuhiro Endo, Yuki Sato, Michihiko Sugawara, Kaito Wada, Kenji Sugisaki, Shu Kanno, Hiroshi C Watanabe, and Haruyuki Nakano, "Optimizing a parameterized controlled gate using free quaternion selection", Quantum Science and Technology 11 1, 015046 (2026).

[23] Zidu Liu, L.-M. Duan, and Dong-Ling Deng, "Solving quantum master equations with deep quantum neural networks", Physical Review Research 4 1, 013097 (2022).

[24] Muhammad Kashif and Saif Al-Kuwari, "The unified effect of data encoding, ansatz expressibility and entanglement on the trainability of HQNNs", International Journal of Parallel, Emergent and Distributed Systems 38 5, 362 (2023).

[25] Ratun Rahman and Dinh C. Nguyen, "Escaping Barren Plateaus in Variational Quantum Algorithms Using Negative Learning Rate in Quantum Internet of Things", IEEE Internet of Things Journal 13 4, 7417 (2026).

[26] Vu Tuan Hai, Le Vu Trung Duong, Pham Hoai Luan, and Yasuhiko Nakashima, 2024 Twelfth International Symposium on Computing and Networking (CANDAR) 238 (2024) ISBN:979-8-3315-2836-2.

[27] Quynh T. Nguyen, Louis Schatzki, Paolo Braccia, Michael Ragone, Patrick J. Coles, Frédéric Sauvage, Martín Larocca, and M. Cerezo, "Theory for Equivariant Quantum Neural Networks", PRX Quantum 5 2, 020328 (2024).

[28] Antonio Sannia, Francesco Tacchino, Ivano Tavernelli, Gian Luca Giorgi, and Roberta Zambrini, "Engineered dissipation to mitigate barren plateaus", npj Quantum Information 10 1, 81 (2024).

[29] Korbinian Kottmann, Friederike Metz, Joana Fraxanet, and Niccolò Baldelli, "Variational quantum anomaly detection: Unsupervised mapping of phase diagrams on a physical quantum computer", Physical Review Research 3 4, 043184 (2021).

[30] Phattharaporn Singkanipa and Daniel A. Lidar, "Beyond unital noise in variational quantum algorithms: noise-induced barren plateaus and limit sets", Quantum 9, 1617 (2025).

[31] Francesco Scala, Stefano Mangini, Chiara Macchiavello, Daniele Bajoni, and Dario Gerace, 2022 International Joint Conference on Neural Networks (IJCNN) 1 (2022) ISBN:978-1-7281-8671-9.

[32] Hiroshi C. Watanabe, Rudy Raymond, Yu-Ya Ohnishi, Eriko Kaminishi, and Michihiko Sugawara, "Optimizing Parameterized Quantum Circuits With Free-Axis Single-Qubit Gates", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[33] Lucas Friedrich and Jonas Maziero, "Restricting to the chip architecture maintains the quantum neural network accuracy", Quantum Information Processing 23 4, 131 (2024).

[34] Laura Gentini, Alessandro Cuccoli, and Leonardo Banchi, "Variational Adiabatic Gauge Transformation on Real Quantum Hardware for Effective Low-Energy Hamiltonians and Accurate Diagonalization", Physical Review Applied 18 3, 034025 (2022).

[35] Zidu Liu, Li-Wei Yu, L.-M. Duan, and Dong-Ling Deng, "Presence and Absence of Barren Plateaus in Tensor-Network Based Machine Learning", Physical Review Letters 129 27, 270501 (2022).

[36] Silvirianti, Bhaskara Narottama, and Soo Young Shin, "Layerwise Quantum Deep Reinforcement Learning for Joint Optimization of UAV Trajectory and Resource Allocation", IEEE Internet of Things Journal 11 1, 430 (2024).

[37] Andrew Arrasmith, Zoë Holmes, M Cerezo, and Patrick J Coles, "Equivalence of quantum barren plateaus to cost concentration and narrow gorges", Quantum Science and Technology 7 4, 045015 (2022).

[38] M Schumann, F K Wilhelm, and A Ciani, "Emergence of noise-induced barren plateaus in arbitrary layered noise models", Quantum Science and Technology 9 4, 045019 (2024).

[39] Adam Callison and Nicholas Chancellor, "Hybrid quantum-classical algorithms in the noisy intermediate-scale quantum era and beyond", Physical Review A 106 1, 010101 (2022).

[40] Marvin Bechtold, Johanna Barzen, Frank Leymann, Alexander Mandl, Julian Obst, Felix Truger, and Benjamin Weder, "Investigating the effect of circuit cutting in QAOA for the MaxCut problem on NISQ devices", Quantum Science and Technology 8 4, 045022 (2023).

[41] Lixue Cheng, Yu-Qin Chen, Shi-Xin Zhang, and Shengyu Zhang, "Quantum approximate optimization via learning-based adaptive optimization", Communications Physics 7 1, 83 (2024).

[42] Vojtěch Novák, Tomáš Bezděk, Ivan Zelinka, Swagatam Das, and Martin Beseda, "A longitudinal analysis of the CEC single-objective competitions (2010–2024) and implications for variational quantum optimization", Swarm and Evolutionary Computation 107, 102469 (2026).

[43] G. Paradezhenko, A. Pervishko, and D. Yudin, "Variatsionnaya kvantovaya optimizatsiya otkrytogo kar'era", Pisʹma v žurnal êksperimentalʹnoj i teoretičeskoj fiziki 119 5-6, 459 (2024).

[44] Soyoung Shin, Ha Eum Kim, Jaewan Kim, Hyeonjun Yeo, Wonho Jhe, and Kabgyun Jeong, "Designing minimalistic variational quantum ansatz inspired by algorithmic cooling", Journal of the Korean Physical Society 89 1, 1 (2026).

[45] Gaoyuan Wang, Jonathan Warrell, Prashant S. Emani, and Mark Gerstein, "Quantum variational autoencoder utilizing regularized mixed-state latent representations", Physical Review A 111 4, 042416 (2025).

[46] Francesco Scala, Andrea Ceschini, Massimo Panella, and Dario Gerace, "A General Approach to Dropout in Quantum Neural Networks", Advanced Quantum Technologies 8 12, 2300220 (2025).

[47] Eric R. Anschuetz and Bobak T. Kiani, "Quantum variational algorithms are swamped with traps", Nature Communications 13 1, 7760 (2022).

[48] André Sequeira, Luis Paulo Santos, and Luis Soares Barbosa, "Trainability issues in quantum policy gradients", Machine Learning: Science and Technology 5 3, 035037 (2024).

[49] A. Monroy-Azpeitia, J. Y. Montiel-Pérez, A. Maldonado-Romo, A. A. Arvizu-Velazquez, J. L. López-Bonilla, and S. Vidal-Beltrán, Studies in Computational Intelligence 1200, 193 (2025) ISBN:978-3-031-85613-6.

[50] Zeyi Tao, Jindi Wu, Qi Xia, and Qun Li, 2023 IEEE International Conference on Quantum Software (QSW) 76 (2023) ISBN:979-8-3503-0479-4.

[51] Tuan Hai Vu, Vu Trung Duong Le, Hoai Luan Pham, and Yasuhiko Nakashima, "Benchmarking Variants of the Adam Optimizer for Quantum Machine Learning Applications", IEEE Open Journal of the Computer Society 6, 1146 (2025).

[52] Tobias Haug and Kishor Bharti, "Generalized quantum assisted simulator", Quantum Science and Technology 7 4, 045019 (2022).

[53] Lucas Q. Galvão, Anna Beatriz M. de Souza, Marcelo A. Moret, and Clebson Cruz, "Variational Quantum Computing for Quantum Simulation: Principles, Implementations, and Challenges", Brazilian Journal of Physics 56 1, 32 (2026).

[54] Ching-Yi Lai, Kao-Yueh Kuo, and Bo-Jyun Liao, "Syndrome decoding by quantum approximate optimization", Quantum Information Processing 23 11, 368 (2024).

[55] Bikram Khanal, Pablo Rivas, Arun Sanjel, Korn Sooksatra, Ernesto Quevedo, and Alejandro Rodriguez, "Generalization error bound for quantum machine learning in NISQ era—a survey", Quantum Machine Intelligence 6 2, 90 (2024).

[56] Timur Gubaev, Philipp Pfeffer, Christian Dreßler, and Jörg Schumacher, "Variational quantum algorithm for anion exchange across an electrolyzer membrane", Physical Review Applied 26 1, 014051 (2026).

[57] Qiang Miao and Thomas Barthel, "Equivalence of cost concentration and gradient vanishing for quantum circuits: an elementary proof in the Riemannian formulation", Quantum Science and Technology 9 4, 045039 (2024).

[58] Thomas Spriggs, Arash Ahmadi, Bokai Chen, and Eliska Greplova, "Quantum resources of quantum and classical variational methods", Machine Learning: Science and Technology 6 1, 015042 (2025).

[59] Elias X. Huber, Benjamin Y. L. Tan, Paul R. Griffin, and Dimitris G. Angelakis, "Exponential qubit reduction in optimization for financial transaction settlement", EPJ Quantum Technology 11 1, 52 (2024).

[60] Nikolay V. Tkachenko, Hang Ren, Wendy M. Billings, Rebecca Tomann, K. Birgitta Whaley, and Martin Head-Gordon, "Beyond real: alternative unitary cluster Jastrow models for molecular electronic structure calculations on near-term quantum computers", Chemical Science 16 47, 22299 (2025).

[61] Hiroshi Ohno, "Bayesian network structure learning using quantum generative models", Quantum Machine Intelligence 6 2, 74 (2024).

[62] Jeongwoo Jae, Jeonghoon Hong, Jinho Choo, and Yeong‐Dae Kwon, "Reinforcement Learning to Learn Quantum States for Heisenberg Scaling Accuracy", Advanced Quantum Technologies 8 10, e2500206 (2025).

[63] Frédéric Sauvage, Martín Larocca, Patrick J Coles, and M Cerezo, "Building spatial symmetries into parameterized quantum circuits for faster training", Quantum Science and Technology 9 1, 015029 (2024).

[64] Fabio Zoratti, Giacomo De Palma, Bobak Kiani, Quynh T. Nguyen, Milad Marvian, Seth Lloyd, and Vittorio Giovannetti, "Improving the speed of variational quantum algorithms for quantum error correction", Physical Review A 108 2, 022611 (2023).

[65] Karthik Meduri, Ruthvik Yedla, Santosh Reddy Addula, Guna Sekhar Sajja, Shaila Rana, Elyson De La Cruz, Mohan Harish Maturi, and Hari Gonaygunta, "Hybrid Quantum-Classical Neural Networks for Healthcare Prediction Powered by Automated Scientific Discovery", Informatics 13 6, 98 (2026).

[66] Kavitha Yogaraj, Brian Quanz, Tarun Vikas, Arijit Mondal, and Samrat Mondal, "Post-variational classical quantum transfer learning for binary classification", Scientific Reports 15 1, 23682 (2025).

[67] Tyler Volkoff, Zoë Holmes, and Andrew Sornborger, "Universal Compiling and (No-)Free-Lunch Theorems for Continuous-Variable Quantum Learning", PRX Quantum 2 4, 040327 (2021).

[68] David A. Kreplin and Marco Roth, "Reduction of finite sampling noise in quantum neural networks", Quantum 8, 1385 (2024).

[69] Alberto Di Meglio, Karl Jansen, Ivano Tavernelli, Constantia Alexandrou, Srinivasan Arunachalam, Christian W. Bauer, Kerstin Borras, Stefano Carrazza, Arianna Crippa, Vincent Croft, Roland de Putter, Andrea Delgado, Vedran Dunjko, Daniel J. Egger, Elias Fernández-Combarro, Elina Fuchs, Lena Funcke, Daniel González-Cuadra, Michele Grossi, Jad C. Halimeh, Zoë Holmes, Stefan Kühn, Denis Lacroix, Randy Lewis, Donatella Lucchesi, Miriam Lucio Martinez, Federico Meloni, Antonio Mezzacapo, Simone Montangero, Lento Nagano, Vincent R. Pascuzzi, Voica Radescu, Enrique Rico Ortega, Alessandro Roggero, Julian Schuhmacher, Joao Seixas, Pietro Silvi, Panagiotis Spentzouris, Francesco Tacchino, Kristan Temme, Koji Terashi, Jordi Tura, Cenk Tüysüz, Sofia Vallecorsa, Uwe-Jens Wiese, Shinjae Yoo, and Jinglei Zhang, "Quantum Computing for High-Energy Physics: State of the Art and Challenges", PRX Quantum 5 3, 037001 (2024).

[70] Shakib Daryanoosh, "Quantum approximate optimization of finite-state bosonic systems", Physical Review E 113 2, 025305 (2026).

[71] David A. Herrera-Martí, "Policy Gradient Approach to Compilation of Variational Quantum Circuits", Quantum 6, 797 (2022).

[72] Paul San Sebastian Sein, Mikel Cañizo, and Román Orús, "Image classification with rotation-invariant variational quantum circuits", Physical Review Research 7 1, 013082 (2025).

[73] Reyhaneh Aghaei Saem, Behrang Tafreshi, Zoë Holmes, and Supanut Thanasilp, "Pitfalls when tackling the exponential concentration of parameterized quantum models", Quantum Science and Technology 11 1, 015049 (2026).

[74] Zhaoxuan Ding, Fengchuang Xing, Yu Xia, and Peiyuan Zeng, 2026 3rd International Conference on Image Processing and Artificial Intelligence (ICIPAI) 269 (2026) ISBN:979-8-3315-8374-3.

[75] Yunfei Wang and Junyu Liu, "A comprehensive review of quantum machine learning: from NISQ to fault tolerance", Reports on Progress in Physics 87 11, 116402 (2024).

[76] Brian Coyle, Mina Doosti, Elham Kashefi, and Niraj Kumar, "Progress toward practical quantum cryptanalysis by variational quantum cloning", Physical Review A 105 4, 042604 (2022).

[77] Bikram Khanal and Pablo Rivas, "Data-dependent generalization bounds for parameterized quantum models under noise", The Journal of Supercomputing 81 4, 611 (2025).

[78] Daniel Huerga, "Variational Quantum Simulation of Valence-Bond Solids", Quantum 6, 874 (2022).

[79] Lucas Friedrich and Jonas Maziero, "Quantum neural network cost function concentration dependency on the parametrization expressivity", Scientific Reports 13 1, 9978 (2023).

[80] Raoul Heese, Patricia Bickert, and Astrid Elisa Niederle, "Representation of binary classification trees with binary features by quantum circuits", Quantum 6, 676 (2022).

[81] Francesca Schiavello, Edoardo Altamura, Ivano Tavernelli, Stefano Mensa, and Benjamin Symons, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 11 (2025) ISBN:979-8-3315-5736-2.

[82] Pierre Decoodt, Tan Jun Liang, Soham Bopardikar, Hemavathi Santhanam, Alfaxad Eyembe, Begonya Garcia-Zapirain, and Daniel Sierra-Sosa, "Hybrid Classical–Quantum Transfer Learning for Cardiomegaly Detection in Chest X-rays", Journal of Imaging 9 7, 128 (2023).

[83] J. Wayne Mullinax and Norm M. Tubman, "Large-scale sparse wave function circuit simulator for applications with the variational quantum eigensolver", The Journal of Chemical Physics 162 7, 074114 (2025).

[84] Weiyuan Gong, Dong Yuan, Weikang Li, and Dong-Ling Deng, "Enhancing quantum adversarial robustness by randomized encodings", Physical Review Research 6 2, 023020 (2024).

[85] Farzaneh Shayeganfar, Ali Ramazani, Veera Sundararaghavan, and Yuhua Duan, "Quantum graph learning and algorithms applied in quantum computer sciences and image classification", Applied Physics Reviews 12 2, 021327 (2025).

[86] Cristian L. Cortes and Stephen K. Gray, "Quantum Krylov subspace algorithms for ground- and excited-state energy estimation", Physical Review A 105 2, 022417 (2022).

[87] Y. S. Teo, "Optimized numerical gradient and Hessian estimation for variational quantum algorithms", Physical Review A 107 4, 042421 (2023).

[88] Xin-Yu Chen, Pan Gao, Chu-Dan Qiu, Ya-Nan Lu, Fan Yang, Yuanyuan Zhao, Hang Li, Jiang Zhang, Shijie Wei, Tonghao Xing, Xin-Yu Pan, Dong Ruan, Feihao Zhang, Keren Li, and Guilu Long, "A noise-robust quantum dynamics learning protocol based on Choi–Jamiolkowski isomorphism: theory and experiment", New Journal of Physics 26 3, 033023 (2024).

[89] Abdelrahman S. Abdelrahman, Shuvro Chowdhury, Flaviano Morone, and Kerem Y. Camsari, "Generalized Probabilistic Approximate Optimization Algorithm", Nature Communications 17 1, 498 (2025).

[90] Eric R. Anschuetz and Xun Gao, "Arbitrary Polynomial Separations in Trainable Quantum Machine Learning", Quantum 10, 1976 (2026).

[91] Rafael Pereira da Silva, "Hamiltonian Minimization in the NISQ Era", SSRN Electronic Journal (2023).

[92] Unpil Baek, Diptarka Hait, James Shee, Oskar Leimkuhler, William J. Huggins, Torin F. Stetina, Martin Head-Gordon, and K. Birgitta Whaley, "Say NO to Optimization: A Nonorthogonal Quantum Eigensolver", PRX Quantum 4 3, 030307 (2023).

[93] Cristian L. Cortes, A. Eugene DePrince, and Stephen K. Gray, "Fast-forwarding quantum simulation with real-time quantum Krylov subspace algorithms", Physical Review A 106 4, 042409 (2022).

[94] Kang-Min Hu, Min Namkung, and Hyang-Tag Lim, "Photonic variational quantum eigensolver for NISQ-compatible quantum technology", Nano Convergence 12 1, 60 (2025).

[95] Kun Wang, Zhixin Song, Xuanqiang Zhao, Zihe Wang, and Xin Wang, "Detecting and quantifying entanglement on near-term quantum devices", npj Quantum Information 8 1, 52 (2022).

[96] Matteo Robbiati, Alejandro Sopena, Andrea Papaluca, and Stefano Carrazza, "Real-time error mitigation for variational optimization on quantum hardware", Physical Review Research 8 1, 013262 (2026).

[97] Samson Wang, Enrico Fontana, M. Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J. Coles, "Noise-induced barren plateaus in variational quantum algorithms", Nature Communications 12 1, 6961 (2021).

[98] Shuyan Chao, Yuxin Deng, Zhanou Liu, and Yuwei Zhang, 2025 IEEE International Conference on Systems, Man, and Cybernetics (SMC) 4831 (2025) ISBN:979-8-3315-3358-8.

[99] Irene Kahvazadeh and A.C. Fong, 2025 International Conference on Intelligent & Innovative Practices in Engineering & Management (IIPEM) 1 (2025) ISBN:979-8-3315-9604-0.

[100] Ju-Young Ryu, Eyuel Elala, and June-Koo Kevin Rhee, "Quantum Graph Neural Network Models for Materials Search", Materials 16 12, 4300 (2023).

[101] Minsu Yeo, Hansol Kim, and Yoonchan Jeong, "Phase Control Algorithms for CBC Systems Based on Evolutionary Strategy for Implementation on FPGA Boards", Journal of Lightwave Technology 43 18, 8563 (2025).

[102] Mujahidul Islam, Serkan Turkeli, and Fatih Ozaydin, "Quantum generative adversarial networks: architectures, use cases, and real-world implementations", Quantum Information Processing 25 1, 11 (2026).

[103] Xiao Shi and Yun Shang, "Avoiding barren plateaus via Gaussian mixture model", New Journal of Physics 27 10, 104501 (2025).

[104] Chiara Leadbeater, Louis Sharrock, Brian Coyle, and Marcello Benedetti, "F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits", Entropy 23 10, 1281 (2021).

[105] Tianyun Wang, Ze Wu, Yuquan Chen, Yanjun Hou, Zhaokai Li, and Xinhua Peng, "Experimental demonstration of entanglement forging for solving Max-Cut problem", Physical Review Research 8 1, 013277 (2026).

[106] Maximilian Balthasar Mansky, Miguel Armayor Martinez, Alejandro Bravo de la Serna, Santiago Londoño Castillo, Dimitra Nikolaidou, Gautham Sathish, Zhihao Wang, Sebastian Wölckert, and Claudia Linnhoff-Popien, "Scaling of symmetry-restricted Lie groups", Physica Scripta 100 8, 085222 (2025).

[107] Lucas Friedrich, Marcos L. W. Basso, Alberto B. P. Junior, Joab M. Varela, Leandro Morais, Rafael Chaves, and Jonas Maziero, "Variational quantum algorithm for entanglement quantification", Physical Review A 112 5, 052452 (2025).

[108] Francesco Di Marcantonio, Massimiliano Incudini, Davide Tezza, and Michele Grossi, "Quantum Advantage Seeker with Kernels (QuASK): a software framework to speed up the research in quantum machine learning", Quantum Machine Intelligence 5 1, 20 (2023).

[109] Yudai Suzuki and Muyuan Li, "Effect of alternating layered Ansätze on trainability of projected quantum kernels", Physical Review A 110 1, 012409 (2024).

[110] Haiyan Wang, "Several fitness functions and entanglement gates in quantum kernel generation", Quantum Machine Intelligence 7 1, 7 (2025).

[111] Lucas Friedrich and Jonas Maziero, "Avoiding barren plateaus with classical deep neural networks", Physical Review A 106 4, 042433 (2022).

[112] Pranav Kairon, Jonas Jäger, and Roman V. Krems, "Equivalence between exponential concentration in quantum machine learning kernels and barren plateaus in variational algorithms", Physical Review A 114 2, 022417 (2026).

[113] Yunlong Yu, Chenfeng Cao, Xiang-Bin Wang, Nic Shannon, and Robert Joynt, "Solution of SAT problems with the adaptive-bias quantum approximate optimization algorithm", Physical Review Research 5 2, 023147 (2023).

[114] Matej Črepinšek, Miha Ravber, Luka Mernik, and Marjan Mernik, "Tackling Blind Spot Challenges in Metaheuristics Algorithms Through Exploration and Exploitation", Mathematics 13 10, 1580 (2025).

[115] Gabriele Agliardi and Enrico Prati, "Optimal Tuning of Quantum Generative Adversarial Networks for Multivariate Distribution Loading", Quantum Reports 4 1, 75 (2022).

[116] Li-Wei Yu, Weikang Li, Qi Ye, Zhide Lu, Zizhao Han, and Dong-Ling Deng, "Expressibility-induced concentration of quantum neural tangent kernels", Reports on Progress in Physics 87 11, 110501 (2024).

[117] Gwonhak Lee, Seonghoon Choi, Joonsuk Huh, and Artur F. Izmaylov, "Efficient strategies for reducing sampling error in quantum Krylov subspace diagonalization", Digital Discovery 4 4, 954 (2025).

[118] Shreeya Sanjeev Gokhale, Rishit Ravi Dhote, and Radhakrishnan Delhibabu, "A review of quantum machine learning algorithms, applications, and emerging advantages", Discover Computing 29 1, 226 (2026).

[119] Y. S. Teo, "Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms", Physical Review A 109 1, 012620 (2024).

[120] Martín Larocca, Supanut Thanasilp, Samson Wang, Kunal Sharma, Jacob Biamonte, Patrick J. Coles, Lukasz Cincio, Jarrod R. McClean, Zoë Holmes, and M. Cerezo, "Barren plateaus in variational quantum computing", Nature Reviews Physics 7 4, 174 (2025).

[121] Andrea Skolik, Michele Cattelan, Sheir Yarkoni, Thomas Bäck, and Vedran Dunjko, "Equivariant quantum circuits for learning on weighted graphs", npj Quantum Information 9 1, 47 (2023).

[122] Lucas Friedrich and Jonas Maziero, "Evolution strategies: application in hybrid quantum-classical neural networks", Quantum Information Processing 22 3, 132 (2023).

[123] Elies Gil-Fuster, Jens Eisert, and Carlos Bravo-Prieto, "Understanding quantum machine learning also requires rethinking generalization", Nature Communications 15 1, 2277 (2024).

[124] Yidong Liao, Min-Hsiu Hsieh, and Chris Ferrie, "Quantum optimization for training quantum neural networks", Quantum Machine Intelligence 6 1, 33 (2024).

[125] Han Qi, Lei Wang, Hongsheng Zhu, Abdullah Gani, and Changqing Gong, "The barren plateaus of quantum neural networks: review, taxonomy and trends", Quantum Information Processing 22 12, 435 (2023).

[126] Beng Yee Gan, Daniel Leykam, and Dimitris G. Angelakis, "Fock state-enhanced expressivity of quantum machine learning models", EPJ Quantum Technology 9 1, 16 (2022).

[127] Haijian Shao, Yujie Wu, Xing Deng, and Yingtao Jiang, "Lie-geometric trainability of quantum dynamical systems: avoiding barren plateaus via low-dimensional Lie subalgebras", Physica Scripta 101 25, 255107 (2026).

[128] Yury Chernyak, "Classical Regularization in Variational Quantum Eigensolvers", (2026).

[129] Chufan Lyu, Zuoheng Zou, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Variational simulation of higher-spin systems on qubit-based quantum simulators", Physical Review Research 8 3, 033119 (2026).

[130] Huan-Yu Liu, Tai-Ping Sun, Yu-Chun Wu, Yong-Jian Han, and Guo-Ping Guo, "Mitigating barren plateaus with transfer-learning-inspired parameter initializations", New Journal of Physics 25 1, 013039 (2023).

[131] Poornima Kumaresan, Shwetha Singaravelu, Lakshmi Rajendran, and Santhosh Sivasubramani, "Eliminating vendor lock-in in quantum machine learning via framework-agnostic neural networks", Frontiers in Quantum Science and Technology 5, 1862700 (2026).

[132] Ze‐Tong Li, Fan‐Xu Meng, Han Zeng, Zhai‐Rui Gong, Zai‐Chen Zhang, and Xu‐Tao Yu, "A Gradient‐Cost Multiobjective Alternate Framework for Variational Quantum Eigensolver with Variable Ansatz", Advanced Quantum Technologies 6 5, 2200130 (2023).

[133] S. Shin, Y. S. Teo, and H. Jeong, "Exponential data encoding for quantum supervised learning", Physical Review A 107 1, 012422 (2023).

[134] Mark R. Hirsbrunner, J. Wayne Mullinax, Yizhi Shen, David B. Williams-Young, Katherine Klymko, Roel Van Beeumen, and Norm M. Tubman, "A circuit-generated quantum subspace algorithm for the variational quantum eigensolver", The Journal of Chemical Physics 161 16, 164103 (2024).

[135] Kasidit Srimahajariyapong, Supanut Thanasilp, and Thiparat Chotibut, "Connecting phases of matter to the flatness of the loss landscape in analog variational quantum algorithms", Communications Physics 9 1, 111 (2026).

[136] Samson Wang, Piotr Czarnik, Andrew Arrasmith, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Can Error Mitigation Improve Trainability of Noisy Variational Quantum Algorithms?", Quantum 8, 1287 (2024).

[137] Mark R. Hirsbrunner, Diana Chamaki, J. Wayne Mullinax, and Norm M. Tubman, "Beyond MP2 initialization for unitary coupled cluster quantum circuits", Quantum 8, 1538 (2024).

[138] Stefan H. Sack, Raimel A. Medina, Alexios A. Michailidis, Richard Kueng, and Maksym Serbyn, "Avoiding Barren Plateaus Using Classical Shadows", PRX Quantum 3 2, 020365 (2022).

[139] Zong-Liang Li and Shi-Xin Zhang, "Dual role of low-weight Pauli propagation: A flawed simulator but a powerful initializer for variational quantum algorithms", Physical Review Research 8 1, 013266 (2026).

[140] Bruno O. Fernandez, Rodrigo Bloot, and Marcelo A. Moret, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 1976 (2025) ISBN:979-8-3315-5736-2.

[141] Jack Cunningham and Jun Zhuang, "Investigating and mitigating barren plateaus in variational quantum circuits: a survey", Quantum Information Processing 24 2, 48 (2025).

[142] Oluwadara Ogunkoya, Kirsten Morris, and Dog̃a Murat Kürkçüog̃lu, "Investigating parameter trainability in the SNAP-displacement protocol of a qudit system", Physica Scripta 100 7, 075109 (2025).

[143] Guilherme Adamatti Bridi, Debbie Lim, Lirandë Pira, Raqueline Azevedo Medeiros Santos, Franklin de Lima Marquezino, and Soumik Adhikary, "Expressivity limits in quantum walk-based optimization", Physical Review A 113 2, 022423 (2026).

[144] Wei Zhao, Tangjie Weng, Yue Ruan, Zhi Liu, Xuangou Wu, Xiao Zheng, and Nei Kato, "Quantum Computing in Wireless Communications and Networking: A Tutorial-cum-Survey", IEEE Communications Surveys & Tutorials 27 4, 2378 (2025).

[145] Joe Gibbs, Kaitlin Gili, Zoë Holmes, Benjamin Commeau, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, "Long-time simulations for fixed input states on quantum hardware", npj Quantum Information 8 1, 135 (2022).

[146] Lucas Friedrich, Tiago de Souza Farias, and Jonas Maziero, "Barren Plateaus Induced by the Dimension of Qudits", (2024).

[147] Yang Qian, Yuxuan Du, and Dacheng Tao, "Shuffle-QUDIO: accelerate distributed VQE with trainability enhancement and measurement reduction", Quantum Machine Intelligence 6 1, 32 (2024).

[148] Xiaozhen Ge, Re-Bing Wu, and Herschel Rabitz, "The optimization landscape of hybrid quantum–classical algorithms: From quantum control to NISQ applications", Annual Reviews in Control 54, 314 (2022).

[149] Savvas Varsamopoulos, Evan Philip, Vincent E. Elfving, Herman W. T. van Vlijmen, Sairam Menon, Ann Vos, Natalia Dyubankova, Bert Torfs, and Anthony Rowe, "Quantum extremal learning", Quantum Machine Intelligence 6 2, 42 (2024).

[150] Zitong Li, Tailong Xiao, Xiaoyang Deng, Guihua Zeng, and Weimin Li, "Optimizing Variational Quantum Neural Networks Based on Collective Intelligence", Mathematics 12 11, 1627 (2024).

[151] Zidu Liu, Pei-Xin Shen, Weikang Li, L-M Duan, and Dong-Ling Deng, "Quantum capsule networks", Quantum Science and Technology 8 1, 015016 (2023).

[152] Ramin Fakhimi and Hamidreza Validi, Encyclopedia of Optimization 1 (2023) ISBN:978-3-030-54621-2.

[153] Sasan Moradi, "Hybrid classical-quantum computation of heat diffusion in multilayer materials", (2025).

[154] J. Gidi, B. Candia, A. D. Muñoz-Moller, A. Rojas, L. Pereira, M. Muñoz, L. Zambrano, and A. Delgado, "Stochastic optimization algorithms for quantum applications", Physical Review A 108 3, 032409 (2023).

[155] Lucas Friedrich and Jonas Maziero, "Learning to learn with an evolutionary strategy applied to variational quantum algorithms", Physical Review A 111 2, 022630 (2025).

[156] Valentin Heyraud, Zejian Li, Kaelan Donatella, Alexandre Le Boité, and Cristiano Ciuti, "Efficient Estimation of Trainability for Variational Quantum Circuits", PRX Quantum 4 4, 040335 (2023).

[157] N. A. Nemkov, "Statistical Models of Barren Plateaus and Anti-Concentration of Pauli Observables", JETP Letters 122 3, 190 (2025).

[158] Daniel Bultrini, Samson Wang, Piotr Czarnik, Max Hunter Gordon, M. Cerezo, Patrick J. Coles, and Lukasz Cincio, "The battle of clean and dirty qubits in the era of partial error correction", Quantum 7, 1060 (2023).

[159] Hao-Kai Zhang, Shuo Liu, and Shi-Xin Zhang, "Absence of Barren Plateaus in Finite Local-Depth Circuits with Long-Range Entanglement", Physical Review Letters 132 15, 150603 (2024).

[160] Daniel Faílde, José Daniel Viqueira, Mariamo Mussa Juane, and Andrés Gómez, "Using Differential Evolution to avoid local minima in Variational Quantum Algorithms", Scientific Reports 13 1, 16230 (2023).

[161] Sofya Manko and Dmitry Frolovtsev, "Classification and reconstruction for single-pixel imaging with classical and quantum neural networks", Signal, Image and Video Processing 19 4, 277 (2025).

[162] Shun Okumura and Masayuki Ohzeki, "Fourier Analysis of Parameterized Quantum Circuits and the Barren Plateau Problem", Journal of the Physical Society of Japan 95 3, 033002 (2026).

[163] Stefano Markidis, "On physics-informed neural networks for quantum computers", Frontiers in Applied Mathematics and Statistics 8, 1036711 (2022).

[164] Gereon Koßmann, Lennart Binkowski, Lauritz van Luijk, Timo Ziegler, and René Schwonnek, "Deep-Circuit QAOA", Quantum 9, 1882 (2025).

[165] Alexey Melnikov, Mohammad Kordzanganeh, Alexander Alodjants, and Ray-Kuang Lee, "Quantum machine learning: from physics to software engineering", Advances in Physics: X 8 1, 2165452 (2023).

[166] Junyu Liu, Zexi Lin, and Liang Jiang, "Laziness, barren plateau, and noises in machine learning", Machine Learning: Science and Technology 5 1, 015058 (2024).

[167] Ayush Mahanta, Jasmeet Kaur, Kapil Kumar, and Manav Jain, 2026 International Conference on Smart Electronic Devices and Intelligent Systems (ICSEDIS) 461 (2026) ISBN:979-8-3315-8824-3.

[168] Supanut Thanasilp, Samson Wang, Nhat Anh Nghiem, Patrick Coles, and Marco Cerezo, "Subtleties in the trainability of quantum machine learning models", Quantum Machine Intelligence 5 1, 21 (2023).

[169] M. Bilkis, M. Cerezo, Guillaume Verdon, Patrick J. Coles, and Lukasz Cincio, "A semi-agnostic ansatz with variable structure for variational quantum algorithms", Quantum Machine Intelligence 5 2, 43 (2023).

[170] Rasyid Ustman Ramadhan, Luthfiya Kurnia Permatahati, Teguh Budi Prayitno, and Yanoar P. Sarwono, "Expressibility and Trainability Analysis of Hardware-Efficient Ansatz Variants in Variational Quantum Eigensolver with a Linear Mixing Model", The Journal of Physical Chemistry A 130 15, 3101 (2026).

[171] M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles, "Cost function dependent barren plateaus in shallow parametrized quantum circuits", Nature Communications 12 1, 1791 (2021).

[172] V. A. Zaytsev, M. E. Groshev, I. A. Maltsev, A. V. Durova, and V. M. Shabaev, "Calculation of the moscovium ground‐state energy by quantum algorithms", International Journal of Quantum Chemistry 124 1, e27232 (2024).

[173] Yong-Hwan Lee and Wan-Bum Lee, "Quantum-Enhanced DNA Image Compression: Theoretical Framework and NISQ Implementation Strategy", Applied Sciences 16 3, 1502 (2026).

[174] Manuel S. Rudolph, Sacha Lerch, Supanut Thanasilp, Oriel Kiss, Oxana Shaya, Sofia Vallecorsa, Michele Grossi, and Zoë Holmes, "Trainability barriers and opportunities in quantum generative modeling", npj Quantum Information 10 1, 116 (2024).

[175] Dmitry A. Fedorov, Bo Peng, Niranjan Govind, and Yuri Alexeev, "VQE method: a short survey and recent developments", Materials Theory 6 1, 2 (2022).

[176] Giuseppe Buonaiuto, Francesco Gargiulo, Giuseppe De Pietro, Massimo Esposito, and Marco Pota, "The effects of quantum hardware properties on the performances of variational quantum learning algorithms", Quantum Machine Intelligence 6 1, 9 (2024).

[177] Aman Mittal, Kasturi Venkata Sai Srikanth, Ferdin Sagai Don Bosco, Abhishek Singh, and Abhishek Chopra, 2025 Supercomputing India (SCI) 1 (2025) ISBN:979-8-3315-5758-4.

[178] Raphael César de Souza Pimenta and Anibal Thiago Bezerra, "Revisiting semiconductor bulk hamiltonians using quantum computers", Physica Scripta 98 4, 045804 (2023).

[179] Christopher K. Long, Kieran Dalton, Crispin H. W. Barnes, David R. M. Arvidsson-Shukur, and Normann Mertig, "Layering and subpool exploration for adaptive variational quantum eigensolvers: Reducing circuit depth, runtime, and susceptibility to noise", Physical Review A 109 4, 042413 (2024).

[180] Vu-Quoc-Minh Nguyen, Trung-Khanh Le, Trong-Tu Bui, and Duc-Hung Le, 2025 10th IEEE International Conference on Integrated Circuits, Design, and Verification (ICDV) 7 (2025) ISBN:979-8-3315-1549-2.

[181] Harriet L. Dao, "Exploring new variational quantum circuit ansatzes for solving SU(2) matrix models", The European Physical Journal C 85 6, 705 (2025).

[182] Ihor Sokolov and Jacek Dziarmaga, "Bang-bang preparation of a quantum many-body ground state in a finite lattice: Optimization of the algorithm with a tensor network", Physical Review B 111 24, 245144 (2025).

[183] Han-Xiao Tao and Re-Bing Wu, 2024 IEEE 63rd Conference on Decision and Control (CDC) 13 (2024) ISBN:979-8-3503-1633-9.

[184] Matthias Möller, Computational Methods in Applied Sciences 58, 357 (2023) ISBN:978-3-031-29081-7.

[185] Jingxuan Chen, Hanna Westerheim, Zoë Holmes, Ivy Luo, Theshani Nuradha, Dhrumil Patel, Soorya Rethinasamy, Kathie Wang, and Mark M. Wilde, "Slack-variable approach for variational quantum semidefinite programming", Physical Review A 112 2, 022607 (2025).

[186] Oriel Kiss, Michele Grossi, Pavel Lougovski, Federico Sanchez, Sofia Vallecorsa, and Thomas Papenbrock, "Quantum computing of the Li6 nucleus via ordered unitary coupled clusters", Physical Review C 106 3, 034325 (2022).

[187] Aakarsh Etar, Sarvapriya Tripathi, Jayesh Soni, Himanshu Upadhyay, and Alexander Perez-Pons, 2025 Conference on AI, Science, Engineering, and Technology (AIxSET) 111 (2025) ISBN:979-8-3315-7124-5.

[188] Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo, "Diagnosing Barren Plateaus with Tools from Quantum Optimal Control", Quantum 6, 824 (2022).

[189] Steven Herbert, "Quantum computing for data-centric engineering and science", Data-Centric Engineering 3, e36 (2022).

[190] Daniel Mastropietro, Georgios Korpas, Vyacheslav Kungurtsev, and Jakub Marecek, "Parallel Variational Quantum Algorithms With Gradient-Informed Restart to Speed Up Optimization in the Presence of Barren Plateaus", IEEE Transactions on Quantum Engineering 7, 1 (2026).

[191] Shengbin Wang, Guihui Li, Zhimin Wang, Zhaoyun Chen, Peng Wang, Yongjian Gu, Yu-Chun Wu, and Guo-Ping Guo, "Improving the trainability of variational quantum eigensolvers on noisy intermediate-scale quantum computers for solving portfolio optimization using convex interpolation", Physical Review A 113 5, 052441 (2026).

[192] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, "Variational quantum algorithms", Nature Reviews Physics 3 9, 625 (2021).

[193] Subhadeep Mondal and Amit Kumar Dutta, "Relative Entropy-Based Training of Quantum Neural Networks", IEEE Transactions on Quantum Engineering 7, 1 (2026).

[194] Bruno O. Fernandez, Rodrigo Bloot, and Marcelo A. Moret, "Quantum circuit evolutionary framework applied to the set partitioning problem", The Journal of Supercomputing 82 7, 411 (2026).

[195] Muhammad Kashif and Muhammad Shafique, 2024 IEEE International Conference on Rebooting Computing (ICRC) 1 (2024) ISBN:979-8-3315-4127-9.

[196] Yuxuan Zhang, Roeland Wiersema, Juan Carrasquilla, Lukasz Cincio, and Yong Baek Kim, "Scalable quantum dynamics compilation via quantum machine learning", Physical Review Research 8 2, 023128 (2026).

[197] Weijie Xiong, Giorgio Facelli, Mehrad Sahebi, Owen Agnel, Thiparat Chotibut, Supanut Thanasilp, and Zoë Holmes, "On fundamental aspects of quantum extreme learning machines", Quantum Machine Intelligence 7 1, 20 (2025).

[198] Ifan Williams and Mathieu Pellen, "A general approach to quantum integration of cross sections in high-energy physics", Quantum Science and Technology 10 4, 045017 (2025).

[199] Marco Ballarin, Stefano Mangini, Simone Montangero, Chiara Macchiavello, and Riccardo Mengoni, "Entanglement entropy production in Quantum Neural Networks", Quantum 7, 1023 (2023).

[200] Shuo Liu, Shi-Xin Zhang, Shao-Kai Jian, and Hong Yao, "Training variational quantum algorithms with random gate activation", Physical Review Research 5 3, L032040 (2023).

[201] Simone Tibaldi, Davide Vodola, Edoardo Tignone, and Elisa Ercolessi, "Bayesian Optimization for QAOA", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[202] Kushagra Goyal and Avinash Pathak, 2025 5th International Conference on AI-ML-Systems (AIMLSystems) 1 (2025) ISBN:979-8-3315-7318-8.

[203] Joe Gibbs, Zoë Holmes, Matthias C. Caro, Nicholas Ezzell, Hsin-Yuan Huang, Lukasz Cincio, Andrew T. Sornborger, and Patrick J. Coles, "Dynamical simulation via quantum machine learning with provable generalization", Physical Review Research 6 1, 013241 (2024).

[204] Han-Xiao Tao, Xin Wang, and Re-Bing Wu, "Design of expressive and trainable pulse-based quantum machine learning models", Physical Review Research 8 1, 013338 (2026).

[205] Charles Moussa, Max Hunter Gordon, Michal Baczyk, M Cerezo, Lukasz Cincio, and Patrick J Coles, "Resource frugal optimizer for quantum machine learning", Quantum Science and Technology 8 4, 045019 (2023).

[206] Matthias C. Caro, Hsin-Yuan Huang, M. Cerezo, Kunal Sharma, Andrew Sornborger, Lukasz Cincio, and Patrick J. Coles, "Generalization in quantum machine learning from few training data", Nature Communications 13 1, 4919 (2022).

[207] Chenfeng Cao, Yeqing Zhou, Swamit Tannu, Nic Shannon, and Robert Joynt, "Exploiting many-body localization for scalable variational quantum simulation", Quantum 9, 1942 (2025).

[208] Kaito Wada, Rudy Raymond, Yuki Sato, and Hiroshi C Watanabe, "Sequential optimal selections of single-qubit gates in parameterized quantum circuits", Quantum Science and Technology 9 3, 035030 (2024).

[209] Chisomo Daka and Somnath Bhattacharyya, "A novel quantum convolutional neural network framework for quantum-enhanced classification of pixelated colour images", Scientific Reports 16 1, 10828 (2026).

[210] Huili Zhang, Yibin Guo, Guanglei Xu, Yulong Feng, Jingning Zhang, Hai-feng Yu, and S.P. Zhao, "Simultaneous determination of multiple low-lying energy levels on a superconducting quantum processor", Physical Review Applied 25 3, 034066 (2026).

[211] Vladimir Vargas-Calderón, Fabio A. González, and Herbert Vinck-Posada, "Optimisation-free density estimation and classification with quantum circuits", Quantum Machine Intelligence 4 2, 16 (2022).

[212] Jakab Nádori, Gregory Morse, Barna Fülöp Villám, Zita Majnay-Takács, Zoltán Zimborás, and Péter Rakyta, "Batched Line Search Strategy for Navigating through Barren Plateaus in Quantum Circuit Training", Quantum 9, 1841 (2025).

[213] Lorenzo Leone, Salvatore F. E. Oliviero, Stefano Piemontese, Sarah True, and Alioscia Hamma, "Retrieving information from a black hole using quantum machine learning", Physical Review A 106 6, 062434 (2022).

[214] Nathan X. Kodama, Alex Bocharov, and Marcus P. da Silva, "Image classification by combining quantum kernel learning and tensor networks", Physical Review A 111 1, 012630 (2025).

[215] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).

[216] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson, "The Variational Quantum Eigensolver: A review of methods and best practices", Physics Reports 986, 1 (2022).

[217] Zoë Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles, "Connecting Ansatz Expressibility to Gradient Magnitudes and Barren Plateaus", PRX Quantum 3 1, 010313 (2022).

[218] Michael Broughton, Guillaume Verdon, Trevor McCourt, Antonio J. Martinez, Jae Hyeon Yoo, Sergei V. Isakov, Philip Massey, Ramin Halavati, Murphy Yuezhen Niu, Alexander Zlokapa, Evan Peters, Owen Lockwood, Andrea Skolik, Sofiene Jerbi, Vedran Dunjko, Martin Leib, Michael Streif, David Von Dollen, Hongxiang Chen, Shuxiang Cao, Roeland Wiersema, Hsin-Yuan Huang, Jarrod R. McClean, Ryan Babbush, Sergio Boixo, Dave Bacon, Alan K. Ho, Hartmut Neven, and Masoud Mohseni, "TensorFlow Quantum: A Software Framework for Quantum Machine Learning", arXiv:2003.02989, (2020).

[219] M. Cerezo and Patrick J. Coles, "Higher order derivatives of quantum neural networks with barren plateaus", Quantum Science and Technology 6 3, 035006 (2021).

[220] Zoë Holmes, Andrew Arrasmith, Bin Yan, Patrick J. Coles, Andreas Albrecht, and Andrew T. Sornborger, "Barren Plateaus Preclude Learning Scramblers", Physical Review Letters 126 19, 190501 (2021).

[221] Taylor L. Patti, Khadijeh Najafi, Xun Gao, and Susanne F. Yelin, "Entanglement devised barren plateau mitigation", Physical Review Research 3 3, 033090 (2021).

[222] Nikolay V. Tkachenko, James Sud, Yu Zhang, Sergei Tretiak, Petr M. Anisimov, Andrew T. Arrasmith, Patrick J. Coles, Lukasz Cincio, and Pavel A. Dub, "Correlation-Informed Permutation of Qubits for Reducing Ansatz Depth in the Variational Quantum Eigensolver", PRX Quantum 2 2, 020337 (2021).

[223] Yong-Xin Yao, Niladri Gomes, Feng Zhang, Cai-Zhuang Wang, Kai-Ming Ho, Thomas Iadecola, and Peter P. Orth, "Adaptive Variational Quantum Dynamics Simulations", PRX Quantum 2 3, 030307 (2021).

[224] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo, "Theory of overparametrization in quantum neural networks", arXiv:2109.11676, (2021).

[225] Weikang Li and Dong-Ling Deng, "Recent advances for quantum classifiers", Science China Physics, Mechanics, and Astronomy 65 2, 220301 (2022).

[226] Andi Gu, Angus Lowe, Pavel A. Dub, Patrick J. Coles, and Andrew Arrasmith, "Adaptive shot allocation for fast convergence in variational quantum algorithms", arXiv:2108.10434, (2021).

[227] Maria Kieferova, Ortiz Marrero Carlos, and Nathan Wiebe, "Quantum Generative Training Using Rényi Divergences", arXiv:2106.09567, (2021).

[228] Bingzhi Zhang and Quntao Zhuang, "Fast decay of classification error in variational quantum circuits", Quantum Science and Technology 7 3, 035017 (2022).

[229] Ranyiliu Chen, Zhixin Song, Xuanqiang Zhao, and Xin Wang, "Variational quantum algorithms for trace distance and fidelity estimation", Quantum Science and Technology 7 1, 015019 (2022).

[230] Akash Kundu, "Reinforcement learning-assisted quantum architecture search for variational quantum algorithms", arXiv:2402.13754, (2024).

[231] Abhinav Anand, Matthias Degroote, and Alán Aspuru-Guzik, "Natural Evolutionary Strategies for Variational Quantum Computation", arXiv:2012.00101, (2020).

[232] Ranyiliu Chen, Zhixin Song, Xuanqiang Zhao, and Xin Wang, "Variational Quantum Algorithms for Trace Distance and Fidelity Estimation", arXiv:2012.05768, (2020).

[233] Christa Zoufal, "Generative Quantum Machine Learning", arXiv:2111.12738, (2021).

[234] Ali Rad, Alireza Seif, and Norbert M. Linke, "Surviving The Barren Plateau in Variational Quantum Circuits with Bayesian Learning Initialization", arXiv:2203.02464, (2022).

[235] Simone Foderà, Gloria Turati, Riccardo Nembrini, Maurizio Ferrari Dacrema, and Paolo Cremonesi, "Reinforcement Learning for Variational Quantum Circuits Design", arXiv:2409.05475, (2024).

[236] Owen Lockwood, "Optimizing Quantum Variational Circuits with Deep Reinforcement Learning", arXiv:2109.03188, (2021).

[237] Gloria Turati, Alessia Marruzzo, Maurizio Ferrari Dacrema, and Paolo Cremonesi, "An Empirical Analysis on the Effectiveness of the Variational Quantum Linear Solver", arXiv:2409.06339, (2024).

[238] Lucas Slattery, Ruslan Shaydulin, Shouvanik Chakrabarti, Marco Pistoia, Sami Khairy, and Stefan M. Wild, "Numerical evidence against advantage with quantum fidelity kernels on classical data", Physical Review A 107 6, 062417 (2023).

[239] Michael R. Geller, Zoë Holmes, Patrick J. Coles, and Andrew Sornborger, "Experimental quantum learning of a spectral decomposition", Physical Review Research 3 3, 033200 (2021).

[240] Yidong Liao, Min-Hsiu Hsieh, and Chris Ferrie, "Quantum Optimization for Training Quantum Neural Networks", arXiv:2103.17047, (2021).

[241] Stefano Mangini, "Variational quantum algorithms for machine learning: theory and applications", arXiv:2306.09984, (2023).

[242] Antoine Michel, "Quantum simulation for strongly interacting fermions with neutral atoms array: towards the simulation of materials of interest", arXiv:2406.13343, (2024).

[243] Lukas Franken, Bogdan Georgiev, Sascha Mücke, Moritz Wolter, Raoul Heese, Christian Bauckhage, and Nico Piatkowski, "Quantum Circuit Evolution on NISQ Devices", arXiv:2012.13453, (2020).

[244] Hao-Kai Zhang, Chengkai Zhu, Geng Liu, and Xin Wang, "Fundamental limitations on optimization in variational quantum algorithms", arXiv:2205.05056, (2022).

[245] Vignesh Anantharamakrishnan and Márcio M. Taddei, "Quantum Time-Series Learning with Evolutionary Algorithms", arXiv:2412.17580, (2024).

[246] Zidu Liu, Qi Ye, Li-Wei Yu, L.-M. Duan, and Dong-Ling Deng, "Theory on variational high-dimensional tensor networks", arXiv:2303.17452, (2023).

[247] Sofya Manko and Dmitry Frolovtsev, "Classification and reconstruction for single-pixel imaging with classical and quantum neural networks", arXiv:2407.12506, (2024).

[248] Brian Coyle, "Machine learning applications for noisy intermediate-scale quantum computers", arXiv:2205.09414, (2022).

[249] Korbinian Kottmann, "Investigating Quantum Many-Body Systems with Tensor Networks, Machine Learning and Quantum Computers", arXiv:2210.11130, (2022).

[250] Fan-Xu Meng, Ze-Tong Li, Xu-Tao Yu, and Zai-Chen Zhang, "Quantum Circuit Architecture Optimization for Variational Quantum Eigensolver via Monto Carlo Tree Search", IEEE Transactions on Quantum Engineering 2, TQE.2021 (2021).

[251] Nahum Sá, Ivan S. Oliveira, and Itzhak Roditi, "Towards solving the BCS Hamiltonian gap in near-term quantum computers", Results in Physics 44, 106131 (2023).

[252] Adrián Pérez-Salinas, "Algorithmic Strategies for seizing Quantum Computing", arXiv:2112.15175, (2021).

[253] Mirko Consiglio, "Variational Quantum Algorithms for Many-Body Systems", arXiv:2502.11985, (2025).

[254] Filippo Brozzi, Gloria Turati, Maurizio Ferrari Dacrema, Filippo Caruso, and Paolo Cremonesi, "Hamiltonian Expressibility for Ansatz Selection in Variational Quantum Algorithms", arXiv:2507.22550, (2025).

[255] Bruno Oziel Fernandez, Rodrigo Bloot, and Marcelo Moret, "Quantum circuit evolutionary framework applied on set partitioning problem", arXiv:2507.20777, (2025).

[256] Marco Knipfer, Alexander Roman, Konstantin T. Matchev, Katia Matcheva, and Sergei Gleyzer, "AI Agents for Variational Quantum Circuit Design", arXiv:2602.19387, (2026).

[257] Luiz Henrique Prudencio dos Santos, Eliane F. Chinaglia, Jessica Fleury Curado, Marcilei A. Guazzelli, Mariana Pojar, Sueli Hatsumi Masunaga, and Roberto Baginski Batista Santos, "Quantum Machine Learning Applied to the Sinking of the Titanic", arXiv:2509.00916, (2025).

[258] Gloria Turati, Simone Foderà, Riccardo Nembrini, Maurizio Ferrari Dacrema, and Paolo Cremonesi, "Separating Ansatz Discovery from Deployment on Larger Problems: Reinforcement Learning for Modular Circuit Design", arXiv:2507.16001, (2025).

[259] Bruno Oziel Fernandez, Rodrigo Bloot, and Marcelo Moret, "Avoiding convergence stagnation in a quantum circuit evolutionary framework through an adaptive cost function", arXiv:2503.22404, (2025).

[260] Pınar Çifci, Serkan Akkoyun, and Lloyd Ronde, "Systematic VQE Benchmarking of the Deuteron, Triton, and Helium-3 within Lattice Pionless Effective Field Theory", arXiv:2604.20908, (2026).

[261] Poornima Kumaresan, Shwetha Singaravelu, Lakshmi Rajendran, and Santhosh Sivasubramani, "Eliminating Vendor Lock-In in Quantum Machine Learning via Framework-Agnostic Neural Networks", arXiv:2604.04414, (2026).

[262] Shehbaz Tariq, Muhammad Talha, Arshid Ali, Muhammad Diyan, and Symeon Chatzinotas, "Quantum Reservoir Computing: Recent Advances and Future Directions", arXiv:2607.18552, (2026).

[263] Philipp Altmann, Maximilian Mansky, Maximilian Zorn, Jonas Stein, and Claudia Linnhoff-Popien, "Quantum Generator Kernels", arXiv:2602.00361, (2026).

[264] Hikaru Wakaura and Takao Tomono, "Genetic-multi-initial generalized VQE: Advanced VQE method using genetic algorithms then local search", International Journal of Quantum Information 24 4, 2550038 (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-13 19:09:14) and SAO/NASA ADS (last updated successfully 2026-08-13 19:09:16). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.