Quantum Natural Gradient
1Center for Computational Quantum Physics and Center for Computational Mathematics, Flatiron Institute, New York, NY 10010 USA
2Xanadu, 777 Bay Street, Toronto, Canada
3Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010 USA
| Published: | 2020-05-25, volume 4, page 269 |
| Eprint: | arXiv:1909.02108v3 |
| Doi: | https://doi.org/10.22331/q-2020-05-25-269 |
| Citation: | Quantum 4, 269 (2020). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
A quantum generalization of Natural Gradient Descent is presented as part of a general-purpose optimization framework for variational quantum circuits. The optimization dynamics is interpreted as moving in the steepest descent direction with respect to the Quantum Information Geometry, corresponding to the real part of the Quantum Geometric Tensor (QGT), also known as the Fubini-Study metric tensor. An efficient algorithm is presented for computing a block-diagonal approximation to the Fubini-Study metric tensor for parametrized quantum circuits, which may be of independent interest.
► BibTeX data
► References
[1] Shun-Ichi Amari. Natural gradient works efficiently in learning. Neural Computation, 10 (2): 251–276, 1998. 10.1162/089976698300017746.
https://doi.org/10.1162/089976698300017746
[2] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, M. Sohaib Alam, Shahnawaz Ahmed, Juan Miguel Arrazola, Carsten Blank, Alain Delgado, Soran Jahangiri, Keri McKiernan, Johannes Jakob Meyer, Zeyue Niu, Antal Szàva, and Nathan Killoran. Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968, 2018.
arXiv:1811.04968
[3] Marin Bukov, Dries Sels, and Anatoli Polkovnikov. Geometric speed limit of accessible many-body state preparation. Physical Review X, 9 (1): 011034, 2019. 10.1103/PhysRevX.9.011034.
https://doi.org/10.1103/PhysRevX.9.011034
[4] Giuseppe Carleo, Federico Becca, Marco Schiró, and Michele Fabrizio. Localization and glassy dynamics of many-body quantum systems. Scientific reports, 2: 243, 2012. 10.1038/srep00243.
https://doi.org/10.1038/srep00243
[5] Giuseppe Carleo, Federico Becca, Laurent Sanchez-Palencia, Sandro Sorella, and Michele Fabrizio. Light-cone effect and supersonic correlations in one-and two-dimensional bosonic superfluids. Physical Review A, 89 (3): 031602, 2014. 10.1103/PhysRevA.89.031602.
https://doi.org/10.1103/PhysRevA.89.031602
[6] Ming-Cheng Chen, Ming Gong, Xiao-Si Xu, Xiao Yuan, Jian-Wen Wang, Can Wang, Chong Ying, Jin Lin, Yu Xu, Yulin Wu, et al. Demonstration of adiabatic variational quantum computing with a superconducting quantum coprocessor. arXiv preprint arXiv:1905.03150, 2019.
arXiv:1905.03150
[7] Ophelia Crawford, Barnaby van Straaten, Daochen Wang, Thomas Parks, Earl Campbell, and Stephen Brierley. Efficient quantum measurement of pauli operators. arXiv preprint arXiv:1908.06942, 2019.
arXiv:1908.06942
[8] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, and Dacheng Tao. The expressive power of parameterized quantum circuits. arXiv preprint arXiv:1810.11922, 2018.
arXiv:1810.11922
[9] Edward Farhi and Hartmut Neven. Classification with quantum neural networks on near term processors. arXiv preprint arXiv:1802.06002, 2018.
arXiv:1802.06002
[10] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028, 2014.
arXiv:1411.4028
[11] Pranav Gokhale, Olivia Angiuli, Yongshan Ding, Kaiwen Gui, Teague Tomesh, Martin Suchara, Margaret Martonosi, and Frederic T Chong. Minimizing state preparations in variational quantum eigensolver by partitioning into commuting families. arXiv preprint arXiv:1907.13623, 2019.
arXiv:1907.13623
[12] Gian Giacomo Guerreschi and Mikhail Smelyanskiy. Practical optimization for hybrid quantum-classical algorithms. arXiv preprint arXiv:1701.01450, 2017.
arXiv:1701.01450
[13] Aram Harrow and John Napp. Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms. arXiv preprint arXiv:1901.05374, 2019.
arXiv:1901.05374
[14] William James Huggins, Piyush Patil, Bradley Mitchell, K Birgitta Whaley, and Miles Stoudenmire. Towards quantum machine learning with tensor networks. Quantum Science and Technology, 4: 024001, 2018. 10.1088/2058-9565/aaea94.
https://doi.org/10.1088/2058-9565/aaea94
[15] Stanislaw Jastrzebski, Zachary Kenton, Devansh Arpit, Nicolas Ballas, Asja Fischer, Yoshua Bengio, and Amos Storkey. Three factors influencing minima in sgd. arXiv preprint arXiv:1711.04623, 2017.
arXiv:1711.04623
[16] Tyson Jones and Simon C Benjamin. Quantum compilation and circuit optimisation via energy dissipation. arXiv preprint arXiv:1811.03147, 2018.
arXiv:1811.03147
[17] Tyson Jones, Suguru Endo, Sam McArdle, Xiao Yuan, and Simon C Benjamin. Variational quantum algorithms for discovering hamiltonian spectra. Physical Review A, 99 (6): 062304, 2019. 10.1103/PhysRevA.99.062304.
https://doi.org/10.1103/PhysRevA.99.062304
[18] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
arXiv:1412.6980
[19] Michael Kolodrubetz, Dries Sels, Pankaj Mehta, and Anatoli Polkovnikov. Geometry and non-adiabatic response in quantum and classical systems. Physics Reports, 697: 1–87, 2017. 10.1016/j.physrep.2017.07.001.
https://doi.org/10.1016/j.physrep.2017.07.001
[20] PH Kramer and Marcos Saraceno. Geometry of the time-dependent variational principle in quantum mechanics. Springer, 1981. 10.1007/3-540-10271-X_317.
https://doi.org/10.1007/3-540-10271-X_317
[21] Ying Li and Simon C Benjamin. Efficient variational quantum simulator incorporating active error minimization. Physical Review X, 7 (2): 021050, 2017. 10.1103/PhysRevX.7.021050.
https://doi.org/10.1103/PhysRevX.7.021050
[22] Tengyuan Liang, Tomaso Poggio, Alexander Rakhlin, and James Stokes. Fisher-rao metric, geometry, and complexity of neural networks. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 888–896, 2019. arXiv preprint arXiv:1711.01530.
arXiv:1711.01530
[23] Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon C Benjamin, and Xiao Yuan. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Information, 5 (1): 1–6, 2019. 10.1038/s41534-019-0187-2.
https://doi.org/10.1038/s41534-019-0187-2
[24] Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nature communications, 9 (1): 4812, 2018. 10.1038/s41467-018-07090-4.
https://doi.org/10.1038/s41467-018-07090-4
[25] Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum circuit learning. Physical Review A, 98 (3): 032309, 2018. 10.1103/PhysRevA.98.032309.
https://doi.org/10.1103/PhysRevA.98.032309
[26] Behnam Neyshabur, Ruslan R Salakhutdinov, and Nati Srebro. Path-SGD: Path-normalized optimization in deep neural networks. In Advances in Neural Information Processing Systems, pages 2422–2430, 2015. arXiv preprint arXiv:1506.02617.
arXiv:1506.02617
[27] 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. 10.1038/ncomms5213.
https://doi.org/10.1038/ncomms5213
[28] Dénes Petz. Information-geometry of quantum states. In Quantum Probability Communications: Volume X, pages 135–157. World Scientific, 1998. 10.1142/9789812816054_0006.
https://doi.org/10.1142/9789812816054_0006
[29] John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2: 79, 2018. 10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79
[30] Maria Schuld, Alex Bocharov, Krysta Svore, and Nathan Wiebe. Circuit-centric quantum classifiers. arXiv preprint arXiv:1804.00633, 2018. 10.1103/PhysRevA.101.032308.
https://doi.org/10.1103/PhysRevA.101.032308
arXiv:1804.00633
[31] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Physical Review A, 99 (3): 032331, 2019. 10.1103/PhysRevA.99.032331.
https://doi.org/10.1103/PhysRevA.99.032331
[32] Sandro Sorella, Michele Casula, and Dario Rocca. Weak binding between two aromatic rings: Feeling the van der waals attraction by quantum monte carlo methods. The Journal of Chemical Physics, 127 (1): 014105, 2007. 10.1063/1.2746035.
https://doi.org/10.1063/1.2746035
[33] James C Spall et al. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Transactions on Automatic Control, 37 (3): 332–341, 1992. 10.1109/9.119632.
https://doi.org/10.1109/9.119632
[34] F Wilczek and A Shapere. Geometric phases in physics. Geometric Phases In Physics. Series: Advanced Series in Mathematical Physics, ISBN: 978-9971-5-0621-6. WORLD SCIENTIFIC, Edited by F Wilczek and A Shapere, vol. 5, 5, 1989. 10.1142/0613.
https://doi.org/10.1142/0613
[35] Xanadu Quantum Technologies. PennyLane source code. https://github.com/XanaduAI/pennylane, 2019. [Online; accessed 3-Mar-2020].
https://github.com/XanaduAI/pennylane
[36] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C Benjamin. Theory of variational quantum simulation. Quantum, 3: 191, 2019. 10.22331/q-2019-10-07-191.
https://doi.org/10.22331/q-2019-10-07-191
Cited by
[1] Werner Dobrautz, Igor O. Sokolov, Ke Liao, Pablo López Ríos, Martin Rahm, Ali Alavi, and Ivano Tavernelli, "Toward Real Chemical Accuracy on Current Quantum Hardware Through the Transcorrelated Method", Journal of Chemical Theory and Computation 20 10, 4146 (2024).
[2] 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).
[3] Pablo Bermejo, Borja Aizpurua, and Román Orús, "Improving gradient methods via coordinate transformations: Applications to quantum machine learning", Physical Review Research 6 2, 023069 (2024).
[4] Yi-Ming Ding, Yan-Cheng Wang, Shi-Xin Zhang, and Zheng Yan, "Exploring the topological sector optimization on quantum computers", Physical Review Applied 22 3, 034031 (2024).
[5] Suguru Endo, Zhenyu Cai, Simon C. Benjamin, and Xiao Yuan, "Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation", Journal of the Physical Society of Japan 90 3, 032001 (2021).
[6] Daniil S. Bagaev, Maxim A. Gavreev, Alena S. Mastiukova, Aleksey K. Fedorov, and Nikita A. Nemkov, "Regularizing quantum loss landscapes by noise injection", Physical Review A 112 3, 032417 (2025).
[7] Xing-Yu Zhang, Qi Yang, Philippe Corboz, Jutho Haegeman, and Wei Tang, "Accelerating two-dimensional tensor network optimization by preconditioning", Physical Review B 113 12, 125111 (2026).
[8] Liang Zhang, Yin Xu, Mohan Wu, Liang Wang, and Hua Xu, "Quantum long short-term memory for drug discovery", EPJ Quantum Technology 13 1, 14 (2026).
[9] Ljubomir Budinski, "Quantum algorithm for the advection–diffusion equation simulated with the lattice Boltzmann method", Quantum Information Processing 20 2, 57 (2021).
[10] Nico Meyer, Daniel D. Scherer, Axel Plinge, Christopher Mutschler, and Michael J. Hartmann, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 36 (2023) ISBN:979-8-3503-4323-6.
[11] Hideyuki Miyahara, "Information geometry of nonmonotonic quantum natural gradient", Quantum Machine Intelligence 7 2, 98 (2025).
[12] Ningyi Xie, Xinwei Lee, Tiejin Chen, Yoshiyuki Saito, Nobuyoshi Asai, and Dongsheng Cail, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 208 (2025) ISBN:979-8-3315-5736-2.
[13] Johannes Jakob Meyer, "Fisher Information in Noisy Intermediate-Scale Quantum Applications", Quantum 5, 539 (2021).
[14] Michele Minervini, Dhrumil Patel, and Mark M. Wilde, "Evolved quantum Boltzmann machines", Physical Review A 113 3, 032427 (2026).
[15] Serpil Yalcin Kuzu and Ayben Karasu Uysal, "On the integration of quantum machine learning into hybrid frameworks for high energy particle physics", The European Physical Journal C 85 12, 1457 (2025).
[16] Akshay Mittal, Krishna Kandi, Anusha Nagineni, and Vamsi Alla, 2025 Cyber Awareness and Research Symposium (CARS) 1 (2025) ISBN:979-8-3315-9628-6.
[17] Yusuke Nomura, "Boltzmann machines and quantum many-body problems", Journal of Physics: Condensed Matter 36 7, 073001 (2024).
[18] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, "Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information", Quantum 5, 567 (2021).
[19] Hongsheng Zhu, Changqing Gong, Abdullah Gani, and Han Qi, Proceedings of the 2023 15th International Conference on Machine Learning and Computing 485 (2023) ISBN:9781450398411.
[20] André Sequeira, Luis Paulo Santos, and Luis Soares Barbosa, "Policy gradients using variational quantum circuits", Quantum Machine Intelligence 5 1, 18 (2023).
[21] Kaelan Donatella, Zakari Denis, Alexandre Le Boité, and Cristiano Ciuti, "Dynamics with autoregressive neural quantum states: Application to critical quench dynamics", Physical Review A 108 2, 022210 (2023).
[22] Xinglan Zhang, Feng Zhang, Yankun Guo, and Fei Chen, "Variational quantum multidimensional scaling algorithm", Quantum Information Processing 23 3, 77 (2024).
[23] Andrew Arrasmith, M. Cerezo, Piotr Czarnik, Lukasz Cincio, and Patrick J. Coles, "Effect of barren plateaus on gradient-free optimization", Quantum 5, 558 (2021).
[24] Kang-Min Hu, Min Namkung, and Hyang-Tag Lim, "Photonic variational quantum eigensolver for NISQ-compatible quantum technology", Nano Convergence 12 1, 60 (2025).
[25] Dominik S. Kufel, Jack Kemp, DinhDuy Vu, Simon M. Linsel, Chris R. Laumann, and Norman Y. Yao, "Approximately Symmetric Neural Networks for Quantum Spin Liquids", Physical Review Letters 135 5, 056702 (2025).
[26] Jinjing Shi, Ren-Xin Zhao, Wenxuan Wang, Shichao Zhang, and Xuelong Li, "QSAN: A Near-Term Achievable Quantum Self-Attention Network", IEEE Transactions on Neural Networks and Learning Systems 36 8, 13995 (2025).
[27] Ananda Roy, Robert M. Konik, and David Rogerson, "Universal Euler-Cartan circuits for quantum field theories", Physical Review A 113 5, 052605 (2026).
[28] Ran-Yu Chang, Yu-Cheng Lin, Pei-Che Hsu, Tsung-Wei Huang, and En-Jui Kuo, "Accelerating Parameter Initialization in Quantum Chemical Simulations via LSTM-FC-VQE", IEEE Access 13, 189725 (2025).
[29] Lukas Broers and Ludwig Mathey, "Mitigated barren plateaus in the time-nonlocal optimization of analog quantum-algorithm protocols", Physical Review Research 6 1, 013076 (2024).
[30] Massimiliano Incudini, Fabio Tarocco, Riccardo Mengoni, Alessandra Di Pierro, and Antonio Mandarino, "Computing graph edit distance on quantum devices", Quantum Machine Intelligence 4 2, 24 (2022).
[31] 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).
[32] Paulo César Galarza-Sánchez, Alex Fernando Erazo-Luzuriaga, and Miguel Fabricio Boné-Andrade, "Uso de computación cuántica en la mejora de algoritmos de aprendizaje automático", Revista Científica Ciencia y Método 1 4, 16 (2023).
[33] Hantao Zhang, Dong Bai, and Zhongzhou Ren, "Iterative Harrow-Hassidim-Lloyd quantum algorithm for solving resonances with eigenvector continuation", Physics Letters B 873, 140174 (2026).
[34] Tobias Haug and M.S. Kim, "Scalable Measures of Magic Resource for Quantum Computers", PRX Quantum 4 1, 010301 (2023).
[35] 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).
[36] Mohannad M. Ibrahim, Hamed Mohammadbagherpoor, Cynthia Rios, Nicholas T. Bronn, and Gregory T. Byrd, "Evaluation of Parameterized Quantum Circuits With Cross-Resonance Pulse-Driven Entanglers", IEEE Transactions on Quantum Engineering 3, 1 (2022).
[37] Jonas Beck, Jonathan Bodky, Johannes Motruk, Tobias Müller, Ronny Thomale, and Pratyay Ghosh, "Phase diagram of the J−Jd Heisenberg model on the maple leaf lattice: Neural networks and density matrix renormalization group", Physical Review B 109 18, 184422 (2024).
[38] David Barral, F. Javier Cardama, Guillermo Díaz-Camacho, Daniel Faílde, Iago F. Llovo, Mariamo Mussa-Juane, Jorge Vázquez-Pérez, Juan Villasuso, César Piñeiro, Natalia Costas, Juan C. Pichel, Tomás F. Pena, and Andrés Gómez, "Review of Distributed Quantum Computing: From single QPU to High Performance Quantum Computing", Computer Science Review 57, 100747 (2025).
[39] Hiroshi Ohno, "Adaptive pruning algorithm using a quantum Fisher information matrix for parameterized quantum circuits", Quantum Machine Intelligence 6 2, 77 (2024).
[40] Yagnik Chatterjee, Zaid Allybokus, Marko J. Rančić, Eric Bourreau, and Arpan Hazra, "A Hybrid Quantum‐Assisted Column Generation Algorithm for the Fleet Conversion Problem", Quantum Engineering 2025 1, 9984337 (2025).
[41] Dhrumil Patel and Mark M. Wilde, "Natural gradient and parameter estimation for quantum Boltzmann machines", Physical Review A 112 5, 052421 (2025).
[42] Leonardo Banchi and Gavin E. Crooks, "Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule", Quantum 5, 386 (2021).
[43] Ke Wang, Weikang Li, Shibo Xu, Mengyao Hu, Jiachen Chen, Yaozu Wu, Chuanyu Zhang, Feitong Jin, Xuhao Zhu, Yu Gao, Ziqi Tan, Zhengyi Cui, Aosai Zhang, Ning Wang, Yiren Zou, Tingting Li, Fanhao Shen, Jiarun Zhong, Zehang Bao, Zitian Zhu, Zixuan Song, Jinfeng Deng, Hang Dong, Xu Zhang, Pengfei Zhang, Wenjie Jiang, Zhide Lu, Zheng-Zhi Sun, Hekang Li, Qiujiang Guo, Zhen Wang, Patrick Emonts, Jordi Tura, Chao Song, H. Wang, and Dong-Ling Deng, "Probing Many-Body Bell Correlation Depth with Superconducting Qubits", Physical Review X 15 2, 021024 (2025).
[44] Hao-En Li, Xiang Li, Jia-Cheng Huang, Guang-Ze Zhang, Zhu-Ping Shen, Chen Zhao, Jun Li, and Han-Shi Hu, "Variational quantum imaginary time evolution for matrix product state Ansatz with tests on transcorrelated Hamiltonians", The Journal of Chemical Physics 161 14, 144104 (2024).
[45] Ilia Luchnikov, Alexander Ryzhov, Sergey Filippov, and Henni Ouerdane, "QGOpt: Riemannian optimization for quantum technologies", SciPost Physics 10 3, 079 (2021).
[46] Juneseo Lee, Alicia B. Magann, Herschel A. Rabitz, and Christian Arenz, "Progress toward favorable landscapes in quantum combinatorial optimization", Physical Review A 104 3, 032401 (2021).
[47] I-Chi Chen, Aleksei Khindanov, Carlos Munoz Salazar, Humberto Munoz Barona, Ghaidaa Harrabi, Feng Zhang, Cai-Zhuang Wang, Thomas Iadecola, Nicola Lanatà, and Yong-Xin Yao, "Quantum-classical embedding via ghost Gutzwiller approximation for enhanced simulations of correlated electron systems", npj Quantum Information 12 1, 102 (2026).
[48] Roeland Wiersema and Nathan Killoran, "Optimizing quantum circuits with Riemannian gradient flow", Physical Review A 107 6, 062421 (2023).
[49] Tomonori Shirakawa, Hiroshi Ueda, and Seiji Yunoki, "Automatic quantum circuit encoding of a given arbitrary quantum state", Physical Review Research 6 4, 043008 (2024).
[50] Ao Chen and Markus Heyl, "Empowering deep neural quantum states through efficient optimization", Nature Physics 20 9, 1476 (2024).
[51] 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).
[52] Maximilian Balthasar Mansky, Tobias Rohe, Jonas Stein, Dmytro Bondarenko, Linus Menzel, and Claudia Linnhoff-Popien, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 95 (2025) ISBN:979-8-3315-5736-2.
[53] 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).
[54] Kunal Pal, "Generalised state space geometry in Hermitian and non-Hermitian quantum systems", Journal of Physics A: Mathematical and Theoretical 59 4, 045302 (2026).
[55] Jane Kim, Gabriel Pescia, Bryce Fore, Jannes Nys, Giuseppe Carleo, Stefano Gandolfi, Morten Hjorth-Jensen, and Alessandro Lovato, "Neural-network quantum states for ultra-cold Fermi gases", Communications Physics 7 1, 148 (2024).
[56] Chae-Yeun Park, "Efficient ground state preparation in variational quantum eigensolver with symmetry-breaking layers", APL Quantum 1 1, 016101 (2024).
[57] Vu Tuan Hai and Le Bin Ho, "Universal compilation for quantum state tomography", Scientific Reports 13 1, 3750 (2023).
[58] Junyu Liu, Zimu Li, Han Zheng, Xiao Yuan, and Jinzhao Sun, "Towards a variational Jordan–Lee–Preskill quantum algorithm", Machine Learning: Science and Technology 3 4, 045030 (2022).
[59] Yuxuan Du, Xinbiao Wang, Naixu Guo, Zhan Yu, Yang Qian, Kaining Zhang, Min-Hsiu Hsieh, Patrick Rebentrost, and Dacheng Tao, A Gentle Introduction to Quantum Machine Learning 1 (2025) ISBN:978-981-95-1283-6.
[60] 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).
[61] 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).
[62] Chen Gong, Zhuo-Yu Wen, Yun-Wei Deng, Nan-Run Zhou, and Qing-Wei Zeng, "Unrolled generative adversarial network for continuous distributions under hybrid quantum-classical model", Laser Physics Letters 21 12, 125207 (2024).
[63] 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).
[64] Ranyiliu Chen, Benchi Zhao, and Xin Wang, "Near-Term Efficient Quantum Algorithms for Entanglement Analysis", Physical Review Applied 20 2, 024071 (2023).
[65] Kimchhor Chiv, Leanghok Hour, Sanghyeon Lee, Tara Kit, Tae-Kyung Kim, and Youngsun Han, "Strategies for Noise-Resilient Quantum Approximate Optimization Algorithms: A Review and Classification of Error Mitigation", IEEE Access 13, 216916 (2025).
[66] Rodrigo Araiza Bravo, Jorge Garcia Ponce, Hong-Ye Hu, and Susanne F. Yelin, "Circumventing traps in analog quantum machine learning algorithms through co-design", APL Quantum 1 4, 046121 (2024).
[67] Hang Zou, Martin Rahm, Anton Frisk Kockum, and Simon Olsson, "Generative flow-based warm start of the variational quantum eigensolver", npj Quantum Information 12 1, 5 (2025).
[68] Matija Medvidović and Giuseppe Carleo, "Classical variational simulation of the Quantum Approximate Optimization Algorithm", npj Quantum Information 7 1, 101 (2021).
[69] Yi Teng, David D. Dai, and Liang Fu, "Solving the fractional quantum Hall problem with self-attention neural network", Physical Review B 111 20, 205117 (2025).
[70] Tanja Đurić, Jia Hui Chung, Bo Yang, and Pinaki Sengupta, "Spin- 1/2 Kagome Heisenberg Antiferromagnet: Machine Learning Discovery of the Spinon Pair-Density-Wave Ground State", Physical Review X 15 1, 011047 (2025).
[71] 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).
[72] Oleksandr Kyriienko, Annie E. Paine, and Vincent E. Elfving, "Solving nonlinear differential equations with differentiable quantum circuits", Physical Review A 103 5, 052416 (2021).
[73] 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).
[74] Hannah Lange, Fabian Döschl, Juan Carrasquilla, and Annabelle Bohrdt, "Neural network approach to quasiparticle dispersions in doped antiferromagnets", Communications Physics 7 1, 187 (2024).
[75] Jun Qi and Min-Hsiu Hsieh, Federated Learning 329 (2024) ISBN:9780443190377.
[76] Marc Illa, Caroline E. P. Robin, and Martin J. Savage, "Quantum simulations of SO(5) many-fermion systems using qudits", Physical Review C 108 6, 064306 (2023).
[77] 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).
[78] Frederik F Flöther, Jan Mikolon, and Maria Longobardi, "Accelerating the drive towards energy-efficient generative AI with quantum computing algorithms", Quantum Science and Technology 10 4, 040501 (2025).
[79] Damian Hofmann, Giammarco Fabiani, Johan Mentink, Giuseppe Carleo, and Michael Sentef, "Role of stochastic noise and generalization error in the time propagation of neural-network quantum states", SciPost Physics 12 5, 165 (2022).
[80] 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).
[81] Difa Farhani Hakim, Teguh Budi Prayitno, Pak Shen Choong, Yanoar Pribadi Sarwono, and Leong-Chuan Kwek, "Quantum Slater-exponent optimization for electronic-structure calculations on simulated noisy intermediate-scale quantum computers", Physical Review A 113 2, 022610 (2026).
[82] Nishant Jain, Brian Coyle, Elham Kashefi, and Niraj Kumar, "Graph neural network initialisation of quantum approximate optimisation", Quantum 6, 861 (2022).
[83] Yifan Zhou and Peng Zhang, "Noise-Resilient Quantum Machine Learning for Stability Assessment of Power Systems", IEEE Transactions on Power Systems 38 1, 475 (2023).
[84] Shouvanik Chakrabarti, Rajiv Krishnakumar, Guglielmo Mazzola, Nikitas Stamatopoulos, Stefan Woerner, and William J. Zeng, "A Threshold for Quantum Advantage in Derivative Pricing", Quantum 5, 463 (2021).
[85] Shraddha Mishra and Chi-Yi Tsai, "QSurfNet: a hybrid quantum convolutional neural network for surface defect recognition", Quantum Information Processing 22 5, 179 (2023).
[86] Raphael César de Souza Pimenta and Anibal Thiago Bezerra, "Revisiting semiconductor bulk hamiltonians using quantum computers", Physica Scripta 98 4, 045804 (2023).
[87] Rubén Darío Guerrero, "Bee-yond the plateau: Training QNNs with swarm algorithms", The Journal of Chemical Physics 162 1, 011101 (2025).
[88] Vu Tuan Hai, Le Vu Trung Duong, Pham Hoai Luan, and Yasuhiko Nakashima, 2024 International Conference on Advanced Technologies for Communications (ATC) 449 (2024) ISBN:979-8-3503-5398-3.
[89] Vijayarangan Natarajan, Quantum Artificial Intelligence 201 (2025) ISBN:978-981-96-5050-7.
[90] Christopher Roth, Attila Szabó, and Allan H. MacDonald, "High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks", Physical Review B 108 5, 054410 (2023).
[91] Eimantas Ledinauskas and Egidijus Anisimovas, "Scalable imaginary time evolution with neural network quantum states", SciPost Physics 15 6, 229 (2023).
[92] Julien Gacon, Jannes Nys, Riccardo Rossi, Stefan Woerner, and Giuseppe Carleo, "Variational quantum time evolution without the quantum geometric tensor", Physical Review Research 6 1, 013143 (2024).
[93] J. J. Postema, P. Bonizzi, G. Koekoek, R. L. Westra, and S. J. J. M. F. Kokkelmans, "Hybrid quantum singular spectrum decomposition for time series analysis", AVS Quantum Science 5 2, 023803 (2023).
[94] Sirui Lu, Lu-Ming Duan, and Dong-Ling Deng, "Quantum adversarial machine learning", Physical Review Research 2 3, 033212 (2020).
[95] Nihal Shetty, Ashil Shetty, and B Karthik Udupa, "A Geometric Analysis of Quantum-Inspired LocalTensor Regression: Formal Proofs of Convergence and Tractability", (2025).
[96] 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).
[97] Abdul Basit, Ali Hassan Jamal, Kashif Hameed, Saif Ullah, and Rab Nawaz, 2023 20th International Bhurban Conference on Applied Sciences and Technology (IBCAST) 606 (2023) ISBN:979-8-3503-0825-9.
[98] Saahil Patel, Benjamin Collis, William Duong, Daniel Koch, Massimiliano Cutugno, Laura Wessing, and Paul Alsing, "Information loss and run time from practical application of quantum data compression", Physica Scripta 98 4, 045111 (2023).
[99] Mario Motta and Julia E. Rice, "Emerging quantum computing algorithms for quantum chemistry", WIREs Computational Molecular Science 12 3, e1580 (2022).
[100] Takashi Tsuchimochi, Yoohee Ryo, Seiichiro L. Ten-no, and Kazuki Sasasako, "Improved Algorithms of Quantum Imaginary Time Evolution for Ground and Excited States of Molecular Systems", Journal of Chemical Theory and Computation 19 2, 503 (2023).
[101] Weikang Li and Dong-Ling Deng, "Recent advances for quantum classifiers", Science China Physics, Mechanics & Astronomy 65 2, 220301 (2022).
[102] Jun Qi, Xiao-Lei Zhang, and Javier Tejedor, ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) 1 (2023) ISBN:978-1-7281-6327-7.
[103] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "An Adaptive Optimizer for Measurement-Frugal Variational Algorithms", Quantum 4, 263 (2020).
[104] Bojia Duan and Chang-Yu Hsieh, "Hamiltonian-based data loading with shallow quantum circuits", Physical Review A 106 5, 052422 (2022).
[105] Axel Pappalardo, Pierre-Emmanuel Emeriau, Giovanni de Felice, Brian Ventura, Hugo Jaunin, Richie Yeung, Bob Coecke, and Shane Mansfield, "Photonic parameter-shift rule: Enabling gradient computation for photonic quantum computers", Physical Review A 111 3, 032429 (2025).
[106] Ananda Roy, Sameer Erramilli, and Robert M. Konik, "Efficient quantum circuits based on the quantum natural gradient", Physical Review Research 6 4, 043083 (2024).
[107] Daniel Claudino, Alexander J. McCaskey, and Dmitry I. Lyakh, " A Backend-agnostic, Quantum-classical Framework for Simulations of Chemistry in C ++ ", ACM Transactions on Quantum Computing 4 1, 1 (2023).
[108] Benjamin MacLellan, Piotr Roztocki, Stefanie Czischek, and Roger G. Melko, "End-to-end variational quantum sensing", npj Quantum Information 10 1, 118 (2024).
[109] Yuheng Xie, Yuanchen Hao, Yuefeng Lin, Yuchen Sun, Ding Wang, Cong Guo, Na Chen, Yang Liu, and Jianjun Tang, "Efficient routing algorithm for trusted relay quantum key distribution networks via quantum reinforcement learning", Optics Express 33 22, 46545 (2025).
[110] Alexey Pyrkov, Alex Aliper, Dmitry Bezrukov, and Alex Zhavoronkov, Applied Artificial Intelligence for Drug Discovery 289 (2026) ISBN:978-3-031-98021-3.
[111] Dirk Heimann, Hans Hohenfeld, Gunnar Schönhoff, Elie Mounzer, and Frank Kirchner, "Learning Fourier series with parametrized quantum circuits", Physical Review Research 7 2, 023151 (2025).
[112] Kaining Zhang, Min-Hsiu Hsieh, and Dacheng Tao, "Deep Variational Quantum Circuits with Barren-Plateau-Free Architectures", Artificial Intelligence Science and Engineering 2 1, 66 (2026).
[113] Maximilian Amsler, Peter Deglmann, Matthias Degroote, Michael P. Kaicher, Matthew Kiser, Michael Kühn, Chandan Kumar, Andreas Maier, Georgy Samsonidze, Anna Schroeder, Michael Streif, Davide Vodola, and Christopher Wever, "Classical and quantum trial wave functions in auxiliary-field quantum Monte Carlo applied to oxygen allotropes and a CuBr2 model system", The Journal of Chemical Physics 159 4, 044119 (2023).
[114] Dian Wu, Riccardo Rossi, Filippo Vicentini, Nikita Astrakhantsev, Federico Becca, Xiaodong Cao, Juan Carrasquilla, Francesco Ferrari, Antoine Georges, Mohamed Hibat-Allah, Masatoshi Imada, Andreas M. Läuchli, Guglielmo Mazzola, Antonio Mezzacapo, Andrew Millis, Javier Robledo Moreno, Titus Neupert, Yusuke Nomura, Jannes Nys, Olivier Parcollet, Rico Pohle, Imelda Romero, Michael Schmid, J. Maxwell Silvester, Sandro Sorella, Luca F. Tocchio, Lei Wang, Steven R. White, Alexander Wietek, Qi Yang, Yiqi Yang, Shiwei Zhang, and Giuseppe Carleo, "Variational benchmarks for quantum many-body problems", Science 386 6719, 296 (2024).
[115] Chenyu Shi, Vedran Dunjko, and Hao Wang, "Weighted approximate quantum natural gradient for variational quantum eigensolver", Quantum Science and Technology 11 1, 015060 (2026).
[116] J. Cortés-Vega, J. F. Barra, L. Pereira, and A. Delgado, "Detecting entanglement of unknown states by violating the Clauser–Horne–Shimony–Holt inequality", Quantum Information Processing 22 5, 203 (2023).
[117] Sidhartha Dash, Luca Gravina, Filippo Vicentini, Michel Ferrero, and Antoine Georges, "Efficiency of neural quantum states in light of the quantum geometric tensor", Communications Physics 8 1, 92 (2025).
[118] David Fitzek, Robert S. Jonsson, Werner Dobrautz, and Christian Schäfer, "Optimizing Variational Quantum Algorithms with qBang: Efficiently Interweaving Metric and Momentum to Navigate Flat Energy Landscapes", Quantum 8, 1313 (2024).
[119] Eimantas Ledinauskas and Egidijus Anisimovas, "Universal performance gap of neural quantum states applied to the Hofstadter-Bose-Hubbard model", SciPost Physics 18 1, 011 (2025).
[120] Guang Yang, Su-Ya Chao, Min Nie, Yuan-Hua Liu, and Mei-Ling Zhang, "Construction method of hybrid quantum long-short term memory neural network for image classification", Acta Physica Sinica 72 5, 058901 (2023).
[121] Mateo Alonso, Guillermo Rubiños Rodríguez, Pablo Díez-Valle, Ana Garbayo, Xela García-Santiago, and Gonzalo Blázquez Gil, "Modeling Energy Communities: A Case Study of Quantum Approximate Optimization on a Superconducting Processor", IEEE Access 13, 106140 (2025).
[122] 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).
[123] R. R. Ferguson, L. Dellantonio, A. Al Balushi, K. Jansen, W. Dür, and C. A. Muschik, "Measurement-Based Variational Quantum Eigensolver", Physical Review Letters 126 22, 220501 (2021).
[124] Nathan A. McMahon, Mahum Pervez, and Christian Arenz, "Equating quantum imaginary time evolution, Riemannian gradient flows, and stochastic implementations", Physical Review Research 8 2, 023024 (2026).
[125] Toi Sasaki and Hideyuki Miyahara, "Quantum natural gradient without monotonicity", Physical Review A 110 2, 022439 (2024).
[126] 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).
[127] Joonho Kim, Jaedeok Kim, and Dario Rosa, "Universal effectiveness of high-depth circuits in variational eigenproblems", Physical Review Research 3 2, 023203 (2021).
[128] Seung Park, Kyunghyun Baek, Seungjin Lee, and Mahn-Soo Choi, "Global optimization in variational quantum algorithms via dynamic tunneling method", New Journal of Physics 26 7, 073053 (2024).
[129] 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).
[130] Junyuan He, Yin Kan, and Cheng Xue, 2024 16th International Conference on Wireless Communications and Signal Processing (WCSP) 139 (2024) ISBN:979-8-3503-9064-3.
[131] 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).
[132] Mengzhen Ren, Yu-Cheng Chen, Ching-Jui Lai, Min-Hsiu Hsieh, and Alice Hu, "Hybrid quantum-classical clustering for preparing a prior distribution of eigenspectrum", npj Quantum Information 12 1, 56 (2026).
[133] 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).
[134] Nawres A. Alwan, Suzan J. Obaiys, Nadia M. G. Al-Saidi, and Rayane Chadli, Lecture Notes in Computer Science 16758, 284 (2027) ISBN:978-3-032-30496-4.
[135] Jianshe Xie, Chen Xu, Chenhao Yin, Yumin Dong, and Zhirong Zhang, "Natural Evolutionary Gradient Descent Strategy for Variational Quantum Algorithms", Intelligent Computing 2, 0042 (2023).
[136] Kathleen E. Hamilton, Emily Lynn, and Raphael C. Pooser, " Mode connectivity in the loss landscape of parameterized quantum circuits", Quantum Machine Intelligence 4 1, 10 (2022).
[137] Xin Wang, Bo Qi, Yabo Wang, and Daoyi Dong, "Entanglement-variational hardware-efficient ansatz for eigensolvers", Physical Review Applied 21 3, 034059 (2024).
[138] Kevin J Sung, Jiahao Yao, Matthew P Harrigan, Nicholas C Rubin, Zhang Jiang, Lin Lin, Ryan Babbush, and Jarrod R McClean, "Using models to improve optimizers for variational quantum algorithms", Quantum Science and Technology 5 4, 044008 (2020).
[139] Jack Y. Araz and Michael Spannowsky, "Classical versus quantum: Comparing tensor-network-based quantum circuits on Large Hadron Collider data", Physical Review A 106 6, 062423 (2022).
[140] Vu Tuan Hai, Nguyen Tan Viet, and Le Bin Ho, "〈qo|op〉: A quantum object optimizer", SoftwareX 26, 101726 (2024).
[141] 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).
[142] Petr Ivashkov, Po-Wei Huang, Kelvin Koor, Lirandë Pira, and Patrick Rebentrost, "QKAN: quantum Kolmogorov-Arnold networks with applications in machine learning and multivariate state preparation", npj Quantum Information 12 1, 73 (2026).
[143] Yijie Zhu, Vaneet Aggarwal, Debanjan Konar, Yuri Pashkin, Plamen Angelov, and Richard Jiang, "Toward Quantum Image Generation on Single Qubit Using Quantum Information Bottleneck", IEEE Transactions on Artificial Intelligence 7 4, 2321 (2026).
[144] 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.
[145] 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", Advanced Quantum Technologies 4 12, 2100114 (2021).
[146] Gregory Boyd and Bálint Koczor, "Training Variational Quantum Circuits with CoVaR: Covariance Root Finding with Classical Shadows", Physical Review X 12 4, 041022 (2022).
[147] 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).
[148] 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).
[149] Andrew Blance and Michael Spannowsky, "Quantum machine learning for particle physics using a variational quantum classifier", Journal of High Energy Physics 2021 2, 212 (2021).
[150] Nikita Astrakhantsev, Guglielmo Mazzola, Ivano Tavernelli, and Giuseppe Carleo, "Phenomenological theory of variational quantum ground-state preparation", Physical Review Research 5 3, 033225 (2023).
[151] 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).
[152] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 129 (2023) ISBN:979-8-3503-4323-6.
[153] Federico Dell'Anna, Rafael Gómez-Lurbe, Armando Pérez, and Elisa Ercolessi, "Quantum natural gradient optimizer on noisy platforms: Quantum approximate optimization algorithm as a case study", Physical Review A 112 2, 022612 (2025).
[154] Michael Kaicher, Florian Dommert, Christopher Wever, Maximilian Amsler, and Michael Kühn, "Generating approximate ground states of strongly correlated quantum many-body systems through quantum imaginary time evolution", Journal of Physics Communications 9 7, 075002 (2025).
[155] Yiming Huang, Xiao Yuan, Huiyuan Wang, and Yuxuan Du, "Coreset selection can accelerate quantum machine learning models with provable generalization", Physical Review Applied 22 1, 014074 (2024).
[156] Ioannis Kolotouros and Petros Wallden, "Random Natural Gradient", Quantum 8, 1503 (2024).
[157] Philip Easom-Mccaldin, Ahmed Bouridane, Ammar Belatreche, and Richard Jiang, "On Depth, Robustness and Performance Using the Data Re-Uploading Single-Qubit Classifier", IEEE Access 9, 65127 (2021).
[158] 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).
[159] James Stokes, Javier Robledo Moreno, Eftychios A. Pnevmatikakis, and Giuseppe Carleo, "Phases of two-dimensional spinless lattice fermions with first-quantized deep neural-network quantum states", Physical Review B 102 20, 205122 (2020).
[160] Robert J. Webber and Michael Lindsey, "Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo", Physical Review Research 4 3, 033099 (2022).
[161] Mourad Halla, "Quantum natural gradient with geodesic corrections for small shallow quantum circuits", Physica Scripta 100 5, 055121 (2025).
[162] Bo Peng and Karol Kowalski, "Variational quantum solver employing the PDS energy functional", Quantum 5, 473 (2021).
[163] Christiane P. Koch, Ugo Boscain, Tommaso Calarco, Gunther Dirr, Stefan Filipp, Steffen J. Glaser, Ronnie Kosloff, Simone Montangero, Thomas Schulte-Herbrüggen, Dominique Sugny, and Frank K. Wilhelm, "Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe", EPJ Quantum Technology 9 1, 19 (2022).
[164] Jia-Cheng Huo, Ling Fan, Ru Zhang, and Cong Cao, "A full-quantum algorithm for solving the exact cover problem via quantum gradient descent iteration", Laser Physics 36 3, 035202 (2026).
[165] Stefano Markidis, "Programming Quantum Neural Networks on NISQ Systems: An Overview of Technologies and Methodologies", Entropy 25 4, 694 (2023).
[166] Scott E. Smart and Prineha Narang, "Many-body eigenstates from quantum manifold optimization", Physical Review A 110 5, 052430 (2024).
[167] Mohammad Aamir Sohail, Mohsen Heidari, and S. Sandeep Pradhan, "Quantum natural stochastic pairwise coordinate descent", npj Quantum Information 11 1, 109 (2025).
[168] Zakari Denis and Giuseppe Carleo, "Accurate neural quantum states for interacting lattice bosons", Quantum 9, 1772 (2025).
[169] Tomonori Shirakawa, Kazuhiro Seki, and Seiji Yunoki, "Discretized quantum adiabatic process for free fermions and comparison with the imaginary-time evolution", Physical Review Research 3 1, 013004 (2021).
[170] Bikram Khanal and Pablo Rivas, "Data-dependent generalization bounds for parameterized quantum models under noise", The Journal of Supercomputing 81 4, 611 (2025).
[171] Maria Schuld and Francesco Petruccione, Encyclopedia of Machine Learning and Data Science 1 (2023) ISBN:978-1-4899-7502-7.
[172] Mourad Halla, "Estimation of Quantum Fisher Information via Stein's Identity in Variational Quantum Algorithms", Quantum 9, 1798 (2025).
[173] Han Qi, Sihui Xiao, Zhuo Liu, Changqing Gong, and Abdullah Gani, "Variational quantum algorithms: fundamental concepts, applications and challenges", Quantum Information Processing 23 6, 224 (2024).
[174] Mingshu Zhao and Zhanyuan Yan, "Interpretable neural network quantum states for solving the steady states of the nonlinear Schrödinger equation", Chaos: An Interdisciplinary Journal of Nonlinear Science 35 11, 113122 (2025).
[175] A. V. Berezutskii, I. A. Luchnikov, and A. K. Fedorov, "Simulating quantum circuits using the multi-scale entanglement renormalization ansatz", Physical Review Research 7 1, 013063 (2025).
[176] Yiyou Chen, Hideyuki Miyahara, Louis-S. Bouchard, and Vwani Roychowdhury, "Quantum approximation of normalized Schatten norms and applications to learning", Physical Review A 106 5, 052409 (2022).
[177] Richard Meister, Cica Gustiani, and Simon C Benjamin, "Exploring ab initio machine synthesis of quantum circuits", New Journal of Physics 25 7, 073018 (2023).
[178] Amara Katabarwa, Sukin Sim, Dax Enshan Koh, and Pierre-Luc Dallaire-Demers, "Connecting geometry and performance of two-qubit parameterized quantum circuits", Quantum 6, 782 (2022).
[179] Kosuke Ito, Wataru Mizukami, and Keisuke Fujii, "Universal noise-precision relations in variational quantum algorithms", Physical Review Research 5 2, 023025 (2023).
[180] Ilya Piatrenka and Marian Rusek, Lecture Notes in Computer Science 13353, 247 (2022) ISBN:978-3-031-08759-2.
[181] Davide Castaldo, Marta Rosa, and Stefano Corni, "Fast-forwarding molecular ground state preparation with optimal control on analog quantum simulators", The Journal of Chemical Physics 161 1, 014105 (2024).
[182] Songwei Zhang, Tie Qiu, Xiaobo Zhou, and Yusheng Ji, "A Quantum-Driven Efficient Learning Model for Enhancing Robustness of IoT Topology", IEEE Transactions on Mobile Computing 24 11, 12391 (2025).
[183] Y. S. Teo, "Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms", Physical Review A 109 1, 012620 (2024).
[184] Shaojun Gui, Tak-San Ho, and Herschel Rabitz, "Discrete real-time learning of quantum-state subspace evolution of many-body systems in the presence of time-dependent control fields", Physical Review A 110 5, 052412 (2024).
[185] Ibsal Assi, Michael Vogl, Meenu Kumari, and J. P. F. LeBlanc, "Beyond trotterization: Variational product formulas for quantum simulation", Physical Review B 113 21, 214314 (2026).
[186] Jan Hermann, James Spencer, Kenny Choo, Antonio Mezzacapo, W. M. C. Foulkes, David Pfau, Giuseppe Carleo, and Frank Noé, "Ab initio quantum chemistry with neural-network wavefunctions", Nature Reviews Chemistry 7 10, 692 (2023).
[187] Xianzhi Huang, Lili Niu, Jie Chen, Lichao Li, Kashif Hayat, and Weiping Liu, "Quantum and classical computational synergy for emerging contaminants management: Advanced insights into cytochrome P450 metabolic mechanisms", Critical Reviews in Environmental Science and Technology 54 24, 1827 (2024).
[188] 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).
[189] 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).
[190] 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).
[191] Rodica Ioana Lung and Florin Sebastian Duma, "A Noisy Optimization mechanism for variational quantum classifiers", Knowledge-Based Systems 340, 115659 (2026).
[192] Leonardo Alchieri, Davide Badalotti, Pietro Bonardi, and Simone Bianco, "An introduction to quantum machine learning: from quantum logic to quantum deep learning", Quantum Machine Intelligence 3 2, 28 (2021).
[193] Yuxuan Du, Zhuozhuo Tu, Xiao Yuan, and Dacheng Tao, "Efficient Measure for the Expressivity of Variational Quantum Algorithms", Physical Review Letters 128 8, 080506 (2022).
[194] André Sequeira, Luis Paulo Santos, and Luis Soares Barbosa, "On Quantum Natural Policy Gradients", IEEE Transactions on Quantum Engineering 5, 1 (2024).
[195] Muhammad Kashif and Muhammad Shafique, 2025 International Joint Conference on Neural Networks (IJCNN) 1 (2025) ISBN:979-8-3315-1042-8.
[196] Maniraman Periyasamy, Axel Plinge, Christopher Mutschler, Daniel D. Scherer, and Wolfgang Mauerer, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 1504 (2024) ISBN:979-8-3315-4137-8.
[197] Shaojun Gui, Tak-San Ho, and Herschel Rabitz, "Control simulations of many-body quantum systems by a synergism of discrete real-time learning and optimal control theory", The Journal of Chemical Physics 163 10, 104108 (2025).
[198] 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).
[199] Umut Çalikyilmaz, Sven Groppe, Jinghua Groppe, Tobias Winker, Stefan Prestel, Farida Shagieva, Daanish Arya, Florian Preis, and Le Gruenwald, "Opportunities for Quantum Acceleration of Databases: Optimization of Queries and Transaction Schedules", Proceedings of the VLDB Endowment 16 9, 2344 (2023).
[200] Giorgio Tosti Balducci, Boyang Chen, Matthias Möller, Marc Gerritsma, and Roeland De Breuker, "Review and perspectives in quantum computing for partial differential equations in structural mechanics", Frontiers in Mechanical Engineering 8, 914241 (2022).
[201] Shui-Yuan Huang, Wan-Jia An, De-Shun Zhang, and Nan-Run Zhou, "Image classification and adversarial robustness analysis based on hybrid quantum–classical convolutional neural network", Optics Communications 533, 129287 (2023).
[202] Abhishek Dixit, Ashish Mani, and Sergey Gorbachev, 2024 IEEE International Conference on Intelligent Signal Processing and Effective Communication Technologies (INSPECT) 1 (2024) ISBN:979-8-3503-7952-5.
[203] Duc-Truyen Le, Vu-Linh Nguyen, Ha C. Nguyen, Hung Q. Nguyen, and Van-Duy Nguyen, "Variational Quantum Eigensolver: A Comparative Analysis of Classical and Quantum Optimization Methods", (2025).
[204] 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).
[205] Ryan L'Abbate, Anthony D'Onofrio, Samuel Stein, Samuel Yen-Chi Chen, Ang Li, Pin-Yu Chen, Juntao Chen, and Ying Mao, "A Quantum-Classical Collaborative Training Architecture Based on Quantum State Fidelity", IEEE Transactions on Quantum Engineering 5, 1 (2024).
[206] Phillip C. Lotshaw, Travis S. Humble, Rebekah Herrman, James Ostrowski, and George Siopsis, "Empirical performance bounds for quantum approximate optimization", Quantum Information Processing 20 12, 403 (2021).
[207] Oleksandr Borysenko, Mykhailo Bratchenko, Ilya Lukin, Mykola Luhanko, Ihor Omelchenko, Andrii Sotnikov, and Alessandro Lomi, "Application of Langevin dynamics to advance the Quantum Natural Gradient optimization algorithm", Physica A: Statistical Mechanics and its Applications 682, 131158 (2026).
[208] Huan Zhang, Robert J. Webber, Michael Lindsey, Timothy C. Berkelbach, and Jonathan Weare, "Improved energies and local energies with weighted variational Monte Carlo", Physical Review Research 8 1, 013213 (2026).
[209] 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).
[210] E. Ercolessi, R. Fioresi, and T. Weber, "The geometry of quantum computing", International Journal of Geometric Methods in Modern Physics 21 10, 2440011 (2024).
[211] Volkan Erol, "Variational Optimization of Quantum Fisher Information: A Hybrid Quantum-Classical Approach to Noise-Resilient Metrology", (2025).
[212] Filippo Vicentini, Damian Hofmann, Attila Szabó, Dian Wu, Christopher Roth, Clemens Giuliani, Gabriel Pescia, Jannes Nys, Vladimir Vargas-Calderón, Nikita Astrakhantsev, and Giuseppe Carleo, "NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems", SciPost Physics Codebases 7 (2022).
[213] Ashutosh Singh, Pooja Siwach, and P. Arumugam, "Quantum simulations of nuclear resonances with variational methods", Physical Review C 112 2, 024323 (2025).
[214] Tatiana A. Bespalova and Oleksandr Kyriienko, "Hamiltonian Operator Approximation for Energy Measurement and Ground-State Preparation", PRX Quantum 2 3, 030318 (2021).
[215] Yuxuan Du and Dacheng Tao, "On Exploring the Potential of Quantum Auto-Encoder for Learning Quantum Systems", IEEE Transactions on Neural Networks and Learning Systems 36 7, 12454 (2025).
[216] Joseph Bowles, David Wierichs, and Chae-Yeun Park, "Backpropagation scaling in parameterised quantum circuits", Quantum 9, 1873 (2025).
[217] He-Liang Huang, Xiao-Yue Xu, Chu Guo, Guojing Tian, Shi-Jie Wei, Xiaoming Sun, Wan-Su Bao, and Gui-Lu Long, "Near-term quantum computing techniques: Variational quantum algorithms, error mitigation, circuit compilation, benchmarking and classical simulation", Science China Physics, Mechanics & Astronomy 66 5, 250302 (2023).
[218] Vu Truc Quynh, Vu Tuan Hai, Le Vu Trung Duong, Pham Hoai Luan, and Yasuhiko Nakashima, 2024 International Technical Conference on Circuits/Systems, Computers, and Communications (ITC-CSCC) 1 (2024) ISBN:979-8-3503-7905-1.
[219] Brian García Sarmina, Jorge Saavedra Benavides, Guo-Hua Sun, and Shi-Hai Dong, "Probing entanglement and parameter sensitivity in QAOA via Quantum Fisher Information", Quantum Review Letters 2, 1 (2026).
[220] Jack Y. Araz, Sebastian Schenk, and Michael Spannowsky, "Toward a quantum simulation of nonlinear sigma models with a topological term", Physical Review A 107 3, 032619 (2023).
[221] Benedikt Fauseweh, "Quantum many-body simulations on digital quantum computers: State-of-the-art and future challenges", Nature Communications 15 1, 2123 (2024).
[222] Maja Franz, Tobias Winker, Sven Groppe, and Wolfgang Mauerer, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 409 (2024) ISBN:979-8-3315-4137-8.
[223] Amandeep Singh Bhatia, Mandeep Kaur Saggi, and Sabre Kais, 2024 IEEE EMBS International Conference on Biomedical and Health Informatics (BHI) 1 (2024) ISBN:979-8-3503-5155-2.
[224] Lars Simon, Holger Eble, and Manuel Radons, "Introducing the kernel descent optimizer for variational quantum algorithms", Scientific Reports 15 1, 28247 (2025).
[225] Mauro Rigo, Benjamin Hall, Morten Hjorth-Jensen, Alessandro Lovato, and Francesco Pederiva, "Solving the nuclear pairing model with neural network quantum states", Physical Review E 107 2, 025310 (2023).
[226] Tianchen Zhao, Chuhao Sun, Asaf Cohen, James Stokes, and Shravan Veerapaneni, "Quantum-inspired variational algorithms for partial differential equations: application to financial derivative pricing", Quantitative Finance 24 1, 1 (2024).
[227] Yudai Suzuki, Hiroshi Yano, Rudy Raymond, and Naoki Yamamoto, 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 1 (2021) ISBN:978-1-6654-1691-7.
[228] Ioannis Kolotouros, David Joseph, and Anand Kumar Narayanan, "Accelerating quantum imaginary-time evolution with random measurements", Physical Review A 111 1, 012424 (2025).
[229] 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).
[230] João C. Getelina, Cai-Zhuang Wang, Thomas Iadecola, Yong-Xin Yao, and Peter P. Orth, "Adaptive variational ground state preparation for spin-1 models on qubit-based architectures", Physical Review B 109 8, 085128 (2024).
[231] 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).
[232] V. Armaos, Dimitrios A. Badounas, Paraskevas Deligiannis, Konstantinos Lianos, and Yordan S. Yordanov, "Efficient Parabolic Optimisation Algorithm for Adaptive VQE Implementations", SN Computer Science 3 6, 443 (2022).
[233] 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).
[234] Zi-Xiao Zhang, Yi-Long Yang, Wan-Bing He, Peng-Wei Zhao, Bing-Nan Lu, and Yu-Gang Ma, "Machine learning the single-Λ hypernuclei with neural-network quantum states", Physics Letters B 874, 140285 (2026).
[235] Tobias Haug, Kishor Bharti, and M.S. Kim, "Capacity and Quantum Geometry of Parametrized Quantum Circuits", PRX Quantum 2 4, 040309 (2021).
[236] Erika Magnusson, Aaron Fitzpatrick, Stefan Knecht, Martin Rahm, and Werner Dobrautz, "Towards efficient quantum computing for quantum chemistry: reducing circuit complexity with transcorrelated and adaptive ansatz techniques", Faraday Discussions 254, 402 (2024).
[237] Kyle M. Sherbert, Hisham Amer, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall, "Parametrization and optimizability of pulse-level variational quantum eigensolvers", Physical Review Applied 23 2, 024036 (2025).
[238] Galan Moody, Volker J Sorger, Daniel J Blumenthal, Paul W Juodawlkis, William Loh, Cheryl Sorace-Agaskar, Alex E Jones, Krishna C Balram, Jonathan C F Matthews, Anthony Laing, Marcelo Davanco, Lin Chang, John E Bowers, Niels Quack, Christophe Galland, Igor Aharonovich, Martin A Wolff, Carsten Schuck, Neil Sinclair, Marko Lončar, Tin Komljenovic, David Weld, Shayan Mookherjea, Sonia Buckley, Marina Radulaski, Stephan Reitzenstein, Benjamin Pingault, Bartholomeus Machielse, Debsuvra Mukhopadhyay, Alexey Akimov, Aleksei Zheltikov, Girish S Agarwal, Kartik Srinivasan, Juanjuan Lu, Hong X Tang, Wentao Jiang, Timothy P McKenna, Amir H Safavi-Naeini, Stephan Steinhauer, Ali W Elshaari, Val Zwiller, Paul S Davids, Nicholas Martinez, Michael Gehl, John Chiaverini, Karan K Mehta, Jacquiline Romero, Navin B Lingaraju, Andrew M Weiner, Daniel Peace, Robert Cernansky, Mirko Lobino, Eleni Diamanti, Luis Trigo Vidarte, and Ryan M Camacho, "2022 Roadmap on integrated quantum photonics", Journal of Physics: Photonics 4 1, 012501 (2022).
[239] 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).
[240] Lennart Bittel and Martin Kliesch, "Training Variational Quantum Algorithms Is NP-Hard", Physical Review Letters 127 12, 120502 (2021).
[241] 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).
[242] Chang Yu Hsieh, Qiming Sun, Shengyu Zhang, and Chee Kong Lee, "Unitary-coupled restricted Boltzmann machine ansatz for quantum simulations", npj Quantum Information 7 1, 19 (2021).
[243] Joseph C. Aulicino, Trevor Keen, and Bo Peng, "State preparation and evolution in quantum computing: A perspective from Hamiltonian moments", International Journal of Quantum Chemistry 122 5, e26853 (2022).
[244] Yoga A. Darmawan, Angga D. Fauzi, Yanoar P. Sarwono, and Rui-Qin Zhang, "Improving qubit reduction for molecular simulations with randomized orbital sampling", AAPPS Bulletin 35 1, 27 (2025).
[245] Ibrahim Gad, Aboul Ella Hassanien, Ashraf Darwish, and Mincong Tang, Lecture Notes in Operations Research 693 (2022) ISBN:978-981-16-8655-9.
[246] Bing Han, Jian Kang, Meng Zhang, and Qian Wu, "Research on Space-Time Data Prediction Model of Quantum Long Short-Term Memory Network Fusion", Photonics 13 5, 477 (2026).
[247] 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).
[248] Kouhei Nakaji and Naoki Yamamoto, "Expressibility of the alternating layered ansatz for quantum computation", Quantum 5, 434 (2021).
[249] Laura Gentini, Alessandro Cuccoli, Stefano Pirandola, Paola Verrucchi, and Leonardo Banchi, "Noise-resilient variational hybrid quantum-classical optimization", Physical Review A 102 5, 052414 (2020).
[250] Ruojing Peng and Garnet Kin-Lic Chan, "First- and quasi-second-order optimization algorithms in variational Monte Carlo", Physical Review Research 7 4, 043351 (2025).
[251] Suguru Endo, Iori Kurata, and Yuya O. Nakagawa, "Calculation of the Green's function on near-term quantum computers", Physical Review Research 2 3, 033281 (2020).
[252] Senwei Liang, Linghua Zhu, Xiaosong Li, and Chao Yang, "QuGStep: Refining step size selection in gradient estimation for variational quantum algorithms", APL Computational Physics 1 2, 026110 (2025).
[253] Muhammad AbuGhanem, "Toward scalable fault-tolerant photonic quantum computers", The Journal of Supercomputing 82 2, 51 (2026).
[254] Dwi Cahyo Mariyanto, Hadyan L. Prihadi, Angga Dito Fauzi, L. T. Handoko, Yanoar P. Sarwono, and Rui‐Qin Zhang, "Fubini‐Study Metric Tensor Evolution and Reduced State Distinguishability in Geometry‐Aware Optimization for Reliable Near‐Term Variational Quantum Eigensolvers", International Journal of Quantum Chemistry 126 13, e70255 (2026).
[255] Junxia YANG, Qizhi CAI, Jinrong GUO, and Guangwei DENG, "Quantum artificial intelligence: a review of the bidirectional empowerment mechanisms and frontier progress in AI and quantum computing", Acta Physica Sinica 75 10(2026).
[256] Ren-Xin Zhao, Jinjing Shi, and Xuelong Li, "QKSAN: A Quantum Kernel Self-Attention Network", IEEE Transactions on Pattern Analysis and Machine Intelligence 46 12, 10184 (2024).
[257] Anthony N. Ciavarella and Ivan A. Chernyshev, "Preparation of the SU(3) lattice Yang-Mills vacuum with variational quantum methods", Physical Review D 105 7, 074504 (2022).
[258] Kunal Sharma, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Trainability of Dissipative Perceptron-Based Quantum Neural Networks", Physical Review Letters 128 18, 180505 (2022).
[259] Jing-Kai Fang, Yue-Feng Lin, Jun-Han Huang, Yibo Chen, Gao-Ming Fan, Yuhui Sun, Guanru Feng, Cong Guo, Tiejun Meng, Yong Zhang, Xun Xu, Jingen Xiang, and Yuxiang Li, "Divide-and-conquer quantum algorithm for hybridde novogenome assembly of short and long reads", (2023).
[260] Markus Hauru, Maarten Van Damme, and Jutho Haegeman, "Riemannian optimization of isometric tensor networks", SciPost Physics 10 2, 040 (2021).
[261] 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).
[262] Tao 涛 Cheng 程, Run-Sheng 润盛 Zhao 赵, Shuang 爽 Wang 王, Rui 睿 Wang 王, and Hong-Yang 鸿洋 Ma 马, "Analysis of learnability of a novel hybrid quantum–classical convolutional neural network in image classification", Chinese Physics B 33 4, 040303 (2024).
[263] Jing-Kai Fang, Yue-Feng Lin, Jun-Han Huang, Yibo Chen, Gao-Ming Fan, Yuhui Sun, Guanru Feng, Cong Guo, Tiejun Meng, Yong Zhang, Xun Xu, Jingen Xiang, and Yuxiang Li, "Divide-and-Conquer Quantum Algorithm for Hybrid de novo Genome Assembly of Short and Long Reads", PRX Life 2 2, 023006 (2024).
[264] Chen Zhao and Xiao-Shan Gao, "Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus", Quantum 5, 466 (2021).
[265] Dmitrii Khitrin, Kenneth R Brown, and Abhinav Anand, "Unbiased observable estimation with approximate channels in fault-tolerant quantum computation", Quantum Science and Technology 11 1, 015034 (2026).
[266] Hao Zhang, Shaojun Dong, Chao Wang, Meng Zhang, and Lixin He, "TNSP: A framework supporting symmetry and fermion tensors for tensor network state methods", Computer Physics Communications 305, 109355 (2024).
[267] Fumiyoshi Kobayashi, Kosuke Mitarai, and Keisuke Fujii, "Parent Hamiltonian as a benchmark problem for variational quantum eigensolvers", Physical Review A 105 5, 052415 (2022).
[268] Guglielmo Mazzola, "Sampling, rates, and reaction currents through reverse stochastic quantization on quantum computers", Physical Review A 104 2, 022431 (2021).
[269] Danny Paulson, Luca Dellantonio, Jan F. Haase, Alessio Celi, Angus Kan, Andrew Jena, Christian Kokail, Rick van Bijnen, Karl Jansen, Peter Zoller, and Christine A. Muschik, "Simulating 2D Effects in Lattice Gauge Theories on a Quantum Computer", PRX Quantum 2 3, 030334 (2021).
[270] Guglielmo Mazzola, "Quantum computing for chemistry and physics applications from a Monte Carlo perspective", The Journal of Chemical Physics 160 1, 010901 (2024).
[271] Bálint Koczor and Simon C. Benjamin, "Quantum natural gradient generalized to noisy and nonunitary circuits", Physical Review A 106 6, 062416 (2022).
[272] Seunghyeok Oh, Jaeho Choi, and Joongheon Kim, 2020 International Conference on Information and Communication Technology Convergence (ICTC) 236 (2020) ISBN:978-1-7281-6758-9.
[273] David Wierichs, Josh Izaac, Cody Wang, and Cedric Yen-Yu Lin, "General parameter-shift rules for quantum gradients", Quantum 6, 677 (2022).
[274] Hantao Zhang, Dong Bai, and Zhongzhou Ren, "Quantum computing for extracting nuclear resonances", Physics Letters B 860, 139187 (2025).
[275] Yang Qian, Xinbiao Wang, Yuxuan Du, Xingyao Wu, and Dacheng Tao, "The Dilemma of Quantum Neural Networks", IEEE Transactions on Neural Networks and Learning Systems 35 4, 5603 (2024).
[276] Tim Weaving, Alexis Ralli, William M. Kirby, Andrew Tranter, Peter J. Love, and Peter V. Coveney, "A Stabilizer Framework for the Contextual Subspace Variational Quantum Eigensolver and the Noncontextual Projection Ansatz", Journal of Chemical Theory and Computation 19 3, 808 (2023).
[277] Jeffmin Lin, Gil Goldshlager, and Lin Lin, "Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation", Journal of Computational Physics 474, 111765 (2023).
[278] Tobias Haug and M. S. Kim, "Natural parametrized quantum circuit", Physical Review A 106 5, 052611 (2022).
[279] Bálint Koczor and Simon C. Benjamin, "Quantum analytic descent", Physical Review Research 4 2, 023017 (2022).
[280] Mario Motta, Kevin J. Sung, and James Shee, "Quantum Algorithms for the Variational Optimization of Correlated Electronic States with Stochastic Reconfiguration and the Linear Method", The Journal of Physical Chemistry A 128 40, 8762 (2024).
[281] Dennis Lima and Saif Al-Kuwari, "Exponential depth decay in Sridhara-compressed VQE simulation for molecular energy ranking", Physical Chemistry Chemical Physics (2026).
[282] Bryce Fore, Jane M. Kim, Giuseppe Carleo, Morten Hjorth-Jensen, Alessandro Lovato, and Maria Piarulli, "Dilute neutron star matter from neural-network quantum states", Physical Review Research 5 3, 033062 (2023).
[283] Alistair W R Smith, A J Paige, and M S Kim, "Faster variational quantum algorithms with quantum kernel-based surrogate models", Quantum Science and Technology 8 4, 045016 (2023).
[284] Quoc Chuong Nguyen, Le Bin Ho, Lan Nguyen Tran, and Hung Q Nguyen, "Qsun: an open-source platform towards practical quantum machine learning applications", Machine Learning: Science and Technology 3 1, 015034 (2022).
[285] Tobias Winker, Umut Çalikyilmaz, Le Gruenwald, and Sven Groppe, Proceedings of the International Workshop on Big Data in Emergent Distributed Environments 1 (2023) ISBN:9798400700934.
[286] Stefano Barison, Filippo Vicentini, and Giuseppe Carleo, "An efficient quantum algorithm for the time evolution of parameterized circuits", Quantum 5, 512 (2021).
[287] Siva Sai and Rajkumar Buyya, "Quantum Artificial Intelligence for mission-critical systems: Foundations, architectural elements, and future directions", Future Generation Computer Systems 184, 108602 (2026).
[288] Yongchun Xu and Heng Hu, "Potential energy minimization for structural analysis via decomposition-free quantum computing", Computers & Structures 330, 108309 (2026).
[289] Pinaki Sen, Amandeep Singh Bhatia, Kamalpreet Singh Bhangu, Ahmed Elbeltagi, and Thippa Reddy Gadekallu, "Variational quantum classifiers through the lens of the Hessian", PLOS ONE 17 1, e0262346 (2022).
[290] Alessandro Carbone, Davide Emilio Galli, Mario Motta, and Barbara Jones, "Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers", Symmetry 14 3, 624 (2022).
[291] Haozhen Situ, Zhengjiang Li, Zhimin He, Qin Li, and Jinjing Shi, "AutoML-driven optimization of variational quantum circuit", Information Sciences 717, 122272 (2025).
[292] Qi-Ming Ding, Yi-Ming Huang, and Xiao Yuan, "Molecular docking via quantum approximate optimization algorithm", Physical Review Applied 21 3, 034036 (2024).
[293] Kazuhiro Seki and Seiji Yunoki, "Spatial, spin, and charge symmetry projections for a Fermi-Hubbard model on a quantum computer", Physical Review A 105 3, 032419 (2022).
[294] Y. S. Teo, "Optimized numerical gradient and Hessian estimation for variational quantum algorithms", Physical Review A 107 4, 042421 (2023).
[295] Daniil Rabinovich, Andrey Kardashin, and Soumik Adhikary, "Role of overparametrization in quantum approximate optimization", Physical Review A 113 6, 062617 (2026).
[296] Emiel Koridon, Joana Fraxanet, Alexandre Dauphin, Lucas Visscher, Thomas E. O'Brien, and Stefano Polla, "A hybrid quantum algorithm to detect conical intersections", Quantum 8, 1259 (2024).
[297] Qiang Miao and Thomas Barthel, "Convergence and Quantum Advantage of Trotterized MERA for Strongly-Correlated Systems", Quantum 9, 1631 (2025).
[298] Adam Kadi, Aymene Selamnia, Zakaria Abou El Houda, Hajar Moudoud, Bouziane Brik, and Lyes Khoukhi, "An In-Depth Comparative Study of Quantum-Classical Encoding Methods for Network Intrusion Detection", IEEE Open Journal of the Communications Society 6, 1129 (2025).
[299] Andrea Di Donna, Lorenzo Contessi, Alessandro Lovato, and Francesco Pederiva, "Hypernuclei with neural network quantum states", Physical Review Research 8 1, 013160 (2026).
[300] Alexander Avdoshkin, Max Geier, and Liang Fu, "Integrated neural wave-function solver for spinful Fermi systems", Physical Review B 113 19, 195108 (2026).
[301] Maria Schuld and Nathan Killoran, "Is Quantum Advantage the Right Goal for Quantum Machine Learning?", PRX Quantum 3 3, 030101 (2022).
[302] Josephine Hunout, Sylvain Laizet, and Lorenzo Iannucci, "Variational quantum algorithm based on Lagrange polynomial encoding to solve differential equations ", Physical Review A 111 6, 062404 (2025).
[303] Seunghyeok Oh, Jaeho Choi, Jong-Kook Kim, and Joongheon Kim, 2021 International Conference on Information Networking (ICOIN) 50 (2021) ISBN:978-1-7281-9101-0.
[304] Gabriel Matos, Chris N. Self, Zlatko Papić, Konstantinos Meichanetzidis, and Henrik Dreyer, "Characterization of variational quantum algorithms using free fermions", Quantum 7, 966 (2023).
[305] 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.
[306] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, and Dacheng Tao, "Learnability of Quantum Neural Networks", PRX Quantum 2 4, 040337 (2021).
[307] Rinka Miura, "Velocity Verlet-based optimization for variational quantum eigensolvers", Quantum Information Processing 25 2, 34 (2026).
[308] 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).
[309] Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan, "Quantum Algorithms for Quantum Chemistry and Quantum Materials Science", Chemical Reviews 120 22, 12685 (2020).
[310] Jakob S Kottmann, Sumner Alperin-Lea, Teresa Tamayo-Mendoza, Alba Cervera-Lierta, Cyrille Lavigne, Tzu-Ching Yen, Vladyslav Verteletskyi, Philipp Schleich, Abhinav Anand, Matthias Degroote, Skylar Chaney, Maha Kesibi, Naomi Grace Curnow, Brandon Solo, Georgios Tsilimigkounakis, Claudia Zendejas-Morales, Artur F Izmaylov, and Alán Aspuru-Guzik, "TEQUILA: a platform for rapid development of quantum algorithms", Quantum Science and Technology 6 2, 024009 (2021).
[311] Mourad Halla, "Modified conjugate quantum natural gradient", EPJ Quantum Technology 12 1, 123 (2025).
[312] Abu Kaisar Mohammad Masum, Mehran Shoushtari Moghadam, Lida Kouhalvandi, M. Hassan Najafi, and Sercan Aygun, Proceedings of the Great Lakes Symposium on VLSI 2025 575 (2025) ISBN:9798400714962.
[313] Roeland Wiersema, Cunlu Zhou, Juan Felipe Carrasquilla, and Yong Baek Kim, "Measurement-induced entanglement phase transitions in variational quantum circuits", SciPost Physics 14 6, 147 (2023).
[314] Yijie Zhu, Richard Jiang, Qiang Ni, and Ahmed Bouridane, "Enable Quantum Graph Neural Networks on a Single Qubit With Quantum Walk", IEEE Transactions on Artificial Intelligence 7 5, 2496 (2026).
[315] Tobias Haug and M. S. Kim, "Generalization of Quantum Machine Learning Models Using Quantum Fisher Information Metric", Physical Review Letters 133 5, 050603 (2024).
[316] Chufan Lyu, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Symmetry enhanced variational quantum spin eigensolver", Quantum 7, 899 (2023).
[317] Yuhan Huang, Qingyu Li, Xiaokai Hou, Rebing Wu, Man-Hong Yung, Abolfazl Bayat, and Xiaoting Wang, "Robust resource-efficient quantum variational ansatz through an evolutionary algorithm", Physical Review A 105 5, 052414 (2022).
[318] Oluwaseyi Giwa, Muhammad Ahmed Mohsin, Folarin Jubril Adesola, and Muhammad Ali Jamshed, "QPPG: Quantum-Preconditioned Policy Gradient for Link Adaptation in Rayleigh Fading Channels", IEEE Wireless Communications Letters 15, 3343 (2026).
[319] Christa Zoufal, Ryan V. Mishmash, Nitin Sharma, Niraj Kumar, Aashish Sheshadri, Amol Deshmukh, Noelle Ibrahim, Julien Gacon, and Stefan Woerner, "Variational quantum algorithm for unconstrained black box binary optimization: Application to feature selection", Quantum 7, 909 (2023).
[320] Nikita A. Nemkov, Evgeniy O. Kiktenko, Ilia A. Luchnikov, and Aleksey K. Fedorov, "Efficient variational synthesis of quantum circuits with coherent multi-start optimization", Quantum 7, 993 (2023).
[321] Maida Wang, Xiao Xue, Mingyang Gao, and Peter V. Coveney, "Quantum-informed machine learning for predicting spatiotemporal chaos with practical quantum advantage", Science Advances 12 16, eaec5049 (2026).
[322] Kishor Bharti, "Fisher Information: A Crucial Tool for NISQ Research", Quantum Views 5, 61 (2021).
[323] Donghwa Lee, Jinil Lee, Seongjin Hong, Hyang-Tag Lim, Young-Wook Cho, Sang-Wook Han, Hyundong Shin, Junaid ur Rehman, and Yong-Su Kim, "Error-mitigated photonic variational quantum eigensolver using a single-photon ququart", Optica 9 1, 88 (2022).
[324] Aram W. Harrow and John C. Napp, "Low-Depth Gradient Measurements Can Improve Convergence in Variational Hybrid Quantum-Classical Algorithms", Physical Review Letters 126 14, 140502 (2021).
[325] Matthew T. Scoggins and Armin Rahmani, "Topological and geometric patterns in optimal bang-bang protocols for variational quantum algorithms: Application to the XXZ model on the square lattice", Physical Review Research 3 4, 043165 (2021).
[326] Alessandro Sinibaldi, Clemens Giuliani, Giuseppe Carleo, and Filippo Vicentini, "Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution", Quantum 7, 1131 (2023).
[327] 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).
[328] Prasanna Kottapalle, Tan Kuan Tak, Pravin Ramdas Kshirsagar, Gopichand Ginnela, and Vijaya Krishna Akula, "QHF-CS: Quantum-Enhanced Heart Failure Prediction Using Quantum CNN with Optimized Feature Qubit Selection with Cuckoo Search in Skewed Clinical Data", Computers, Materials & Continua 84 2, 3857 (2025).
[329] James Stokes, Brian Chen, and Shravan Veerapaneni, "Numerical and geometrical aspects of flow-based variational quantum Monte Carlo", Machine Learning: Science and Technology 4 2, 021001 (2023).
[330] Nikita A. Zemlevskiy, "Scalable quantum simulations of scattering in scalar field theory on 120 qubits", Physical Review D 112 3, 034502 (2025).
[331] Weikang Li, Zhi-de Lu, and Dong-Ling Deng, "Quantum Neural Network Classifiers: A Tutorial", SciPost Physics Lecture Notes 61 (2022).
[332] Yuan Yao, Pierre Cussenot, Richard A. Wolf, and Filippo Miatto, "Complex natural gradient optimization for optical quantum circuit design", Physical Review A 105 5, 052402 (2022).
[333] Tyson Jones and Simon C. Benjamin, "Robust quantum compilation and circuit optimisation via energy minimisation", Quantum 6, 628 (2022).
[334] Grier M. Jones and Hans-Arno Jacobsen, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 155 (2024) ISBN:979-8-3315-4137-8.
[335] S. Mangini, F. Tacchino, D. Gerace, D. Bajoni, and C. Macchiavello, "Quantum computing models for artificial neural networks", Europhysics Letters 134 1, 10002 (2021).
[336] 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).
[337] M. Drissi, J. W. T Keeble, J. Rozalén Sarmiento, and A. Rios, "Second-order optimization strategies for neural network quantum states", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 382 2275, 20240057 (2024).
[338] Y. Nishida, "Research on Quantum Circuit Learning Models for Molecular Dynamics Simulation", The Journal of Physical Chemistry A 129 40, 9434 (2025).
[339] Luning Li, Xuchen Zhang, Zhicheng Cui, Weiming Xu, Xuesen Xu, Jianyu Wang, and Rong Shu, "An Overview of Quantum Machine Learning Research in China", Applied Sciences 15 5, 2555 (2025).
[340] Bobak Toussi Kiani, Giacomo De Palma, Milad Marvian, Zi-Wen Liu, and Seth Lloyd, "Learning quantum data with the quantum earth mover’s distance", Quantum Science and Technology 7 4, 045002 (2022).
[341] Cica Gustiani, Richard Meister, and Simon C Benjamin, "Exploiting subspace constraints and ab initio variational methods for quantum chemistry", New Journal of Physics 25 7, 073019 (2023).
[342] Juan M Cruz-Martinez, Matteo Robbiati, and Stefano Carrazza, "Multi-variable integration with a variational quantum circuit", Quantum Science and Technology 9 3, 035053 (2024).
[343] Nico Meyer, Christian Ufrecht, Maniraman Periyasamy, Axel Plinge, Christopher Mutschler, Daniel D. Scherer, and Andreas Maier, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 817 (2024) ISBN:979-8-3315-4137-8.
[344] Philip Easom-McCaldin, Ahmed Bouridane, Ammar Belatreche, Richard Jiang, and Somaya Al-Maadeed, "Efficient Quantum Image Classification Using Single Qubit Encoding", IEEE Transactions on Neural Networks and Learning Systems 35 2, 1472 (2024).
[345] Hirofumi Nishi, Taichi Kosugi, and Yu-ichiro Matsushita, "Implementation of quantum imaginary-time evolution method on NISQ devices by introducing nonlocal approximation", npj Quantum Information 7 1, 85 (2021).
[346] 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).
[347] Zhaoqi Leng, Pranav Mundada, Saeed Ghadimi, and Andrew Houck, "Efficient Algorithms for High-Dimensional Quantum Optimal Control of a Transmon Qubit", Physical Review Applied 19 4, 044034 (2023).
[348] Han Qi, Xintong Huo, and Abdullah Gani, 2026 International Conference on Multi-scale Artificial Intelligence (MAI) 1 (2026) ISBN:979-8-3315-4566-6.
[349] Maiyuren Srikumar, Charles D Hill, and Lloyd C L Hollenberg, "Clustering and enhanced classification using a hybrid quantum autoencoder", Quantum Science and Technology 7 1, 015020 (2022).
[350] Jogi Suda Neto, Lluis Quiles Ardila, Thiago Nascimento Nogueira, Felipe Albuquerque, João Paulo Papa, Rodrigo Capobianco Guido, and Felipe Fernandes Fanchini, "Quantum neural networks successfully calibrate language models", Quantum Machine Intelligence 6 1, 8 (2024).
[351] Riccardo Rende, Luciano Loris Viteritti, Federico Becca, Antonello Scardicchio, Alessandro Laio, and Giuseppe Carleo, "Foundation neural-networks quantum states as a unified Ansatz for multiple hamiltonians", Nature Communications 16 1, 7213 (2025).
[352] Maximilian Balthasar Mansky, Jonas Nüßlein, David Bucher, Daniëlle Schuman, Sebastian Zielinski, and Claudia Linnhoff-Popien, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 485 (2023) ISBN:979-8-3503-4323-6.
[353] 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).
[354] Junxu Li, "Knowledge distillation inspired variational quantum eigensolver with virtual annealing ", New Journal of Physics 28 2, 024501 (2026).
[355] David Amaro, Matthias Rosenkranz, Nathan Fitzpatrick, Koji Hirano, and Mattia Fiorentini, "A case study of variational quantum algorithms for a job shop scheduling problem", EPJ Quantum Technology 9 1, 5 (2022).
[356] Kosuke Mitarai, Yasunari Suzuki, Wataru Mizukami, Yuya O. Nakagawa, and Keisuke Fujii, "Quadratic Clifford expansion for efficient benchmarking and initialization of variational quantum algorithms", Physical Review Research 4 3, 033012 (2022).
[357] Laszlo Gyongyosi and Sandor Imre, "Networked Quantum Services†", Quantum Information & Computation 25 2, 97 (2025).
[358] Si-Jing Du, Ao Chen, and Garnet Kin-Lic Chan, "Neuralized fermionic tensor networks for quantum many-body systems", Physical Review B 113 8, 085134 (2026).
[359] Shiwen An and Konstantinos Slavakis, 2025 International Conference on Quantum Communications, Networking, and Computing (QCNC) 568 (2025) ISBN:979-8-3315-3159-1.
[360] Daniel Claudino, Jerimiah Wright, Alexander J. McCaskey, and Travis S. Humble, "Benchmarking Adaptive Variational Quantum Eigensolvers", Frontiers in Chemistry 8, 606863 (2020).
[361] Matija Medvidović and Dries Sels, "Variational Quantum Dynamics of Two-Dimensional Rotor Models", PRX Quantum 4 4, 040302 (2023).
[362] Luca Gravina, Vincenzo Savona, and Filippo Vicentini, "Neural Projected Quantum Dynamics: a systematic study", Quantum 9, 1803 (2025).
[363] Guang-Ze Zhang, Jia-Cheng Huang, Lian-Wei Ye, Jun Li, and Han-Shi Hu, "Scalable quantum simulations of molecular systems via improved optimization of neural quantum states", The Journal of Chemical Physics 162 22, 224108 (2025).
[364] João C. Getelina, Niladri Gomes, Thomas Iadecola, Peter P. Orth, and Yong-Xin Yao, "Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations", SciPost Physics 15 3, 102 (2023).
[365] 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).
[366] Dieter Jaksch, Peyman Givi, Andrew J. Daley, and Thomas Rung, "Variational Quantum Algorithms for Computational Fluid Dynamics", AIAA Journal 61 5, 1885 (2023).
[367] Mandaar B. Pande, Lecture Notes in Electrical Engineering 1221, 3 (2025) ISBN:978-981-97-4710-8.
[368] Matija Medvidović and Javier Robledo Moreno, "Neural-network quantum states for many-body physics", The European Physical Journal Plus 139 7, 631 (2024).
[369] Bisma Majid, Shabir Ahmed Sofi, and Zamrooda Jabeen, "Quantum machine learning: a systematic categorization based on learning paradigms, NISQ suitability, and fault tolerance", Quantum Machine Intelligence 7 1, 39 (2025).
[370] Michele Minervini, Dhrumil Patel, and Mark M. Wilde, "Quantum natural gradient with thermal-state initialization", Physical Review A 112 2, 022424 (2025).
[371] Enrico Fontana, M. Cerezo, Andrew Arrasmith, Ivan Rungger, and Patrick J. Coles, "Non-trivial symmetries in quantum landscapes and their resilience to quantum noise", Quantum 6, 804 (2022).
[372] Lajos Hanzo, Zunaira Babar, Zhenyu Cai, Daryus Chandra, Ivan B. Djordjevic, Balint Koczor, Soon Xin Ng, Mohsen Razavi, and Osvaldo Simeone, "Quantum Information Processing, Sensing, and Communications: Their Myths, Realities, and Futures", Proceedings of the IEEE 113 9, 1024 (2025).
[373] Yizhi Wang, Shichuan Xue, Yaxuan Wang, Jiangfang Ding, Weixu Shi, Dongyang Wang, Yong Liu, Yingwen Liu, Xiang Fu, Guangyao Huang, Anqi Huang, Mingtang Deng, and Junjie Wu, "Experimental quantum natural gradient optimization in photonics", Optics Letters 48 14, 3745 (2023).
[374] Jeihee Cho and Shiho Kim, Advances in Computers 140, 113 (2026) ISBN:9780443223822.
[375] 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).
[376] Fei Li and Xiao-Wei Li, "Hybrid real-imaginary time evolution for low-depth Hamiltonian simulation in quantum optimization", Physica A: Statistical Mechanics and its Applications 693, 131539 (2026).
[377] 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).
[378] Xinglan Zhang and Feng Zhang, "Variational Quantum Computation Integer Factorization Algorithm", International Journal of Theoretical Physics 62 11, 245 (2023).
[379] Ziyang Li, Xiaofei Fu, Lingdong Meng, and Ruishan Du, "A repetitive amplitude encoding method for enhancing the mapping ability of quantum neural networks", Scientific Reports 15 1, 32111 (2025).
[380] Alberto Acevedo, Carmen G Almudéver, Miguel Angel Garcia-March, Rafael Gómez-Lurbe, Luca Ion, Mohit Lal Bera, Rodrigo M Sanz, Somayeh Mehrabankar, Tanmoy Pandit, Armando Pérez, and Andreu Anglés-Castillo, "Adaptive time compressed QITE (ACQ) and its geometrical interpretation", Quantum Science and Technology 11 3, 035009 (2026).
[381] Yagnik Chatterjee, Eric Bourreau, and Marko J. Rančić, "Solving various NP-hard problems using exponentially fewer qubits on a quantum computer", Physical Review A 109 5, 052441 (2024).
[382] Le Bin Ho, "A stochastic evaluation of quantum Fisher information matrix with generic Hamiltonians", EPJ Quantum Technology 10 1, 37 (2023).
[383] Vishal S. Ngairangbam, Michael Spannowsky, and Michihisa Takeuchi, "Anomaly detection in high-energy physics using a quantum autoencoder", Physical Review D 105 9, 095004 (2022).
[384] Y.V.R. Naga Pawan and Bhanu Prakash Kolla, Advances in Computers 140, 271 (2026) ISBN:9780443223822.
[385] Benjamin A. Cordier, Nicolas P. D. Sawaya, Gian Giacomo Guerreschi, and Shannon K. McWeeney, "Biology and medicine in the landscape of quantum advantages", Journal of The Royal Society Interface 19 196, 20220541 (2022).
[386] Chloé Gauvin-Ndiaye, Joseph Tindall, Javier Robledo Moreno, and Antoine Georges, "Mott Transition and Volume Law Entanglement with Neural Quantum States", Physical Review Letters 134 7, 076502 (2025).
[387] Aeishah Ameera Anuar, François Jamet, Fabio Gironella, Fedor Šimkovic IV, and Riccardo Rossi, "Operator-projected variational quantum imaginary time evolution", Journal of Physics A: Mathematical and Theoretical 59 24, 245301 (2026).
[388] Madhusudan Singh, Irish Singh, and Dhananjay Singh, Lecture Notes in Computer Science 14531, 226 (2024) ISBN:978-3-031-53826-1.
[389] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, "Quantum computational chemistry", Reviews of Modern Physics 92 1, 015003 (2020).
[390] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Shahnawaz Ahmed, Vishnu Ajith, M. Sohaib Alam, Guillermo Alonso-Linaje, B. AkashNarayanan, Ali Asadi, Juan Miguel Arrazola, Utkarsh Azad, Sam Banning, Carsten Blank, Thomas R Bromley, Benjamin A. Cordier, Jack Ceroni, Alain Delgado, Olivia Di Matteo, Amintor Dusko, Tanya Garg, Diego Guala, Anthony Hayes, Ryan Hill, Aroosa Ijaz, Theodor Isacsson, David Ittah, Soran Jahangiri, Prateek Jain, Edward Jiang, Ankit Khandelwal, Korbinian Kottmann, Robert A. Lang, Christina Lee, Thomas Loke, Angus Lowe, Keri McKiernan, Johannes Jakob Meyer, J. A. Montañez-Barrera, Romain Moyard, Zeyue Niu, Lee James O'Riordan, Steven Oud, Ashish Panigrahi, Chae-Yeun Park, Daniel Polatajko, Nicolás Quesada, Chase Roberts, Nahum Sá, Isidor Schoch, Borun Shi, Shuli Shu, Sukin Sim, Arshpreet Singh, Ingrid Strandberg, Jay Soni, Antal Száva, Slimane Thabet, Rodrigo A. Vargas-Hernández, Trevor Vincent, Nicola Vitucci, Maurice Weber, David Wierichs, Roeland Wiersema, Moritz Willmann, Vincent Wong, Shaoming Zhang, and Nathan Killoran, "PennyLane: Automatic differentiation of hybrid quantum-classical computations", arXiv:1811.04968, (2018).
[391] Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma, and Patrick J. Coles, "Operator Sampling for Shot-frugal Optimization in Variational Algorithms", arXiv:2004.06252, (2020).
[392] Sam McArdle, Suguru Endo, Alan Aspuru-Guzik, Simon Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402, (2018).
[393] Kazuhiro Seki, Tomonori Shirakawa, and Seiji Yunoki, "Symmetry-adapted variational quantum eigensolver", Physical Review A 101 5, 052340 (2020).
[394] Naoki Yamamoto, "On the natural gradient for variational quantum eigensolver", arXiv:1909.05074, (2019).
[395] Patrick Huembeli and Alexandre Dauphin, "Characterizing the loss landscape of variational quantum circuits", Quantum Science and Technology 6 2, 025011 (2021).
[396] Barnaby van Straaten and Bálint Koczor, "Measurement Cost of Metric-Aware Variational Quantum Algorithms", PRX Quantum 2 3, 030324 (2021).
[397] Lennart Bittel, Jens Watty, and Martin Kliesch, "Fast gradient estimation for variational quantum algorithms", arXiv:2210.06484, (2022).
[398] Di Luo, Jiayu Shen, Rumen Dangovski, and Marin Soljačić, "QuACK: Accelerating Gradient-Based Quantum Optimization with Koopman Operator Learning", arXiv:2211.01365, (2022).
[399] Nguyen Tan Viet, Nguyen Thi Chuong, Vu Thi Ngoc Huyen, and Le Bin Ho, "tqix.pis: A toolbox for quantum dynamics simulation of spin ensembles in Dicke basis", Computer Physics Communications 286, 108686 (2023).
[400] Markus Hauru, Maarten Van Damme, and Jutho Haegeman, "Riemannian optimization of isometric tensor networks", arXiv:2007.03638, (2020).
[401] Korbinian Kottmann, "Investigating Quantum Many-Body Systems with Tensor Networks, Machine Learning and Quantum Computers", arXiv:2210.11130, (2022).
[402] Tianchen Zhao, Giuseppe Carleo, James Stokes, and Shravan Veerapaneni, "Natural evolution strategies and variational Monte Carlo", arXiv:2005.04447, (2020).
[403] Weiyuan Gong and Dong-Ling Deng, "Universal Adversarial Examples and Perturbations for Quantum Classifiers", arXiv:2102.07788, (2021).
[404] M. Muñoz, L. Pereira, C. Vargas, S. Niklitschek, and A. Delgado, "Complex Field Formulation of the Quantum Estimation Theory", arXiv:2203.03064, (2022).
[405] David Rogerson and Ananda Roy, "Quantum Circuit Optimization using Differentiable Programming of Tensor Network States", arXiv:2408.12583, (2024).
[406] Stefano Mangini, "Variational quantum algorithms for machine learning: theory and applications", arXiv:2306.09984, (2023).
[407] Maximilian Balthasar Mansky, Jonas Nüßlein, David Bucher, Daniëlle Schuman, Sebastian Zielinski, and Claudia Linnhoff-Popien, "Sampling Problems on a Quantum Computer", arXiv:2402.16341, (2024).
[408] Kazuki Osawa, Satoki Ishikawa, Rio Yokota, Shigang Li, and Torsten Hoefler, "ASDL: A Unified Interface for Gradient Preconditioning in PyTorch", arXiv:2305.04684, (2023).
[409] Maniraman Periyasamy, Axel Plinge, Christopher Mutschler, Daniel D. Scherer, and Wolfgang Mauerer, "Guided-SPSA: Simultaneous Perturbation Stochastic Approximation assisted by the Parameter Shift Rule", arXiv:2404.15751, (2024).
[410] Eimantas Ledinauskas and Egidijus Anisimovas, "Universal Performance Gap of Neural Quantum States Applied to the Hofstadter-Bose-Hubbard Model", arXiv:2405.01981, (2024).
[411] Mark M. Wilde, "Quantum Fisher information matrices from Rényi relative entropies", arXiv:2510.02218, (2025).
[412] Gaurav Rudra Malik, Amit Kumar Jaiswal, S. Aravinda, and Sunil Kumar Mishra, "Evaluating quantum circuits in the reservoir computing paradigm", arXiv:2605.01253, (2026).
[413] Mirko Consiglio, "Variational Quantum Algorithms for Many-Body Systems", arXiv:2502.11985, (2025).
[414] Aritra Bal, Markus Klute, Benedikt Maier, Melik Oughton, Eric Pezone, and Michael Spannowsky, "QINNs: Quantum-Informed Neural Networks", arXiv:2510.17984, (2025).
[415] Rui-Hao Li, Semeon Valgushev, and Khadijeh Najafi, "Variational Thermal State Preparation on Digital Quantum Processors Assisted by Matrix Product States", arXiv:2510.23546, (2025).
[416] Zhuo Chen, Oriol Mayné i Comas, Zhuotao Jin, Di Luo, and Marin Soljačić, "L$^2$M: Mutual Information Scaling Law for Long-Context Language Modeling", arXiv:2503.04725, (2025).
[417] Shiwen An and Konstantinos Slavakis, "Tensor-Based Binary Graph Encoding for Variational Quantum Classifiers", arXiv:2501.14185, (2025).
[418] 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", arXiv:2603.24140, (2026).
[419] Alessandro Lovato, Giuseppe Carleo, Bryce Fore, Morten Hjorth-Jensen, Jane Kim, Arnau Rios, and Noemi Rocco, "Neural-network quantum states for the nuclear many-body problem", arXiv:2602.13826, (2026).
[420] Thomas Ayral, "Dynamical mean field theory with quantum computing", arXiv:2508.00118, (2025).
[421] Azadeh Alavi, Fatemeh Kouchmeshki, and Abdolrahman Alavi, "Practical Quantum-Classical Feature Fusion for complex data Classification", arXiv:2512.19180, (2025).
[422] Thanveer Shaik, Xiaohui Tao, and Haoran Xie, "Quantum Machine Unlearning: Foundations, Mechanisms, and Taxonomy", arXiv:2511.00406, (2025).
[423] Shiwen An, Jiayi Wang, and Konstantinos Slavakis, "LogosQ: A High-Performance and Type-Safe Quantum Computing Library in Rust", arXiv:2512.23183, (2025).
[424] Michael Poppel, David Bucher, Maximilian Zorn, Markus Baumann, Sebastian Wölckert, Claudia Linnhoff-Popien, Philipp Altmann, and Jonas Stein, "Architecture Shape Governs QNN Trainability: Jacobian Null Space Growth and Parameter Efficiency", arXiv:2605.05942, (2026).
[425] Ernesto Acosta, Guillermo Botella, and Carlos Cano, "QUBO-based training for VQAs on Quantum Annealers", arXiv:2509.01821, (2025).
[426] Subrit Dikshit, Ritu Tiwari, and Priyank Jain, "Multilingual Machine Translation with Quantum Encoder Decoder Attention-based Convolutional Variational Circuits", arXiv:2505.09407, (2025).
[427] Gianluca Scanu, Luca Barletta, and Stefano Rini, "JGRA: Jacobian Geometry Robustness Assessment in NISQ Noise-Aware Quantum Neural Networks", arXiv:2606.09964, (2026).
[428] Yi-Ran Xue, Rui Wang, Baigeng Wang, and Chenan Wei, "Low-variance estimators overcome the phase-gradient bottleneck in complex-valued neural quantum states", arXiv:2606.13912, (2026).
[429] Arthur J. Parzygnat and Andrew Vlasic, "Quantum encodings that preserve persistent homology", arXiv:2605.28927, (2026).
The above citations are from Crossref's cited-by service (last updated successfully 2026-07-15 10:19:28) and SAO/NASA ADS (last updated successfully 2026-07-14 22:16:55). The list may be incomplete as not all publishers provide suitable and complete citation data.
Could not fetch ADS cited-by data during last attempt 2026-07-15 10:19:28: Cannot retrieve data from ADS due to rate limitations.
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.
Pingback: Perspective in Quantum Views by John Napp "Variational quantum algorithms and geometry"