Structure optimization for parameterized quantum circuits

Mateusz Ostaszewski; Edward Grant; Marcello Benedetti

doi:10.22331/q-2021-01-28-391

Structure optimization for parameterized quantum circuits

Mateusz Ostaszewski^1,2, Edward Grant^2,3, and Marcello Benedetti^2,4

¹Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100 Gliwice, Poland
²Department of Computer Science, University College London, WC1E 6BT London, United Kingdom
³Rahko Limited, N4 3JP London, United Kingdom
⁴Cambridge Quantum Computing Limited, CB2 1UB Cambridge, United Kingdom

Published:	2021-01-28, volume 5, page 391
Eprint:	arXiv:1905.09692v3
Doi:	https://doi.org/10.22331/q-2021-01-28-391
Citation:	Quantum 5, 391 (2021).

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Abstract

We propose an efficient method for simultaneously optimizing both the structure and parameter values of quantum circuits with only a small computational overhead. Shallow circuits that use structure optimization perform significantly better than circuits that use parameter updates alone, making this method particularly suitable for noisy intermediate-scale quantum computers. We demonstrate the method for optimizing a variational quantum eigensolver for finding the ground states of Lithium Hydride and the Heisenberg model in simulation, and for finding the ground state of Hydrogen gas on the IBM Melbourne quantum computer.

► BibTeX data

@article{Ostaszewski2021structure,
  doi = {10.22331/q-2021-01-28-391},
  url = {https://doi.org/10.22331/q-2021-01-28-391},
  title = {Structure optimization for parameterized quantum circuits},
  author = {Ostaszewski, Mateusz and Grant, Edward and Benedetti, Marcello},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {391},
  month = jan,
  year = {2021}
}

► References

[1] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien, ``A variational eigenvalue solver on a photonic quantum processor'' Nature Communications 5, 4213 (2014).
https://doi.org/10.1038/ncomms5213

[2] E. Farhi, J. Goldstone, S. Gutmann, and H. Neven, ``Quantum Algorithms for Fixed Qubit Architectures'' (2017).
arXiv:1703.06199

[3] Edward Farhiand Hartmut Neven ``Classification with Quantum Neural Networks on Near Term Processors'' (2018).
arXiv:1802.06002

[4] Marcello Benedetti, Delfina Garcia-Pintos, Oscar Perdomo, Vicente Leyton-Ortega, Yunseong Nam, and Alejandro Perdomo-Ortiz, ``A generative modeling approach for benchmarking and training shallow quantum circuits'' npj Quantum Information 5 (2019).
https://doi.org/10.1038/s41534-019-0157-8

[5] Marcello Benedetti, Edward Grant, Leonard Wossnig, and Simone Severini, ``Adversarial quantum circuit learning for pure state approximation'' New Journal of Physics 21, 043023 (2019).
https://doi.org/10.1088/1367-2630/ab14b5

[6] Hongxiang Chen, Leonard Wossnig, Simone Severini, Hartmut Neven, and Masoud Mohseni, ``Universal discriminative quantum neural networks'' (2018).
arXiv:1805.08654

[7] Edward Grant, Marcello Benedetti, Shuxiang Cao, Andrew Hallam, Joshua Lockhart, Vid Stojevic, Andrew G Green, and Simone Severini, ``Hierarchical quantum classifiers'' npj Quantum Information 4, 1–8 (2018).
https://doi.org/10.1038/s41534-018-0116-9

[8] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini, ``Parameterized quantum circuits as machine learning models'' Quantum Science and Technology 4, 043001 (2019).
https://doi.org/10.1088/2058-9565/ab4eb5

[9] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, ``Quantum circuit learning'' Phys. Rev. A 98, 032309 (2018).
https://doi.org/10.1103/PhysRevA.98.032309

[10] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta, ``Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets'' Nature 549, 242–246 (2017).
https://doi.org/10.1038/nature23879

[11] Kosuke Mitarai, Tennin Yan, and Keisuke Fujii, ``Generalization of the Output of a Variational Quantum Eigensolver by Parameter Interpolation with a Low-depth Ansatz'' Phys. Rev. Applied 11, 044087 (2019).
https://doi.org/10.1103/PhysRevApplied.11.044087

[12] Artur F. Izmaylov, Tzu-Ching Yen, and Ilya G. Ryabinkin, ``Revising the measurement process in the variational quantum eigensolver: is it possible to reduce the number of separately measured operators?'' Chem. Sci. 10, 3746–3755 (2019).
https://doi.org/10.1039/C8SC05592K

[13] Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello 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

[14] Rui Li, Unai Alvarez-Rodriguez, Lucas Lamata, and Enrique Solano, ``Approximate Quantum Adders with Genetic Algorithms: An IBM Quantum Experience'' Quantum Measurements and Quantum Metrology 4, 1 –7 (26 Jul. 2017).
https://doi.org/10.1515/qmetro-2017-0001
https://www.degruyter.com/view/journals/qmetro/4/1/article-p1.xml

[15] Harper R. Grimsley, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall, ``An adaptive variational algorithm for exact molecular simulations on a quantum computer'' Nature Communications 10 (2019).
https://doi.org/10.1038/s41467-019-10988-2

[16] Ken M. Nakanishi, Keisuke Fujii, and Synge Todo, ``Sequential minimal optimization for quantum-classical hybrid algorithms'' Phys. Rev. Research 2, 043158 (2020).
https://doi.org/10.1103/PhysRevResearch.2.043158

[17] Robert M. Parrish, Joseph T. Iosue, Asier Ozaeta, and Peter L. McMahon, ``A Jacobi Diagonalization and Anderson Acceleration Algorithm For Variational Quantum Algorithm Parameter Optimization'' (2019).
arXiv:1904.03206

[18] D.P. Bertsekas ``Nonlinear Programming'' Athena Scientific (1999).

[19] Ankan Sahaand Ambuj Tewari ``On the Finite Time Convergence of Cyclic Coordinate Descent Methods'' (2010).
arXiv:1005.2146

[20] Stephen J Wright ``Coordinate descent algorithms'' Mathematical Programming 151, 3–34 (2015).
https://doi.org/10.1007/s10107-015-0892-3

[21] Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven, ``Barren plateaus in quantum neural network training landscapes'' Nature Communications 9 (2018).
https://doi.org/10.1038/s41467-018-07090-4

[22] Héctor Abraham, Ismail Yunus Akhalwaya, Gadi Aleksandrowicz, Thomas Alexander, Eli Arbel, Abraham Asfaw, Carlos Azaustre, Panagiotis Barkoutsos, George Barron, and Luciano Bello, ``Qiskit: An Open-source Framework for Quantum Computing'' (2019).
https://doi.org/10.5281/zenodo.2562110

[23] Sukin Sim, Peter D. Johnson, and Alán Aspuru-Guzik, ``Expressibility and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum-Classical Algorithms'' Advanced Quantum Technologies 2, 1900070 (2019).
https://doi.org/10.1002/qute.201900070

[24] J. C. Spall ``Multivariate stochastic approximation using a simultaneous perturbation gradient approximation'' IEEE Transactions on Automatic Control 37, 332–341 (1992).
https://doi.org/10.1109/9.119632

[25] Diederik P. Kingmaand Jimmy Ba ``Adam: A Method for Stochastic Optimization'' (2014).
arXiv:1412.6980

[26] Anders Sørensenand Klaus Mølmer ``Quantum Computation with Ions in Thermal Motion'' Phys. Rev. Lett. 82, 1971–1974 (1999).
https://doi.org/10.1103/PhysRevLett.82.1971

[27] Norbert M. Linke, Dmitri Maslov, Martin Roetteler, Shantanu Debnath, Caroline Figgatt, Kevin A. Landsman, Kenneth Wright, and Christopher Monroe, ``Experimental comparison of two quantum computing architectures'' Proceedings of the National Academy of Sciences 114, 3305–3310 (2017).
https://doi.org/10.1073/pnas.1618020114
https://www.pnas.org/content/114/13/3305

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[1] Qingqing Xiong, Jinzhe Jiang, Chen Li, Xin Zhang, and Yaqian Zhao, Proceedings of the 2024 16th International Conference on Machine Learning and Computing 94 (2024) ISBN:9798400709234.

[2] Lukas Schulte, Daniel Hein, Steffen Udluft, and Thomas A. Runkler, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 1727 (2025) ISBN:979-8-3315-5736-2.

[3] Xuan-Bac Nguyen, Hoang-Quan Nguyen, Hugh Churchill, Samee U. Khan, and Khoa Luu, "Quantum visual feature encoding revisited", Quantum Machine Intelligence 6 2, 61 (2024).

[4] Ran-Yi-Liu Chen, Ben-Chi Zhao, Zhi-Xin Song, Xuan-Qiang Zhao, Kun Wang, and Xin Wang, "Hybrid quantum-classical algorithms: Foundation, design and applications", Acta Physica Sinica 70 21, 210302 (2021).

[5] Hsin-Yuan Huang, Yunchao Liu, Michael Broughton, Isaac Kim, Anurag Anshu, Zeph Landau, and Jarrod R. McClean, Proceedings of the 56th Annual ACM Symposium on Theory of Computing 1343 (2024) ISBN:9798400703836.

[6] Roeland Wiersema and Nathan Killoran, "Optimizing quantum circuits with Riemannian gradient flow", Physical Review A 107 6, 062421 (2023).

[7] Dylan Herman, Rudy Raymond, Muyuan Li, Nicolas Robles, Antonio Mezzacapo, and Marco Pistoia, "Expressivity of Variational Quantum Machine Learning on the Boolean Cube", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[8] Yaswitha Gujju, Atsushi Matsuo, and Rudy Raymond, "Quantum machine learning on near-term quantum devices: Current state of supervised and unsupervised techniques for real-world applications", Physical Review Applied 21 6, 067001 (2024).

[9] Maqsudur Rahman, Rui Liu, Anup Majumder, Pintu Chandra Paul, Kangtong Mo, Amena Begum, Kashmi Sultana, Nahida Akter, Lu Wei, Ye Zhang, and Jun Zhuang, "Benchmarking Barren Plateau Mitigation Strategies in Quantum Neural Networks on Standard and Medical Image Datasets", Journal of Imaging 12 7, 275 (2026).

[10] Ranyiliu Chen, Benchi Zhao, and Xin Wang, "Near-Term Efficient Quantum Algorithms for Entanglement Analysis", Physical Review Applied 20 2, 024071 (2023).

[11] Darya Martyniuk, Johannes Jung, and Adrian Paschke, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 1695 (2024) ISBN:979-8-3315-4137-8.

[12] Bhaskara Narottama, Sonia Aïssa, and Hidekazu Murata, "Channel Estimation and Tracking in STAR-RIS Aided Systems: A Recurrent Quantum Learning Framework", IEEE Transactions on Green Communications and Networking 9 4, 2129 (2025).

[13] Shuo Sun, Chandan Kumar, Kevin Shen, Elvira Shishenina, and Christian B. Mendl, "Evaluating Ground State Energies of Chemical Systems with Low-Depth Quantum Circuits and High Accuracy", The Journal of Physical Chemistry A 129 10, 2379 (2025).

[14] Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao, "Differentiable quantum architecture search", Quantum Science and Technology 7 4, 045023 (2022).

[15] Rudy Raymond, Alexander Buts, and Marco Pistoia, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 01 (2025) ISBN:979-8-3315-5736-2.

[16] Xin Wang, Zhixin Song, and Youle Wang, "Variational Quantum Singular Value Decomposition", Quantum 5, 483 (2021).

[17] Akash Kundu, Ludmila Botelho, and Adam Glos, "Hamiltonian-oriented homotopy quantum approximate optimization algorithm", Physical Review A 109 2, 022611 (2024).

[18] James Brown, Marc P Coons, Erika Lloyd, Alexandre Fleury, Krzysztof Bieniasz, Valentin Senicourt, and Arman Zaribafiyan, "Iterative qubit coupled cluster using only clifford circuits", Physica Scripta 100 10, 105401 (2025).

[19] Xavier Bonet-Monroig, Hao Wang, Diederick Vermetten, Bruno Senjean, Charles Moussa, Thomas Bäck, Vedran Dunjko, and Thomas E. O'Brien, "Performance comparison of optimization methods on variational quantum algorithms", Physical Review A 107 3, 032407 (2023).

[20] Yi Fan, Changsu Cao, Xusheng Xu, Zhenyu Li, Dingshun Lv, and Man-Hong Yung, "Circuit-Depth Reduction of Unitary-Coupled-Cluster Ansatz by Energy Sorting", The Journal of Physical Chemistry Letters 14 43, 9596 (2023).

[21] Arpan Choudhury, Sonaldeep Halder, Rahul Maitra, and Debashree Ghosh, "Fragment, Entangle, and Consolidate: Strong Correlation through Bifold Quantum Circuits", Journal of Chemical Theory and Computation 22 11, 5569 (2026).

[22] Lars Simon, Holger Eble, and Manuel Radons, "Introducing the kernel descent optimizer for variational quantum algorithms", Scientific Reports 15 1, 28247 (2025).

[23] Bálint Koczor and Simon C. Benjamin, "Quantum analytic descent", Physical Review Research 4 2, 023017 (2022).

[24] Kentaro Tamura, Yohichi Suzuki, Rudy Raymond, Hiroshi C. Watanabe, Yuki Sato, Ruho Kondo, Michihiko Sugawara, and Naoki Yamamoto, "Noise Robustness of Quantum Relaxation for Combinatorial Optimization", IEEE Transactions on Quantum Engineering 5, 1 (2024).

[25] Yue Ruan, Pengyue Chen, Qi Li, Ling Yang, Zhiqiang Yuan, Xiling Xue, Xi Li, and Zhihao Liu, "XY-mixer ansatz assisted by counterdiabatic driving for combinational optimization", Physical Review Research 7 1, 013243 (2025).

[26] Kirill Plekhanov, Matthias Rosenkranz, Mattia Fiorentini, and Michael Lubasch, "Variational quantum amplitude estimation", Quantum 6, 670 (2022).

[27] Mingrui Jing, Geng Liu, Hongbin Ren, and Xin Wang, "Quantum sequential scattering model for quantum state learning", Physical Review A 109 6, 062425 (2024).

[28] Ruilin Liu, Sebastián V. Romero, Izaskun Oregi, Eneko Osaba, Esther Villar-Rodriguez, and Yue Ban, "Digital Quantum Simulation and Circuit Learning for the Generation of Coherent States", Entropy 24 11, 1529 (2022).

[29] Weitang Li, Yufei Ge, Shi-Xin Zhang, Yu-Qin Chen, and Shengyu Zhang, "Efficient and Robust Parameter Optimization of the Unitary Coupled-Cluster Ansatz", Journal of Chemical Theory and Computation 20 9, 3683 (2024).

[30] Sachin Namdeo, Sachin Khurana, and Manisha J. Nene, Information Systems Engineering and Management 16, 257 (2024) ISBN:978-3-031-69200-0.

[31] César Feniou, Muhammad Hassan, Baptiste Claudon, Axel Courtat, Olivier Adjoua, Yvon Maday, and Jean-Philip Piquemal, "Greedy gradient-free adaptive variational quantum algorithms on a noisy intermediate scale quantum computer", Scientific Reports 15 1, 18689 (2025).

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

[33] Zhimin He, Chuangtao Chen, Lvzhou Li, Shenggen Zheng, and Haozhen Situ, "Quantum Architecture Search with Meta‐Learning", Advanced Quantum Technologies 5 8, 2100134 (2022).

[34] Jinkai Tian, Xiaoyu Sun, Yuxuan Du, Shanshan Zhao, Qing Liu, Kaining Zhang, Wei Yi, Wanrong Huang, Chaoyue Wang, Xingyao Wu, Min-Hsiu Hsieh, Tongliang Liu, Wenjing Yang, and Dacheng Tao, "Recent Advances for Quantum Neural Networks in Generative Learning", IEEE Transactions on Pattern Analysis and Machine Intelligence 45 10, 12321 (2023).

[35] 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).

[36] Shi-Xin Zhang, Jonathan Allcock, Zhou-Quan Wan, Shuo Liu, Jiace Sun, Hao Yu, Xing-Han Yang, Jiezhong Qiu, Zhaofeng Ye, Yu-Qin Chen, Chee-Kong Lee, Yi-Cong Zheng, Shao-Kai Jian, Hong Yao, Chang-Yu Hsieh, and Shengyu Zhang, "TensorCircuit: a Quantum Software Framework for the NISQ Era", Quantum 7, 912 (2023).

[37] Manpreet Singh Jattana, Fengping Jin, Hans De Raedt, and Kristel Michielsen, "Assessment of the Variational Quantum Eigensolver: Application to the Heisenberg Model", Frontiers in Physics 10, 907160 (2022).

[38] Yaochong Li, Jing Zhang, Rigui Zhou, Yi Qu, and Ruiqing Xu, "AQEA-QAS: An Adaptive Quantum Evolutionary Algorithm for Quantum Architecture Search", Entropy 27 7, 733 (2025).

[39] 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).

[40] Aashna Anil Zade, Kenji Sugisaki, Matthias Werner, Ana Palacios, Jordi Riu, Jan Nogué, Artur Garcia-Saez, Arnau Riera, and V. S. Prasannaa, "Quantum annealing eigensolver as a NISQ era tool for probing strong correlation effects in quantum chemistry", The European Physical Journal Plus 140 9, 930 (2025).

[41] Péter Mernyei, Konstantinos Meichanetzidis, and İsmail İlkan Ceylan, "Equivariant quantum graph circuits: constructions for universal approximation over graphs", Quantum Machine Intelligence 5 1, 6 (2023).

[42] Bojia Duan and Chang-Yu Hsieh, "Hamiltonian-based data loading with shallow quantum circuits", Physical Review A 106 5, 052422 (2022).

[43] Zhijian Lai, Jiang Hu, Taehee Ko, Jiayuan Wu, and Dong An, "Interpolation-based coordinate descent method for parameterized quantum circuits", Communications Physics 9 1, 41 (2026).

[44] Shiro Tamiya and Hayata Yamasaki, "Stochastic gradient line Bayesian optimization for efficient noise-robust optimization of parameterized quantum circuits", npj Quantum Information 8 1, 90 (2022).

[45] Ya-Qi Ling, Jun-Hu Zhang, Li-Hua Zhang, Yan-Ran Li, and Hui-Lei Huang, "Image Classification Using Hybrid Classical-Quantum Neutral Networks", International Journal of Theoretical Physics 63 5, 125 (2024).

[46] Jaividhyarthi Vivekanand and Yogesh Kumar B, "A-QAOA: Improved Convergence via Adaptive Parameter Scheduling in Precision Agriculture", (2026).

[47] 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).

[48] Jiawen Dai, Jialiang Gu, Raymond Ho, Gary Man-Tat Man, Kwok Tai Chui, and Kevin Hung, "EAQAS: Embedding-Aware Quantum Architecture Search via cross-attention fusion and hierarchical representation learning", EPJ Quantum Technology 13 1, 22 (2026).

[49] Fu Chen, Qinglin Zhao, Li Feng, Chuangtao Chen, Yangbin Lin, and Jianhong Lin, "Quantum mixed-state self-attention network", Neural Networks 185, 107123 (2025).

[50] Yuxuan Du, Tao Huang, Shan You, Min-Hsiu Hsieh, and Dacheng Tao, "Quantum circuit architecture search for variational quantum algorithms", npj Quantum Information 8 1, 62 (2022).

[51] Trung Q. Duong, Long D. Nguyen, Bhaskara Narottama, James Adu Ansere, Dang Van Huynh, and Hyundong Shin, "Quantum-Inspired Real-Time Optimization for 6G Networks: Opportunities, Challenges, and the Road Ahead", IEEE Open Journal of the Communications Society 3, 1347 (2022).

[52] Benjamin Jaderberg, Alexander Eisfeld, Dieter Jaksch, and Sarah Mostame, "Recompilation-enhanced simulation of electron–phonon dynamics on IBM quantum computers", New Journal of Physics 24 9, 093017 (2022).

[53] Zhimin He, Xuefen Zhang, Chuangtao Chen, Zhiming Huang, Yan Zhou, and Haozhen Situ, "A GNN-based predictor for quantum architecture search", Quantum Information Processing 22 2, 128 (2023).

[54] S. Mangini, F. Tacchino, D. Gerace, D. Bajoni, and C. Macchiavello, "Quantum computing models for artificial neural networks", Europhysics Letters 134 1, 10002 (2021).

[55] Jules Tilly, P. V. Sriluckshmy, Akashkumar Patel, Enrico Fontana, Ivan Rungger, Edward Grant, Robert Anderson, Jonathan Tennyson, and George H. Booth, "Reduced density matrix sampling: Self-consistent embedding and multiscale electronic structure on current generation quantum computers", Physical Review Research 3 3, 033230 (2021).

[56] Dimitris Alevras, Mihir Metkar, Takahiro Yamamoto, Vaibhaw Kumar, Triet Friedhoff, Jae-Eun Park, Mitsuharu Takeori, Mariana LaDue, Wade Davis, and Alexey Galda, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 488 (2024) ISBN:979-8-3315-4137-8.

[57] F. Barratt, James Dborin, Matthias Bal, Vid Stojevic, Frank Pollmann, and A. G. Green, "Parallel quantum simulation of large systems on small NISQ computers", npj Quantum Information 7 1, 79 (2021).

[58] Ben Jaderberg, George Pennington, Kate V. Marshall, Lewis W. Anderson, Abhishek Agarwal, Lachlan P. Lindoy, Ivan Rungger, Stefano Mensa, and Jason Crain, "Variational preparation of normal matrix product states on quantum computers", Physical Review Research 8 1, 013081 (2026).

[59] Shabnam Jabeen, Dmytro Kurdydyk, Aadi Palnitkar, Mihir Talati, Jeffrey Yan, and Jinghong Yang, "Quantum machine learning for state tomography using classical data", Quantum Machine Intelligence 8 1, 40 (2026).

[60] Yuki Sato, Hiroshi C. Watanabe, Rudy Raymond, Ruho Kondo, Kaito Wada, Katsuhiro Endo, Michihiko Sugawara, and Naoki Yamamoto, "Variational quantum algorithm for generalized eigenvalue problems and its application to the finite-element method", Physical Review A 108 2, 022429 (2023).

[61] Hugh G. A. Burton, Daniel Marti-Dafcik, David P. Tew, and David J. Wales, "Exact electronic states with shallow quantum circuits from global optimisation", npj Quantum Information 9 1, 75 (2023).

[62] David Wierichs, Josh Izaac, Cody Wang, and Cedric Yen-Yu Lin, "General parameter-shift rules for quantum gradients", Quantum 6, 677 (2022).

[63] Sandeep Suresh Cranganore, Vincenzo De Maio, Ivona Brandic, and Ewa Deelman, "Paving the way to hybrid quantum–classical scientific workflows", Future Generation Computer Systems 158, 346 (2024).

[64] Nikita A. Zemlevskiy, "Scalable quantum simulations of scattering in scalar field theory on 120 qubits", Physical Review D 112 3, 034502 (2025).

[65] Javier Mancilla and Christophe Pere, "A Preprocessing Perspective for Quantum Machine Learning Classification Advantage in Finance Using NISQ Algorithms", Entropy 24 11, 1656 (2022).

[66] Aikaterini Gratsea and Patrick Huembeli, "The effect of the processing and measurement operators on the expressive power of quantum models", Quantum Machine Intelligence 5 2, 32 (2023).

[67] Anqi 安琪 Zhang 张, Chunhui 春辉 Wu 武, and Shengmei 生妹 Zhao 赵, "Gray code based gradient-free optimization algorithm for parameterized quantum circuit", Chinese Physics B 33 2, 020311 (2024).

[68] Zhiyan Ding, Taehee Ko, Jiahao Yao, Lin Lin, and Xiantao Li, "Random coordinate descent: A simple alternative for optimizing parameterized quantum circuits", Physical Review Research 6 3, 033029 (2024).

[69] 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).

[70] Anqi 安琪 Zhang 张, Kelun 可伦 Wang 王, Yihua 逸华 Wu 吴, and Sheng-Mei 生妹 Zhao 赵, "Single-qubit quantum classifier based on gradient-free optimization algorithm", Chinese Physics B 32 10, 100308 (2023).

[71] Tomislav Piskor, Jan-Michael Reiner, Sebastian Zanker, Nicolas Vogt, Michael Marthaler, Frank K. Wilhelm, and Florian G. Eich, "Using gradient-based algorithms to determine ground-state energies on a quantum computer", Physical Review A 105 6, 062415 (2022).

[72] Tao Jiang, John Rogers, Marius S. Frank, Ove Christiansen, Yong-Xin Yao, and Nicola Lanatà, "Error mitigation in variational quantum eigensolvers using tailored probabilistic machine learning", Physical Review Research 6 3, 033069 (2024).

[73] Maria Schuld and Nathan Killoran, "Is Quantum Advantage the Right Goal for Quantum Machine Learning?", PRX Quantum 3 3, 030101 (2022).

[74] Erik J. Gustafson, Juha Tiihonen, Diana Chamaki, Farshud Sorourifar, J. Wayne Mullinax, Andy C. Y. Li, Filip B. Maciejewski, Nicolas P. D. Sawaya, Jaron T. Krogel, David E. Bernal Neira, and Norm M. Tubman, "Surrogate optimization of variational quantum circuits", Proceedings of the National Academy of Sciences 122 36, e2408530122 (2025).

[75] Marcello Benedetti, Mattia Fiorentini, and Michael Lubasch, "Hardware-efficient variational quantum algorithms for time evolution", Physical Review Research 3 3, 033083 (2021).

[76] Frederik vom Ende, "Finite-Dimensional Stinespring Curves Can Approximate Any Dynamics", Open Systems & Information Dynamics 31 01, 2450004 (2024).

[77] Mark-Oliver Wolf, Tom Ewen, and Ivica Turkalj, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 560 (2023) ISBN:979-8-3503-4323-6.

[78] Bhaskara Narottama, Berk Canberk, Simon L. Cotton, Hyundong Shin, George K. Karagiannidis, and Trung Q. Duong, "Non-Centralized Quantum Neural Networks for Cell-Free MIMO Systems", IEEE Transactions on Wireless Communications 25, 9971 (2026).

[79] Arseny Kovyrshin, Dilhan Manawadu, Edoardo Altamura, George Pennington, Benjamin Jaderberg, Sebastian Brandhofer, Anton Nykänen, Aaron Miller, Walter Talarico, Stefan Knecht, Fabijan Pavošević, Alberto Baiardi, Francesco Tacchino, Ivano Tavernelli, Stefano Mensa, Jason Crain, Lars Tornberg, and Anders Broo, "Approximate quantum circuit compilation for proton-transfer kinetics on quantum processors", Physical Chemistry Chemical Physics 28 4, 3035 (2026).

[80] Adam Glos, Aleksandra Krawiec, and Zoltán Zimborás, "Space-efficient binary optimization for variational quantum computing", npj Quantum Information 8 1, 39 (2022).

[81] Maijie Deng, Zhimin He, Shenggen Zheng, Yan Zhou, Fei Zhang, and Haozhen Situ, "A progressive predictor-based quantum architecture search with active learning", The European Physical Journal Plus 138 10, 905 (2023).

[82] Asahi Chikaoka and Haozhao Liang, "Quantum computing for the Lipkin model with unitary coupled cluster and structure learning ansatz * ", Chinese Physics C 46 2, 024106 (2022).

[83] Sonaldeep Halder, Kartikey Anand, and Rahul Maitra, "Construction of Chemistry-Inspired Dynamic Ansatz Utilizing Generative Machine Learning", The Journal of Physical Chemistry A 129 26, 5889 (2025).

[84] Hsin-Yi Lin, Huan-Hsin Tseng, Samuel Yen-Chi Chen, and Shinjae Yoo, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 1884 (2025) ISBN:979-8-3315-5736-2.

[85] L. C. G. Govia, C. Poole, M. Saffman, and H. K. Krovi, "Freedom of the mixer rotation axis improves performance in the quantum approximate optimization algorithm", Physical Review A 104 6, 062428 (2021).

[86] Li Ding and Lee Spector, "Multi-Objective Evolutionary Architecture Search for Parameterized Quantum Circuits", Entropy 25 1, 93 (2023).

[87] David Amaro, Carlo Modica, Matthias Rosenkranz, Mattia Fiorentini, Marcello Benedetti, and Michael Lubasch, "Filtering variational quantum algorithms for combinatorial optimization", Quantum Science and Technology 7 1, 015021 (2022).

[88] Arianna Crippa, Karl Jansen, and Enrico Rinaldi, "Analysis of the confinement string in (2+1)-dimensional Quantum Electrodynamics with a trapped-ion quantum computer", Communications Physics 9 1, 46 (2026).

[89] Anton Nykänen, Matteo A. C. Rossi, Elsi-Mari Borrelli, Sabrina Maniscalco, Guillermo García-Pérez, and Adam Glos, "Mitigating the measurement overhead of ADAPT-VQE with optimized informationally complete generalized measurements", Physical Review Research 7 4, 043114 (2025).

[90] Run-Sheng 润盛 Zhao 赵, Hong-Yang 鸿洋 Ma 马, Tao 涛 Cheng 程, Shuang 爽 Wang 王, and Xing-Kui 兴奎 Fan 范, "Quantum generative adversarial networks based on a readout error mitigation method with fault tolerant mechanism", Chinese Physics B 33 4, 040304 (2024).

[91] Chayan Patra, Sonaldeep Halder, and Rahul Maitra, "Projective quantum eigensolver via adiabatically decoupled subsystem evolution: A resource efficient approach to molecular energetics in noisy quantum computers", The Journal of Chemical Physics 160 21, 214122 (2024).

[92] Xin Wang, Bo Qi, Yabo Wang, and Daoyi Dong, "Entanglement-variational hardware-efficient ansatz for eigensolvers", Physical Review Applied 21 3, 034059 (2024).

[93] 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).

[94] 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).

[95] Joona V. Pankkonen, "Two-gate extensions of Free Axis and Free Quaternion Selection for sequential optimization of parameterized quantum circuits", Physics Open 28, 100444 (2026).

[96] Nam Nguyen and Kwang-Cheng Chen, "Quantum Embedding Search for Quantum Machine Learning", IEEE Access 10, 41444 (2022).

[97] Leopoldo Sarra, Kevin Ellis, and Florian Marquardt, "Discovering quantum circuit components with program synthesis", Machine Learning: Science and Technology 5 2, 025029 (2024).

[98] Adrián Pérez-Salinas, Hao Wang, and Xavier Bonet-Monroig, "Analyzing variational quantum landscapes with information content", npj Quantum Information 10 1, 27 (2024).

[99] Noah F. Berthusen, Thaís V. Trevisan, Thomas Iadecola, and Peter P. Orth, "Quantum dynamics simulations beyond the coherence time on noisy intermediate-scale quantum hardware by variational Trotter compression", Physical Review Research 4 2, 023097 (2022).

[100] Nikita Astrakhantsev, Sheng-Hsuan Lin, Frank Pollmann, and Adam Smith, "Time evolution of uniform sequential circuits", Physical Review Research 5 3, 033187 (2023).

[101] Benjamin Avramidis, Hari P. Paudel, Dominic Alfonso, Yuhua Duan, and Kenneth D. Jordan, "Ground state property calculations of LiHn complexes using IBM Qiskit’s quantum simulator", AIP Advances 14 3, 035047 (2024).

[102] 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).

[103] Bodo Rosenhahn and Tobias J. Osborne, "Monte Carlo graph search for quantum circuit optimization", Physical Review A 108 6, 062615 (2023).

[104] Bijan Garnayak, Neeraj Marwaha, and Mamata Garanayak, "Performance Assessment of Quantum CNN vs RNN for Medicinal Leaf Classification with UI Sustenance", Journal of Electrical and Electronic Engineering 14 2, 66 (2026).

[105] Meghashrita Das, Arundhuti Naskar, Pabitra Mitra, and Biswajit Basu, "Shallow quantum neural networks (SQNNs) with application to crack identification", Applied Intelligence 54 2, 1247 (2024).

[106] 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).

[107] Tianlong Chen, Zhenyu Zhang, Hanrui Wang, Jiaqi Gu, Zirui Li, David Z. Pan, Frederic T. Chong, Song Han, and Zhangyang Wang, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 51 (2023) ISBN:979-8-3503-4323-6.

[108] Thomas Ayral, Pauline Besserve, Denis Lacroix, and Edgar Andres Ruiz Guzman, "Quantum computing with and for many-body physics", The European Physical Journal A 59 10, 227 (2023).

[109] 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).

[110] 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).

[111] Chayan Patra, Debaarjun Mukherjee, Sonaldeep Halder, Dibyendu Mondal, and Rahul Maitra, "Toward a resource-optimized dynamic quantum algorithm via non-iterative auxiliary subspace corrections", The Journal of Chemical Physics 161 14, 144119 (2024).

[112] Mayank Shekhar Jha, Sameul Yen-Chi Chen, Chetan Kulkarni, and Joongheon Kim, "A comprehensive review on quantum deep neural networks for prognostics and health management: Fundamentals, challenges and opportunities", Engineering Applications of Artificial Intelligence 177, 114991 (2026).

[113] Carlos Bravo-Prieto, Julien Baglio, Marco Cè, Anthony Francis, Dorota M. Grabowska, and Stefano Carrazza, "Style-based quantum generative adversarial networks for Monte Carlo events", Quantum 6, 777 (2022).

[114] Hongshun Yao, Xia Liu, Mingrui Jing, Guangxi Li, and Xin Wang, "LCQNN: linear combination of quantum neural networks", Quantum Machine Intelligence 8 1, 29 (2026).

[115] K. Asnaashari, D. Bondarenko, and R. V. Krems, "Advantages of discrete variable representation in variational quantum eigensolvers for vibrational energy calculations", Physical Chemistry Chemical Physics 28 13, 7900 (2026).

[116] Francesco Preti, Michael Schilling, Sofiene Jerbi, Lea M. Trenkwalder, Hendrik Poulsen Nautrup, Felix Motzoi, and Hans J. Briegel, "Hybrid discrete-continuous compilation of trapped-ion quantum circuits with deep reinforcement learning", Quantum 8, 1343 (2024).

[117] Kaito Wada, Rudy Raymond, Yu-ya Ohnishi, Eriko Kaminishi, Michihiko Sugawara, Naoki Yamamoto, and Hiroshi C. Watanabe, "Simulating time evolution with fully optimized single-qubit gates on parametrized quantum circuits", Physical Review A 105 6, 062421 (2022).

[118] Li Ding and Lee Spector, Proceedings of the Genetic and Evolutionary Computation Conference Companion 2190 (2022) ISBN:9781450392686.

[119] Roeland Wiersema, Dylan Lewis, David Wierichs, Juan Carrasquilla, and Nathan Killoran, "Here comes the SU(N): multivariate quantum gates and gradients", Quantum 8, 1275 (2024).

[120] Sonaldeep Halder, Anish Dey, Chinmay Shrikhande, and Rahul Maitra, "Machine learning assisted construction of a shallow depth dynamic ansatz for noisy quantum hardware", Chemical Science 15 9, 3279 (2024).

[121] Enrico Fontana, Manuel S. Rudolph, Ross Duncan, Ivan Rungger, and Cristina Cîrstoiu, "Classical simulations of noisy variational quantum circuits", npj Quantum Information 11 1, 84 (2025).

[122] Bhaskara Narottama and Sonia Aïssa, 2023 IEEE 34th Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC) 1 (2023) ISBN:978-1-6654-6483-3.

[123] Thomas Hubregtsen, Frederik Wilde, Shozab Qasim, and Jens Eisert, "Single-component gradient rules for variational quantum algorithms", Quantum Science and Technology 7 3, 035008 (2022).

[124] Agustin Silva, Omar Gustavo Zabaleta, and Constancio Miguel Arizmendi, "Learning Mixed Strategies in Quantum Games with Imperfect Information", Quantum Reports 4 4, 462 (2022).

[125] Zhimin He, Maijie Deng, Shenggen Zheng, Lvzhou Li, and Haozhen Situ, "GSQAS: Graph Self-supervised Quantum Architecture Search", Physica A: Statistical Mechanics and its Applications 630, 129286 (2023).

[126] Maitreyee Sarkar, Lisa Roy, Akash Popat Gutal, Atul Kumar, and Manikandan Paranjothy, "Symmetry-Match Fraction for Active-Space Selection: A Simple Route to Accurate Reaction Energies in VQE/UCCSD", The Journal of Physical Chemistry A 130 28, 5594 (2026).

[127] A. M. Romero, J. Engel, Ho Lun Tang, and Sophia E. Economou, "Solving nuclear structure problems with the adaptive variational quantum algorithm", Physical Review C 105 6, 064317 (2022).

[128] Matan Ben-Dov, David Shnaiderov, Adi Makmal, and Emanuele G. Dalla Torre, "Approximate encoding of quantum states using shallow circuits", npj Quantum Information 10 1, 65 (2024).

[129] Xiaodong Ding, Yingying Wang, Haiping Dang, Shiqin Di, and Shunli Zhang, "Electrocardiogram arrhythmia detection based on embedded quantum-enhanced neural network feature extraction", Physica Scripta 100 12, 125115 (2025).

[130] Sergio Altares-López, Juan José García-Ripoll, and Angela Ribeiro, "AutoQML: Automatic generation and training of robust quantum-inspired classifiers by using evolutionary algorithms on grayscale images", Expert Systems with Applications 244, 122984 (2024).

[131] Alisa Haukisalmi, Daniel Paz Ramos, Matti Raasakka, Andrea Marchesin, Lauri Ylinen, and Ilkka Tittonen, "Noisy quantum simulation: Performance and resource considerations for the Tavis-Cummings and Heisenberg models", Physical Review Research 7 4, 043254 (2025).

[132] Eriko Kaminishi, Takashi Mori, Michihiko Sugawara, and Naoki Yamamoto, "Impact of measurement noise on escaping saddles in variational quantum algorithms", Scientific Reports 16 1, 9390 (2026).

[133] Rodrigo Coelho, André Sequeira, and Luís Paulo Santos, "VQC-based reinforcement learning with data re-uploading: performance and trainability", Quantum Machine Intelligence 6 2, 53 (2024).

[134] Seenivasan Hariharan, Sachin Kinge, and Lucas Visscher, "Modeling Heterogeneous Catalysis Using Quantum Computers: An Academic and Industry Perspective", Journal of Chemical Information and Modeling 65 2, 472 (2025).

[135] Palak Chawla, Shweta, K. R. Swain, Tushti Patel, Renu Bala, Disha Shetty, Kenji Sugisaki, Sudhindu Bikash Mandal, Jordi Riu, Jan Nogué, V. S. Prasannaa, and B. P. Das, "Relativistic variational-quantum-eigensolver calculations of molecular electric dipole moments on quantum hardware", Physical Review A 111 2, 022817 (2025).

[136] Rowan Pellow-Jarman, Anban Pillay, Ilya Sinayskiy, and Francesco Petruccione, "Hybrid genetic optimization for quantum feature map design", Quantum Machine Intelligence 6 2, 45 (2024).

[137] Shanchuan Li, Daisuke Tsukayama, Jun-ichi Shirakashi, Tetsuo Shibuya, and Hiroshi Imai, "Quantum architecture search with neural predictor based on ZX-calculus", EPJ Quantum Technology 12 1, 106 (2025).

[138] Yangyang Li, Xiaobin Hao, Guanlong Liu, Ronghua Shang, and Licheng Jiao, "QEA-QCNN: optimization of quantum convolutional neural network architecture based on quantum evolution", Memetic Computing 16 3, 233 (2024).

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

[140] Vu Tuan Hai, Nguyen Tan Viet, Jesus Urbaneja, Nguyen Vu Linh, Lan Nguyen Tran, and Le Bin Ho, "Multi-target quantum compilation algorithm", Machine Learning: Science and Technology 5 4, 045057 (2024).

[141] 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).

[142] 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).

[143] Kentaro Yamamoto, David Zsolt Manrique, Irfan T. Khan, Hideaki Sawada, and David Muñoz Ramo, "Quantum hardware calculations of periodic systems with partition-measurement symmetry verification: Simplified models of hydrogen chain and iron crystals", Physical Review Research 4 3, 033110 (2022).

[144] Izuho Koyasu, Rudy Raymond, and Hiroshi Imai, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 457 (2023) ISBN:979-8-3503-4323-6.

[145] Muhammad Umer, Eleftherios Mastorakis, and Dimitris G Angelakis, "Efficient estimation and sequential optimization of cost functions in variational quantum algorithms", Quantum Science and Technology 10 3, 035022 (2025).

[146] Haozhen Situ, Zhengjiang Li, Zhimin He, Qin Li, and Jinjing Shi, "AutoML-driven optimization of variational quantum circuit", Information Sciences 717, 122272 (2025).

[147] Matt Lourens, Ilya Sinayskiy, Daniel K. Park, Carsten Blank, and Francesco Petruccione, "Hierarchical quantum circuit representations for neural architecture search", npj Quantum Information 9 1, 79 (2023).

[148] 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).

[149] Hiep L Thi, Lecture Notes in Networks and Systems 1180, 111 (2025) ISBN:978-981-97-9323-5.

[150] Sergio Altares-López, Angela Ribeiro, and Juan José García-Ripoll, "Automatic design of quantum feature maps", Quantum Science and Technology 6 4, 045015 (2021).

[151] Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer, "Effect of data encoding on the expressive power of variational quantum-machine-learning models", Physical Review A 103 3, 032430 (2021).

[152] Jonas Jäger, Thierry N. Kaldenbach, Max Haas, and Erik Schultheis, "Fast gradient-free optimization of excitations in variational quantum eigensolvers", Communications Physics 8 1, 418 (2025).

[153] Vicente P. Soloviev, Vedran Dunjko, Concha Bielza, Pedro Larrañaga, and Hao Wang, "Trainability maximization using estimation of distribution algorithms assisted by surrogate modelling for quantum architecture search", EPJ Quantum Technology 11 1, 69 (2024).

[154] Matan Ben Dov, Itai Arad, and Emanuele G Dalla Torre, "Quantum landscape tomography for efficient single-gate optimization on quantum computers", Quantum Science and Technology 11 3, 035025 (2026).

[155] Zhimin He, Junjian Su, Chuangtao Chen, Minghua Pan, and Haozhen Situ, "Search space pruning for quantum architecture search", The European Physical Journal Plus 137 4, 491 (2022).

[156] Bojia Duan and Chang-Yu Hsieh, "Compact and classically preprocessed data-loading quantum circuit as a quantum random access memory", Physical Review A 110 1, 012616 (2024).

[157] Bhaskara Narottama, Abrar Ul Haq, James Adu Ansere, Nidhi Simmons, Berk Canberk, Simon L. Cotton, Hyundong Shin, and Trung Q. Duong, "Quantum Deep Reinforcement Learning for Digital Twin-Enabled 6G Networks and Semantic Communications: Considerations for Adoption and Security", IEEE Transactions on Network Science and Engineering 13, 2053 (2026).

[158] Tatiana A. Bespalova and Oleksandr Kyriienko, "Hamiltonian Operator Approximation for Energy Measurement and Ground-State Preparation", PRX Quantum 2 3, 030318 (2021).

[159] Davide Castaldo, Soran Jahangiri, Alain Delgado, and Stefano Corni, "Quantum Simulation of Molecules in Solution", Journal of Chemical Theory and Computation 18 12, 7457 (2022).

[160] 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).

[161] P. Besserve and T. Ayral, "Unraveling correlated material properties with noisy quantum computers: Natural orbitalized variational quantum eigensolving of extended impurity models within a slave-boson approach", Physical Review B 105 11, 115108 (2022).

[162] Ngoc Nhi Nguyen, Thai T. Vu, John Le, Hoa Khanh Dam, Dung Hoang Duong, and Dinh Thai Hoang, IFIP Advances in Information and Communication Technology 787, 434 (2026) ISBN:978-3-032-27992-7.

[163] Jonas Jäger, Philipp Elsässer, and Elham Torabian, "Quantum feature-map learning with reduced resource overhead", Physical Review Research 8 2, 023247 (2026).

[164] Hrushikesh Patil, Yulun Wang, and Predrag S. Krstić, "Variational quantum linear solver with a dynamic ansatz", Physical Review A 105 1, 012423 (2022).

[165] Hiroshi C. Watanabe, Rudy Raymond, Yu-Ya Ohnishi, Eriko Kaminishi, and Michihiko Sugawara, 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 100 (2021) ISBN:978-1-6654-1691-7.

[166] Tom Ewen, Ivica Turkalj, Patrick Holzer, and Mark-Oliver Wolf, "Application of ZX-calculus to quantum architecture search", Quantum Machine Intelligence 7 1, 34 (2025).

[167] Bowen Li, Ting Li, and Fei Li, "A design method for efficient variational quantum models based on specific Pauli axis", Quantum Information Processing 22 10, 387 (2023).

[168] Katsuhiro Endo, Yuki Sato, Rudy Raymond, Kaito Wada, Naoki Yamamoto, and Hiroshi C. Watanabe, "Optimal parameter configurations for sequential optimization of the variational quantum eigensolver", Physical Review Research 5 4, 043136 (2023).

[169] Kosei Teramoto, Rudy Raymond, and Hiroshi Imai, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 543 (2023) ISBN:979-8-3503-4323-6.

[170] Yuto Takaki, Kosuke Mitarai, Makoto Negoro, Keisuke Fujii, and Masahiro Kitagawa, "Learning temporal data with a variational quantum recurrent neural network", Physical Review A 103 5, 052414 (2021).

[171] WenShan Xu, Ri-Gui Zhou, YaoChong Li, and XiaoXue Zhang, "Towards an efficient variational quantum algorithm for solving linear equations", Communications in Theoretical Physics 76 11, 115103 (2024).

[172] Benjamin D M Jones, Lana Mineh, and Ashley Montanaro, "Benchmarking a wide range of optimisers for solving the Fermi–Hubbard model using the variational quantum eigensolver", Quantum Science and Technology 10 4, 045032 (2025).

[173] Jingwei Wen, Zhengan Wang, Chitong Chen, Junxiang Xiao, Hang Li, Ling Qian, Zhiguo Huang, Heng Fan, Shijie Wei, and Guilu Long, "A full circuit-based quantum algorithm for excited-states in quantum chemistry", Quantum 8, 1219 (2024).

[174] Tohid PayamiFar, Reza Sojoudizadeh, Hadi Azizian, and Loghman Rahimi, "Seismic Optimization of Steel Mega‐Braced Frame With Improved Prairie Dog Metaheuristic Optimization Algorithm", The Structural Design of Tall and Special Buildings 34 3, e2207 (2025).

[175] Zhijian Lai, Jiang Hu, Dong An, and Zaiwen Wen, "Extended parameter-shift rules with minimal derivative variance for parameterized quantum circuits", Physical Review Applied 25 1, 014005 (2026).

[176] Yingli Yang, Zongkang Zhang, Anbang Wang, Xiaosi Xu, Xiaoting Wang, and Ying Li, "Maximizing quantum-computing expressive power through randomized circuits", Physical Review Research 6 2, 023098 (2024).

[177] Shuxiang Cao, Weixi Zhang, Jules Tilly, Abhishek Agarwal, Mustafa Bakr, Giulio Campanaro, Simone D Fasciati, James Wills, Boris Shteynas, Vivek Chidambaram, Peter Leek, and Ivan Rungger, "Encoding optimization for quantum machine learning demonstrated on a superconducting transmon qutrit", Quantum Science and Technology 9 4, 045037 (2024).

[178] Zirui Sheng, Weitang Li, and Zhigang Shuai, "Quantum-computational chemistry in noisy intermediate-scale quantum era: TenCirChem and its application", Chinese Science Bulletin 70 34, 5792 (2025).

[179] Hongxiang Chen, Max Nusspickel, Jules Tilly, and George H. Booth, "Variational quantum eigensolver for dynamic correlation functions", Physical Review A 104 3, 032405 (2021).

[180] Manpreet Singh Jattana, Fengping Jin, Hans De Raedt, and Kristel Michielsen, "Improved Variational Quantum Eigensolver Via Quasidynamical Evolution", Physical Review Applied 19 2, 024047 (2023).

[181] Nils Herrmann, Daanish Arya, Marcus W. Doherty, Angus Mingare, Jason C. Pillay, Florian Preis, and Stefan Prestel, 2023 IEEE International Conference on Quantum Software (QSW) 162 (2023) ISBN:979-8-3503-0479-4.

[182] Jun Zhuang, Jack Cunningham, and Chaowen Guan, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 109 (2025) ISBN:979-8-3315-5736-2.

[183] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, "Quantum computational chemistry", Reviews of Modern Physics 92 1, 015003 (2020).

[184] 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).

[185] Marcello Benedetti, Erika Lloyd, Stefan Sack, and Mattia Fiorentini, "Parameterized quantum circuits as machine learning models", Quantum Science and Technology 4 4, 043001 (2019).

[186] Ken M. Nakanishi, Keisuke Fujii, and Synge Todo, "Sequential minimal optimization for quantum-classical hybrid algorithms", Physical Review Research 2 4, 043158 (2020).

[187] Ryan LaRose and Brian Coyle, "Robust data encodings for quantum classifiers", Physical Review A 102 3, 032420 (2020).

[188] D. Chivilikhin, A. Samarin, V. Ulyantsev, I. Iorsh, A. R. Oganov, and O. Kyriienko, "MoG-VQE: Multiobjective genetic variational quantum eigensolver", arXiv:2007.04424, (2020).

[189] Chris Cade, Lana Mineh, Ashley Montanaro, and Stasja Stanisic, "Strategies for solving the Fermi-Hubbard model on near-term quantum computers", Physical Review B 102 23, 235122 (2020).

[190] David Wierichs, Christian Gogolin, and Michael Kastoryano, "Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer", Physical Review Research 2 4, 043246 (2020).

[191] Andrea Mari, Thomas R. Bromley, and Nathan Killoran, "Estimating the gradient and higher-order derivatives on quantum hardware", Physical Review A 103 1, 012405 (2021).

[192] Arthur G. Rattew, Shaohan Hu, Marco Pistoia, Richard Chen, and Steve Wood, "A Domain-agnostic, Noise-resistant, Hardware-efficient Evolutionary Variational Quantum Eigensolver", arXiv:1910.09694, (2019).

[193] Oinam Romesh Meitei, Bryan T. Gard, George S. Barron, David P. Pappas, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall, "Gate-free state preparation for fast variational quantum eigensolver simulations", npj Quantum Information 7 1, 155 (2021).

[194] Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao, "Neural predictor based quantum architecture search", Machine Learning: Science and Technology 2 4, 045027 (2021).

[195] James Dborin, Fergus Barratt, Vinul Wimalaweera, Lewis Wright, and Andrew G. Green, "Matrix product state pre-training for quantum machine learning", Quantum Science and Technology 7 3, 035014 (2022).

[196] Youle Wang, Guangxi Li, and Xin Wang, "Variational Quantum Gibbs State Preparation with a Truncated Taylor Series", Physical Review Applied 16 5, 054035 (2021).

[197] Sukin Sim, Jonathan Romero, Jérôme F. Gonthier, and Alexander A. Kunitsa, "Adaptive pruning-based optimization of parameterized quantum circuits", Quantum Science and Technology 6 2, 025019 (2021).

[198] I. Rungger, N. Fitzpatrick, H. Chen, C. H. Alderete, H. Apel, A. Cowtan, A. Patterson, D. Munoz Ramo, Y. Zhu, N. H. Nguyen, E. Grant, S. Chretien, L. Wossnig, N. M. Linke, and R. Duncan, "Dynamical mean field theory algorithm and experiment on quantum computers", arXiv:1910.04735, (2019).

[199] Zhide Lu, Pei-Xin Shen, and Dong-Ling Deng, "Markovian Quantum Neuroevolution for Machine Learning", Physical Review Applied 16 4, 044039 (2021).

[200] Andrew Patterson, Hongxiang Chen, Leonard Wossnig, Simone Severini, Dan Browne, and Ivan Rungger, "Quantum state discrimination using noisy quantum neural networks", Physical Review Research 3 1, 013063 (2021).

[201] Jules Tilly, Glenn Jones, Hongxiang Chen, Leonard Wossnig, and Edward Grant, "Computation of molecular excited states on IBM quantum computers using a discriminative variational quantum eigensolver", Physical Review A 102 6, 062425 (2020).

[202] Shuxiang Cao, Leonard Wossnig, Brian Vlastakis, Peter Leek, and Edward Grant, "Cost-function embedding and dataset encoding for machine learning with parametrized quantum circuits", Physical Review A 101 5, 052309 (2020).

[203] Youle Wang, Guangxi Li, and Xin Wang, "Variational quantum Gibbs state preparation with a truncated Taylor series", arXiv:2005.08797, (2020).

[204] Kerstin Beer, Daniel List, Gabriel Müller, Tobias J. Osborne, and Christian Struckmann, "Training Quantum Neural Networks on NISQ Devices", arXiv:2104.06081, (2021).

[205] Thomas Hoffmann and Douglas Brown, "Gradient Estimation with Constant Scaling for Hybrid Quantum Machine Learning", arXiv:2211.13981, (2022).

[206] Hongxiang Chen, Michael Vasmer, Nikolas P. Breuckmann, and Edward Grant, "Machine learning logical gates for quantum error correction", arXiv:1912.10063, (2019).

[207] Mohammad Pirhooshyaran and Tamas Terlaky, "Quantum Circuit Design Search", arXiv:2012.04046, (2020).

[208] Santosh Kumar Radha, "Quantum option pricing using Wick rotated imaginary time evolution", arXiv:2101.04280, (2021).

[209] Kerstin Beer, "Quantum neural networks", arXiv:2205.08154, (2022).

[210] Oinam Romesh Meitei, Bryan T. Gard, George S. Barron, David P. Pappas, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall, "Gate-free state preparation for fast variational quantum eigensolver simulations: ctrl-VQE", arXiv:2008.04302, (2020).

[211] Soumik Adhikary, "Entanglement assisted training algorithm for supervised quantum classifiers", Quantum Information Processing 20 8, 254 (2021).

[212] Kerstin Beer and Gabriel Müller, "Dissipative quantum generative adversarial networks", arXiv:2112.06088, (2021).

[213] Mark-Oliver Wolf, Tom Ewen, and Ivica Turkalj, "Quantum Architecture Search for Quantum Monte Carlo Integration via Conditional Parameterized Circuits with Application to Finance", arXiv:2304.08793, (2023).

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

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

[216] Niladri Gomes, Anirban Mukherjee, Feng Zhang, Thomas Iadecola, Cai-Zhuang Wang, Kai-Ming Ho, Peter P. Orth, and Yong-Xin Yao, "Adaptive Variational Quantum Imaginary Time Evolution Approach for Ground State Preparation", arXiv:2102.01544, (2021).

[217] Xuchen You and Xiaodi Wu, "Exponentially Many Local Minima in Quantum Neural Networks", arXiv:2110.02479, (2021).

[218] Sheng-Hsuan Lin, Rohit Dilip, Andrew G. Green, Adam Smith, and Frank Pollmann, "Real- and imaginary-time evolution with compressed quantum circuits", arXiv:2008.10322, (2020).

[219] Mina Doosti, "Unclonability and Quantum Cryptanalysis: From Foundations to Applications", arXiv:2210.17545, (2022).

[220] Waheeda Saib, Petros Wallden, and Ismail Akhalwaya, "The Effect of Noise on the Performance of Variational Algorithms for Quantum Chemistry", arXiv:2108.12388, (2021).

[221] Rowan Pellow-Jarman, Anban Pillay, Ilya Sinayskiy, and Francesco Petruccione, "Hybrid Genetic Optimisation for Quantum Feature Map Design", arXiv:2302.02980, (2023).

[222] Gaurav Rudra Malik, Amit Kumar Jaiswal, S. Aravinda, and Sunil Kumar Mishra, "Evaluating quantum circuits in the reservoir computing paradigm", arXiv:2605.01253, (2026).

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

[224] 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).

[225] Shabnam Jabeen, Dmytro Kurdydyk, Aadi Palnitkar, Mihir Talati, Jeffrey Yan, and Jinghong Yang, "Quantum Machine Learning for State Tomography Using Classical Data", arXiv:2507.01246, (2025).

[226] Fabian Finger, Frederic Rapp, Pranav Kalidindi, Kerry He, Kante Yin, Alexander Koziell-Pipe, David Zsolt Manrique, Gabriel Greene-Diniz, Stephen Clark, Hamza Fawzi, Bernardino Romera-Paredes, Alhussein Fawzi, and Konstantinos Meichanetzidis, "Automated near-term quantum algorithm discovery for molecular ground states", arXiv:2603.26359, (2026).

[227] 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).

[228] Hao-Kai Zhang, Chenghong Zhu, and Xin Wang, "Predicting quantum learnability from landscape fluctuation", arXiv:2406.11805, (2024).

[229] Luca Arceci, Viacheslav Kuzmin, and Rick Van Bijnen, "Gaussian process model kernels for noisy optimization in variational quantum algorithms", arXiv:2412.13271, (2024).

[230] Tom Ewen, Ivica Turkalj, Patrick Holzer, and Mark-Oliver Wolf, "Application of ZX-calculus to Quantum Architecture Search", arXiv:2406.01095, (2024).

[231] Sandeep Suresh Cranganore, Vincenzo De Maio, Ivona Brandic, and Ewa Deelman, "Paving the Way to Hybrid Quantum-Classical Scientific Workflows", arXiv:2404.10389, (2024).

[232] Jorge J. Martínez de Lejarza, "Quantum computing applications in High Energy Physics: clustering, integration and generative models", arXiv:2512.01597, (2025).

[233] Ngoc Nhi Nguyen, Thai T Vu, John Le, Hoa Khanh Dam, Dung Hoang Duong, and Dinh Thai Hoang, "BRIDG-Q: Barren-Plateau-Resilient Initialisation with Data-Aware LLM-Generated Quantum Circuits", arXiv:2603.23979, (2026).

[234] Sayantan Pramanik and M Girish Chandra, "One Coordinate at a Time: Convergence Guarantees for Rotosolve in Variational Quantum Algorithms", arXiv:2604.25613, (2026).

[235] Max Haas, Thierry N. Kaldenbach, Thomas Hammerschmidt, and Daniel Barragan-Yani, "Efficient Operator Selection and Warm-Start Strategy for Excitations in Variational Quantum Eigensolvers", arXiv:2602.10776, (2026).

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