Theory of variational quantum simulation

Xiao Yuan; Suguru Endo; Qi Zhao; Ying Li; Simon C. Benjamin

doi:10.22331/q-2019-10-07-191

Theory of variational quantum simulation

Xiao Yuan¹, Suguru Endo¹, Qi Zhao², Ying Li³, and Simon C. Benjamin¹

¹Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
²Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
³Graduate School of China Academy of Engineering Physics, Beijing 100193, China

Published:	2019-10-07, volume 3, page 191
Eprint:	arXiv:1812.08767v4
Doi:	https://doi.org/10.22331/q-2019-10-07-191
Citation:	Quantum 3, 191 (2019).

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Abstract

The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.

Popular summary

Universal quantum computers will eventually solve various classically intractable problems, but the exciting challenge is to demonstrate the first real quantum advantage as soon as possible -- with NISQ (for Noisy Intermediate Scaled Quantum) devices. In this regime, we may only be able to manipulate hundreds or thousands of qubits and the operations will be imperfect (or 'noisy'). With such a limited noisy quantum computer, it is unclear how to demonstrate any quantum advantage in any practical task.

This work solves this problem by exploring hybrid algorithms that only solve the core challenging problem with the quantum hardware and the higher level problem with a classical computer. This can be called the quantum coprocessor model: the quantum device handles only the bits that the conventional computer cannot. By considering different variational principles, we show how to simulate real and imaginary time dynamics of closed and open systems. Our work can thus be applied for solving static problems or simulating the dynamics of chemistry and general many-body physics with near-term quantum computers. These are tasks that, until recently, would have been thought to need a full scale fault-tolerant quantum computer in the more distant future.

► BibTeX data

@article{Yuan2019theoryofvariational,
  doi = {10.22331/q-2019-10-07-191},
  url = {https://doi.org/10.22331/q-2019-10-07-191},
  title = {Theory of variational quantum simulation},
  author = {Yuan, Xiao and Endo, Suguru and Zhao, Qi and Li, Ying and Benjamin, Simon C.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {191},
  month = oct,
  year = {2019}
}

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

[108] Arseny Kovyrshin, Mårten Skogh, Lars Tornberg, Anders Broo, Stefano Mensa, Emre Sahin, Benjamin C. B. Symons, Jason Crain, and Ivano Tavernelli, "Nonadiabatic Nuclear–Electron Dynamics: A Quantum Computing Approach", The Journal of Physical Chemistry Letters 14 31, 7065 (2023).

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[110] 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).

[111] 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).

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[114] Martin Mootz and Yong-Xin Yao, "Efficient Berry phase calculation via adaptive variational quantum computing approach", APL Quantum 3 1, 016111 (2026).

[115] Lixing Zhang, Kaijun Shen, Yiying Yan, Kewei Sun, Maxim F. Gelin, and Yang Zhao, "Hamiltonian non-Hermicity: Accurate dynamics with the multiple Davydov D2Ansätze ", The Journal of Chemical Physics 161 19, 194108 (2024).

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[117] Shi-Yao Hou, Guanru Feng, Zipeng Wu, Hongyang Zou, Wei Shi, Jinfeng Zeng, Chenfeng Cao, Sheng Yu, Zikai Sheng, Xin Rao, Bing Ren, Dawei Lu, Junting Zou, Guoxing Miao, Jingen Xiang, and Bei Zeng, "SpinQ Gemini: a desktop quantum computing platform for education and research", EPJ Quantum Technology 8 1, 20 (2021).

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[120] Yuta Shingu, Yuya Seki, Shohei Watabe, Suguru Endo, Yuichiro Matsuzaki, Shiro Kawabata, Tetsuro Nikuni, and Hideaki Hakoshima, "Boltzmann machine learning with a variational quantum algorithm", Physical Review A 104 3, 032413 (2021).

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[122] Yongdan Yang, Zongkang Zhang, Xiaosi Xu, Bing-Nan Lu, and Ying Li, "Quantum algorithms for optimal effective theory of many-body systems", Physical Review A 108 3, 032403 (2023).

[123] Yuta Shingu, Tetsuro Nikuni, Shiro Kawabata, and Yuichiro Matsuzaki, "Quantum annealing with error mitigation", Physical Review A 109 4, 042606 (2024).

[124] Youle Wang, Benchi Zhao, and Xin Wang, "Quantum Algorithms for Estimating Quantum Entropies", Physical Review Applied 19 4, 044041 (2023).

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

[126] Jia‐Wei Ying, Jun‐Chen Shen, Lan Zhou, Wei Zhong, Ming‐Ming Du, and Yu‐Bo Sheng, "Preparing a Fast Pauli Decomposition for Variational Quantum Solving Linear Equations", Annalen der Physik 535 11, 2300212 (2023).

[127] Pooja Siwach, Kaytlin Harrison, and A. Baha Balantekin, "Collective neutrino oscillations on a quantum computer with hybrid quantum-classical algorithm", Physical Review D 108 8, 083039 (2023).

[128] G. V. Paradezhenko, A. A. Pervishko, and D. Yudin, "Probabilistic tensor optimization of quantum circuits for the max−k−cut problem", Physical Review A 109 1, 012436 (2024).

[129] Yuta Shingu, Yuki Takeuchi, Suguru Endo, Shiro Kawabata, Shohei Watabe, Tetsuro Nikuni, Hideaki Hakoshima, and Yuichiro Matsuzaki, "Variational secure cloud quantum computing", Physical Review A 105 2, 022603 (2022).

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

[131] Dawid A. Hryniuk and Marzena H. Szymańska, "Variational approach to open quantum systems with long-range competing interactions", Communications Physics 9 1, 45 (2026).

[132] Carlos Bravo-Prieto, Ryan LaRose, M. Cerezo, Yigit Subasi, Lukasz Cincio, and Patrick J. Coles, "Variational Quantum Linear Solver", Quantum 7, 1188 (2023).

[133] Chandan Sarma and P. D. Stevenson, "A low-circuit-depth quantum computing approach to the nuclear shell model", Discover Quantum Science 2 1, 6 (2026).

[134] Yabo Wang, Bo Qi, Chris Ferrie, and Daoyi Dong, "Trainability enhancement of parameterized quantum circuits via reduced-domain parameter initialization", Physical Review Applied 22 5, 054005 (2024).

[135] Jungyun Lee and Daniel K Park, "Improving generalization and trainability of quantum eigensolvers via graph neural encoding", Machine Learning: Science and Technology 7 4, 045007 (2026).

[136] Shi-Xin Zhang, Zhou-Quan Wan, Chee-Kong Lee, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao, "Variational Quantum-Neural Hybrid Eigensolver", Physical Review Letters 128 12, 120502 (2022).

[137] Zhenyu Li, Jie Liu, Xiangjian Shen, and Feixue Gao, "Challenges and opportunities of quantum-computational chemistry", SCIENTIA SINICA Chimica 53 2, 119 (2023).

[138] Zirui Sheng, Yufei Ge, Jianpeng Chen, Weitang Li, and Zhigang Shuai, "Quantum Computer Simulation of Molecules in Optical Cavity", Precision Chemistry 3 6, 326 (2025).

[139] Hao-Nan Xie, Shi-Jie Wei, Fan Yang, Zheng-An Wang, Chi-Tong Chen, Heng Fan, and Gui-Lu Long, "Probabilistic imaginary-time evolution algorithm based on nonunitary quantum circuits", Physical Review A 109 5, 052414 (2024).

[140] Erik Lötstedt and Kaoru Yamanouchi, Topics in Applied Physics 151, 137 (2024) ISBN:978-3-031-55462-9.

[141] R Au-Yeung, B Camino, O Rathore, and V Kendon, "Quantum algorithms for scientific computing", Reports on Progress in Physics 87 11, 116001 (2024).

[142] Hiroyuki Harada, Kaito Wada, and Naoki Yamamoto, "Doubly Optimal Parallel Wire Cutting without Ancilla Qubits", PRX Quantum 5 4, 040308 (2024).

[143] Barnaby van Straaten and Bálint Koczor, "Measurement Cost of Metric-Aware Variational Quantum Algorithms", PRX Quantum 2 3, 030324 (2021).

[144] Marius Lemm and Oliver Siebert, "Thermal Area Law for Lattice Bosons", Quantum 7, 1083 (2023).

[145] Kishor Bharti, Tobias Haug, Vlatko Vedral, and Leong-Chuan Kwek, "Noisy intermediate-scale quantum algorithm for semidefinite programming", Physical Review A 105 5, 052445 (2022).

[146] M. Cerezo, Kunal Sharma, Andrew Arrasmith, and Patrick J. Coles, "Variational quantum state eigensolver", npj Quantum Information 8 1, 113 (2022).

[147] Nobuyuki Yoshioka, Yuya O. Nakagawa, Kosuke Mitarai, and Keisuke Fujii, "Variational quantum algorithm for nonequilibrium steady states", Physical Review Research 2 4, 043289 (2020).

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

[149] Manas Sajjan, Junxu Li, Raja Selvarajan, Shree Hari Sureshbabu, Sumit Suresh Kale, Rishabh Gupta, Vinit Singh, and Sabre Kais, "Quantum machine learning for chemistry and physics", Chemical Society Reviews 51 15, 6475 (2022).

[150] Taichi Kosugi, Hirofumi Nishi, and Yu-ichiro Matsushita, "Exhaustive search for optimal molecular geometries using imaginary-time evolution on a quantum computer", npj Quantum Information 9 1, 112 (2023).

[151] 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).

[152] Jacopo Rizzo, Francesco Libbi, Francesco Tacchino, Pauline J. Ollitrault, Nicola Marzari, and Ivano Tavernelli, "One-particle Green's functions from the quantum equation of motion algorithm", Physical Review Research 4 4, 043011 (2022).

[153] Stefano Barison, Filippo Vicentini, and Giuseppe Carleo, "An efficient quantum algorithm for the time evolution of parameterized circuits", Quantum 5, 512 (2021).

[154] Conor Mc Keever and Michael Lubasch, "Towards Adiabatic Quantum Computing Using Compressed Quantum Circuits", PRX Quantum 5 2, 020362 (2024).

[155] Qing-Xing Xie, Yi Song, and Yan Zhao, "Power of the Sine Hamiltonian Operator for Estimating the Eigenstate Energies on Quantum Computers", Journal of Chemical Theory and Computation 18 12, 7586 (2022).

[156] 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).

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

[158] Nannan Ma, P. Z. Zhao, and Jiangbin Gong, "Quantum machine learning with indefinite causal order", Physical Review A 110 5, 052406 (2024).

[159] Yulun Wang and Predrag S Krstić, "Multistate transition dynamics by strong time-dependent perturbation in NISQ era", Journal of Physics Communications 7 7, 075004 (2023).

[160] 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).

[161] Benedikt Fauseweh and Jian-Xin Zhu, "Quantum computing Floquet energy spectra", Quantum 7, 1063 (2023).

[162] 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).

[163] Bence Bakó, Tenzan Araki, and Bálint Koczor, "Exponential Distillation of Dominant Eigenproperties", PRX Quantum 7 1, 010334 (2026).

[164] Felipe Gómez-Lozada, Nicolas Perico-García, Nikita Gourianov, Hayder Salman, and Juan José Mendoza-Arenas, "Simulating quantum turbulence with matrix-product states", Physical Review Applied 25 6, 064069 (2026).

[165] Virginia N. Ciriano-Tejel, Michael A. Fogarty, Simon Schaal, Louis Hutin, Benoit Bertrand, Lisa Ibberson, M. Fernando Gonzalez-Zalba, Jing Li, Yann-Michel Niquet, Maud Vinet, and John J.L. Morton, "Spin Readout of a CMOS Quantum Dot by Gate Reflectometry and Spin-Dependent Tunneling", PRX Quantum 2 1, 010353 (2021).

[166] Cristina Cîrstoiu, Zoë Holmes, Joseph Iosue, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, "Variational fast forwarding for quantum simulation beyond the coherence time", npj Quantum Information 6 1, 82 (2020).

[167] Chee Kong Lee, Pranay Patil, Shengyu Zhang, and Chang Yu Hsieh, "Neural-network variational quantum algorithm for simulating many-body dynamics", Physical Review Research 3 2, 023095 (2021).

[168] Jinil Lee, Wooyeong Song, Donghwa Lee, Yosep Kim, Seung-Woo Lee, Hyang-Tag Lim, Hojoong Jung, Sang-Wook Han, and Yong-Su Kim, "Photonic variational quantum eigensolver using entanglement measurements", Quantum Science and Technology 9 4, 045028 (2024).

[169] Ricard Puig, Marc Drudis, Supanut Thanasilp, and Zoë Holmes, "Variational Quantum Simulation: A Case Study for Understanding Warm Starts", PRX Quantum 6 1, 010317 (2025).

[170] Satoshi Ejima, Kazuhiro Seki, Benedikt Fauseweh, and Seiji Yunoki, "Probabilistic imaginary-time evolution in state-vector-based and shot-based simulations and on quantum devices", Physical Review Research 7 4, 043182 (2025).

[171] Fayçal Fedouaki, Mouhsene Fri, Kaoutar Douaioui, and Amellal Asmae, "Quantum Computing for Supply Chain Optimization: Algorithms, Hybrid Frameworks, and Industry Applications", Logistics 10 3, 67 (2026).

[172] 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).

[173] Hans Hon Sang Chan, Richard Meister, Matthew L. Goh, and Bálint Koczor, "Algorithmic Shadow Spectroscopy", PRX Quantum 6 1, 010352 (2025).

[174] Bojko N. Bakalov, João C. Getelina, Raghav G. Jha, Alexander F. Kemper, and Yuan Liu, "Quantum simulation of massive Thirring and Gross-Neveu models for arbitrary number of flavors", Physical Review D 113 11, 114508 (2026).

[175] Luca Crippa, Francesco Tacchino, Mario Chizzini, Antonello Aita, Michele Grossi, Alessandro Chiesa, Paolo Santini, Ivano Tavernelli, and Stefano Carretta, "Simulating Static and Dynamic Properties of Magnetic Molecules with Prototype Quantum Computers", Magnetochemistry 7 8, 117 (2021).

[176] Takaharu Yoshida, Yuta Shingu, Chihaya Shimada, Tetsuro Nikuni, Hideaki Hakoshima, and Yuichiro Matsuzaki, "Hardware-efficient quantum annealing with error mitigation via classical shadow", Physical Review A 113 6, 062430 (2026).

[177] Ting Zhang, Jinzhao Sun, Xiao-Xu Fang, Xiao-Ming Zhang, Xiao Yuan, and He Lu, "Experimental Quantum State Measurement with Classical Shadows", Physical Review Letters 127 20, 200501 (2021).

[178] Guglielmo Mazzola, "Quantum computing for chemistry and physics applications from a Monte Carlo perspective", The Journal of Chemical Physics 160 1, 010901 (2024).

[179] Cun Long, Long Cao, Liwei Ge, Qun-Xiang Li, YiJing Yan, Rui-Xue Xu, Yao Wang, and Xiao Zheng, "Quantum neural network approach to Markovian dissipative dynamics of many-body open quantum systems", The Journal of Chemical Physics 161 8, 084105 (2024).

[180] Michael Rose and David A. Mazziotti, "Many-body time evolution from a correlation-efficient quantum algorithm", Physical Review A 113 6, L060406 (2026).

[181] Steffen Backes, Yuta Murakami, Shiro Sakai, and Ryotaro Arita, "Dynamical mean-field theory for the Hubbard-Holstein model on a quantum device", Physical Review B 107 16, 165155 (2023).

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

[183] 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).

[184] Filipe Fontanela, Antoine Jacquier, and Mugad Oumgari, "Short Communication: A Quantum Algorithm for Linear PDEs Arising in Finance", SIAM Journal on Financial Mathematics 12 4, SC98 (2021).

[185] 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).

[186] Chae-Yeun Park and Nathan Killoran, "Hamiltonian variational ansatz without barren plateaus", Quantum 8, 1239 (2024).

[187] Weitang Li, Shi-Xin Zhang, Zirui Sheng, Cunxi Gong, Jianpeng Chen, and Zhigang Shuai, "Quantum machine learning of molecular energies with hybrid quantum-neural wavefunction", Digital Discovery 4 10, 2697 (2025).

[188] 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.

[189] Tangyou Huang, Yongcheng Ding, Léonce Dupays, Yue Ban, Man-Hong Yung, Adolfo del Campo, and Xi Chen, "Time-optimal control of driven oscillators by variational circuit learning", Physical Review Research 5 2, 023173 (2023).

[190] 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).

[191] Ashutosh Singh, Pooja Siwach, and P. Arumugam, "Quantum simulations of nuclear resonances with variational methods", Physical Review C 112 2, 024323 (2025).

[192] Tatsuhiko Shirai, "Quasiadiabatic thermal ensemble preparation in the thermodynamic limit", Physical Review E 113 5, 054113 (2026).

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

[194] Bingzhi Zhang, Junyu Liu, Xiao-Chuan Wu, Liang Jiang, and Quntao Zhuang, "Dynamical transition in controllable quantum neural networks with large depth", Nature Communications 15 1, 9354 (2024).

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

[196] Yigal Ilin and Itai Arad, "Dissipative Variational Quantum Algorithms for Gibbs State Preparation", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[197] Fong Yew Leong, Dax Enshan Koh, Wei-Bin Ewe, and Jian Feng Kong, "Variational quantum simulation of partial differential equations: applications in colloidal transport", International Journal of Numerical Methods for Heat & Fluid Flow 33 11, 3669 (2023).

[198] Shaojun Wu, Shan Jin, Abolfazl Bayat, and Xiaoting Wang, "Enhancing the reachability of variational quantum algorithms via input-state design", Communications Physics 9 1, 194 (2026).

[199] Alice Barthe and Adrián Pérez-Salinas, "Gradients and frequency profiles of quantum re-uploading models", Quantum 8, 1523 (2024).

[200] Sheng‐Yao Wu, Yan‐Qi Song, Run‐Ze Li, Su‐Juan Qin, Qiao‐Yan Wen, and Fei Gao, "Resource‐Efficient Adaptive Variational Quantum Algorithm for Combinatorial Optimization Problems", Advanced Quantum Technologies 9 2, 2400484 (2026).

[201] 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).

[202] Tirthak Patel, Daniel Silver, and Devesh Tiwari, 2022 Design, Automation & Test in Europe Conference & Exhibition (DATE) 334 (2022) ISBN:978-3-9819263-6-1.

[203] Xiao Xiao, J. K. Freericks, and A. F. Kemper, "Robust measurement of wave function topology on NISQ quantum computers", Quantum 7, 987 (2023).

[204] Giulia Mazzola, Simon V. Mathis, Guglielmo Mazzola, and Ivano Tavernelli, "Gauge-invariant quantum circuits for U (1) and Yang-Mills lattice gauge theories", Physical Review Research 3 4, 043209 (2021).

[205] Martin Mootz, Peter P Orth, Chuankun Huang, Liang Luo, Jigang Wang, and Yong-Xin Yao, "Two-dimensional coherent spectrum of high-spin models via a quantum computing approach", Quantum Science and Technology 9 3, 035054 (2024).

[206] 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).

[207] 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).

[208] Maurice Weber, Abhinav Anand, Alba Cervera-Lierta, Jakob S. Kottmann, Thi Ha Kyaw, Bo Li, Alán Aspuru-Guzik, Ce Zhang, and Zhikuan Zhao, "Toward reliability in the NISQ era: Robust interval guarantee for quantum measurements on approximate states", Physical Review Research 4 3, 033217 (2022).

[209] Gonçalo Pascoal, João Paulo Fernandes, and Rui Abreu, 2024 IEEE International Conference on Quantum Software (QSW) 146 (2024) ISBN:979-8-3503-6847-5.

[210] Xin Yi, Jia-Cheng Huo, Yong-Pan Gao, Ling Fan, Ru Zhang, and Cong Cao, "Iterative quantum algorithm for combinatorial optimization based on quantum gradient descent", Results in Physics 56, 107204 (2024).

[211] Pauline J Ollitrault, Sven Jandura, Alexander Miessen, Irene Burghardt, Rocco Martinazzo, Francesco Tacchino, and Ivano Tavernelli, "Quantum algorithms for grid-based variational time evolution", Quantum 7, 1139 (2023).

[212] Wooseop Hwang and Bálint Koczor, "Preparing ground and excited states using adiabatic CoVaR", New Journal of Physics 27 2, 023025 (2025).

[213] Hirad Alipanah, Feng Zhang, Yong-Xin Yao, Richard Thompson, Nam Nguyen, Junyu Liu, Peyman Givi, Brian J. McDermott, and Juan José Mendoza-Arenas, "Quantum dynamics simulation of the advection-diffusion equation", Physical Review Research 7 4, 043318 (2025).

[214] 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).

[215] Yuhan Huang, Siyuan Jin, Bei Zeng, and Qiming Shao, "Adaptive diversity-based quantum circuit architecture search", Physical Review Research 6 3, 033033 (2024).

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

[217] Gaurav Saxena, Ahmed Shalabi, and Thi Ha Kyaw, "Practical limitations of quantum data propagation on noisy quantum processors", Physical Review Applied 21 5, 054014 (2024).

[218] 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).

[219] Hamza Jnane, Brennan Undseth, Zhenyu Cai, Simon C. Benjamin, and Bálint Koczor, "Multicore Quantum Computing", Physical Review Applied 18 4, 044064 (2022).

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

[221] Tianchen Zhao, Giuseppe Carleo, James Stokes, and Shravan Veerapaneni, "Natural evolution strategies and variational Monte Carlo", Machine Learning: Science and Technology 2 2, 02LT01 (2021).

[222] Simen Kvaal, Håkon Richard Fredheim, Mads Greisen Højlund, and Thomas Bondo Pedersen, "Time-dependent Bivariational Principle: Theoretical Foundation for Real-Time Propagation Methods of Coupled-Cluster Type", The Journal of Physical Chemistry A 129 15, 3508 (2025).

[223] Pejman Jouzdani and Stefan Bringuier, "Hybrid Quantum-Classical Eigensolver without Variation or Parametric Gates", Quantum Reports 3 1, 137 (2021).

[224] Mariane Mangin-Brinet, Jing Zhang, Denis Lacroix, and Edgar Andres Ruiz Guzman, "Efficient solution of the non-unitary time-dependent Schrodinger equation on a quantum computer with complex absorbing potential", Quantum 8, 1311 (2024).

[225] Michael Vogl, "Variational principle for the time evolution operator, its usefulness in effective theories of condensed matter systems, and insight into the role played by the quantum geometry of unitary transformations", The European Physical Journal Plus 140 9, 848 (2025).

[226] Andrey Kardashin, Anastasiia Pervishko, Jacob Biamonte, and Dmitry Yudin, "Numerical hardware-efficient variational quantum simulation of a soliton solution", Physical Review A 104 2, L020402 (2021).

[227] Qingyu Li, Chiranjib Mukhopadhyay, Ludovico Minati, and Abolfazl Bayat, "Quantum reservoir computing for predicting and characterizing chaotic maps", Physical Review Research 8 1, 013304 (2026).

[228] Laszlo Gyongyosi, "Adaptive Problem Solving Dynamics in Gate-Model Quantum Computers", Entropy 24 9, 1196 (2022).

[229] Tobias Haug, Kishor Bharti, and M.S. Kim, "Capacity and Quantum Geometry of Parametrized Quantum Circuits", PRX Quantum 2 4, 040309 (2021).

[230] Hantao Zhang, Dong Bai, and Zhongzhou Ren, "Quantum computing for extracting nuclear resonances", Physics Letters B 860, 139187 (2025).

[231] Yongdan Yang, Ruyu Yang, and Xiaosi Xu, "Quantum-enhanced Green's function Monte Carlo algorithm for excited states of the nuclear shell model", Physical Review A 110 4, 042623 (2024).

[232] Dan-Bo Zhang and Tao Yin, "Collective optimization for variational quantum eigensolvers", Physical Review A 101 3, 032311 (2020).

[233] Pablo Rivas, Liang Zhao, and Javier Orduz, 2021 International Conference on Computational Science and Computational Intelligence (CSCI) 52 (2021) ISBN:978-1-6654-5841-2.

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

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Theory of variational quantum simulation

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