Small quantum computers will soon be available with the capability of performing some computational tasks that are beyond the reach of even the largest classical supercomputers. However, these devices will be noisy, limiting the complexity of the algorithms that they can implement correctly. We present a new quantum algorithm that can calculate the spectrum of complex quantum systems, and show how it may be used to perform useful computations even on the imperfect quantum devices available in the near- future.
The algorithm we have developed - variational quantum deflation - uses a hybrid of both quantum and classical resources to determine the excited state energies of quantum systems. We achieve this at almost no extra cost over the most promising hybrid method for determining ground state energies - the variational quantum eigensolver - by first finding the ground state and then energetically exciting it, allowing the first excited state to be found as the ground state of the new modified system. By repeating this procedure and incorporating techniques to mitigate device errors, our method can determine multiple excited states even on imperfect hardware. Our algorithm is therefore a promising candidate for useful near-term quantum-enhanced computation.

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

[2] Jonathan Wei Zhong Lau, Kian Hwee Lim, Kishor Bharti, Leong-Chuan Kwek, and Sai Vinjanampathy, "Convex Optimization for Nonequilibrium Steady States on a Hybrid Quantum Processor", Physical Review Letters 130 24, 240601 (2023).

[3] Fan Chen, Lei Jiang, Hausi Müller, Philip Richerme, Cheng Chu, Zhenxiao Fu, and Min Yang, "NISQ Quantum Computing: A Security-Centric Tutorial and Survey [Feature]", IEEE Circuits and Systems Magazine 24 1, 14 (2024).

[4] Suguru Endo, Jinzhao Sun, Ying Li, Simon C. Benjamin, and Xiao Yuan, "Variational Quantum Simulation of General Processes", Physical Review Letters 125 1, 010501 (2020).

[5] Hirotoshi Hirai and Sho Koh, "Non-adiabatic quantum wavepacket dynamics simulation based on electronic structure calculations using the variational quantum eigensolver", Chemical Physics 556, 111460 (2022).

[6] Yu-Cheng Chen, Yu-Qin Chen, Alice Hu, Chang-Yu Hsieh, and Shengyu Zhang, "Quantum imaginary-time control for accelerating the ground-state preparation", Physical Review Research 5 2, 023087 (2023).

[7] Alistair W. R. Smith, Johnnie Gray, and M. S. Kim, "Efficient Quantum State Sample Tomography with Basis-Dependent Neural Networks", PRX Quantum 2 2, 020348 (2021).

[8] Takashi Tsuchimochi, Yoohee Ryo, Siu Chung Tsang, and Seiichiro L. Ten-no, "Multi-state quantum simulations via model-space quantum imaginary time evolution", npj Quantum Information 9 1, 113 (2023).

[9] Phillip W. K. Jensen, Peter D. Johnson, and Alexander A. Kunitsa, "Near-term quantum algorithm for computing molecular and materials properties based on recursive variational series methods", Physical Review A 108 2, 022422 (2023).

[10] Duc Tuan Hoang, Friederike Metz, Andreas Thomasen, Tran Duong Anh-Tai, Thomas Busch, and Thomás Fogarty, "Variational quantum algorithm for ergotropy estimation in quantum many-body batteries", Physical Review Research 6 1, 013038 (2024).

[11] Feng Zhang, Niladri Gomes, Noah F. Berthusen, Peter P. Orth, Cai-Zhuang Wang, Kai-Ming Ho, and Yong-Xin Yao, "Shallow-circuit variational quantum eigensolver based on symmetry-inspired Hilbert space partitioning for quantum chemical calculations", Physical Review Research 3 1, 013039 (2021).

[12] Benchen Huang, Marco Govoni, and Giulia Galli, "Simulating the Electronic Structure of Spin Defects on Quantum Computers", PRX Quantum 3 1, 010339 (2022).

[13] Stefano Battaglia, Max Rossmannek, Vladimir V. Rybkin, Ivano Tavernelli, and Jürg Hutter, "A general framework for active space embedding methods with applications in quantum computing", npj Computational Materials 10 1, 297 (2024).

[14] Kenji Sugisaki, Chikako Sakai, Kazuo Toyota, Kazunobu Sato, Daisuke Shiomi, and Takeji Takui, "Bayesian phase difference estimation: a general quantum algorithm for the direct calculation of energy gaps", Physical Chemistry Chemical Physics 23 36, 20152 (2021).

[15] Johanna Barzen, Quantum Computing in the Arts and Humanities 1 (2022) ISBN:978-3-030-95537-3.

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

[17] Pierpaolo Pravatto, Davide Castaldo, Federico Gallina, Barbara Fresch, Stefano Corni, and Giorgio J Moro, "Quantum computing for classical problems: variational quantum eigensolver for activated processes", New Journal of Physics 23 12, 123045 (2021).

[18] Alina Joch, Götz S Uhrig, and Benedikt Fauseweh, "Entanglement-informed construction of variational quantum circuits", Quantum Science and Technology 10 3, 035032 (2025).

[19] Chee-Kong Lee, Chang-Yu Hsieh, Shengyu Zhang, and Liang Shi, "Variational Quantum Simulation of Chemical Dynamics with Quantum Computers", Journal of Chemical Theory and Computation 18 4, 2105 (2022).

[20] Josh J. M. Kirsopp, Cono Di Paola, David Zsolt Manrique, Michal Krompiec, Gabriel Greene‐Diniz, Wolfgang Guba, Agnes Meyder, Detlef Wolf, Martin Strahm, and David Muñoz Ramo, "Quantum computational quantification of protein–ligand interactions", International Journal of Quantum Chemistry 122 22, e26975 (2022).

[21] Eduarda Sangiogo-Gil and Leticia González, "Nonadiabatic molecular dynamics on quantum computers: challenges and opportunities", Pure and Applied Chemistry 97 11, 1647 (2025).

[22] Xu-Dan Xie, Zheng-Yuan Xue, and Dan-Bo Zhang, "Variational quantum algorithms for scanning the complex spectrum of non-Hermitian systems", Frontiers of Physics 19 4, 41202 (2024).

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

[24] Giuseppe Clemente, Arianna Crippa, and Karl Jansen, "Strategies for the determination of the running coupling of ( 2+1 )-dimensional QED with quantum computing", Physical Review D 106 11, 114511 (2022).

[25] Alexandre Drinko, Guilherme I. Correr, Ivan Medina, Pedro C. Azado, Askery Canabarro, and Diogo O. Soares-Pinto, "Benchmarking variational quantum eigensolvers for entanglement detection in many-body Hamiltonian ground states", AVS Quantum Science 7 2, 023802 (2025).

[26] Loris Delafosse, Vincent Robert, and Saad Yalouz, "Orthogonally constrained CASSCF framework: Newton–Raphson orbital optimization and nuclear gradients", The Journal of Chemical Physics 164 24, 241101 (2026).

[27] Kosuke Ito, Wataru Mizukami, and Keisuke Fujii, "Universal noise-precision relations in variational quantum algorithms", Physical Review Research 5 2, 023025 (2023).

[28] Jin-Min Liang, Shu-Qian Shen, Ming Li, and Shao-Ming Fei, "Quantum algorithms for the generalized eigenvalue problem", Quantum Information Processing 21 1, 23 (2022).

[29] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik, "Quantum Chemistry in the Age of Quantum Computing", Chemical Reviews 119 19, 10856 (2019).

[30] Christian Soize, Loïc Joubert-Doriol, and Artur F. Izmaylov, "Quantum computer formulation of the FKP-operator eigenvalue problem for probabilistic learning on manifolds", Computer Methods in Applied Mechanics and Engineering 443, 118080 (2025).

[31] Leonidas Taliadouros, Ilias G. Mavromatis, and Ioannis A. Kougioumtzoglou, "Eigenvalue analysis of stochastic structural systems: A quantum computing approach", Probabilistic Engineering Mechanics 81, 103814 (2025).

[32] Xiaoxiao Xiao, Wei-Hai Fang, and Zhendong Li, "Two-dimensional physics-constrained hardware-efficient ansatz on quantum computers", Chinese Science Bulletin 70 34, 5841 (2025).

[33] 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).

[34] Shigeki Gocho, Hajime Nakamura, Shu Kanno, Qi Gao, Takao Kobayashi, Taichi Inagaki, and Miho Hatanaka, "Excited state calculations using variational quantum eigensolver with spin-restricted ansätze and automatically-adjusted constraints", npj Computational Materials 9 1, 13 (2023).

[35] Lila Cadi Tazi and Alex J. W. Thom, "Folded Spectrum VQE: A Quantum Computing Method for the Calculation of Molecular Excited States", Journal of Chemical Theory and Computation 20 6, 2491 (2024).

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

[37] Dibyendu Mondal and Rahul Maitra, "Determination of molecular excited states via symmetry guided subspace search variational quantum eigensolver", Physical Chemistry Chemical Physics 28 17, 10865 (2026).

[38] Kosuke Mitarai, Yuya O. Nakagawa, and Wataru Mizukami, "Theory of analytical energy derivatives for the variational quantum eigensolver", Physical Review Research 2 1, 013129 (2020).

[39] Corey Jason Trahan, Mark Loveland, Noah Davis, and Elizabeth Ellison, "A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods", Entropy 25 4, 580 (2023).

[40] Henrik Dreyer, Mircea Bejan, and Etienne Granet, "Quantum computing critical exponents", Physical Review A 104 6, 062614 (2021).

[41] Weiwei Zhu, Kai Yang, and Jiawei Zan, 2023 16th International Conference on Advanced Computer Theory and Engineering (ICACTE) 157 (2023) ISBN:979-8-3503-1753-4.

[42] Y. Herasymenko and T.E. O'Brien, "A diagrammatic approach to variational quantum ansatz construction", Quantum 5, 596 (2021).

[43] Tianyi Li, Yumeng Zeng, Qiming Ding, Zixuan Huo, Xiaosi Xu, Jiajun Ren, Diandong Tang, Xiaoxia Cai, and Xiao Yuan, "Efficient quantum simulation of non-adiabatic molecular dynamics with precise electronic structure", Digital Discovery 5 2, 548 (2026).

[44] Juliana Cuéllar-Zuquin, Javier Carmona-García, Miriam Navarrete-Miguel, Luis Cerdán, Antonio Francés-Monerris, Angelo Giussani, Javier Segarra-Martí, and Daniel Roca-Sanjuán, Photochemistry 28 (2022) ISBN:978-1-83916-567-2.

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

[46] Wenyang Qian, Robert Basili, Soham Pal, Glenn Luecke, and James P. Vary, "Solving hadron structures using the basis light-front quantization approach on quantum computers", Physical Review Research 4 4, 043193 (2022).

[47] Atsushi Matsuo, Lecture Notes in Computer Science 12805, 22 (2021) ISBN:978-3-030-79836-9.

[48] 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).

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

[50] Daniel Marti-Dafcik, Hugh G. A. Burton, and David P. Tew, "Spin coupling is all you need: Encoding strong electron correlation in molecules on quantum computers", Physical Review Research 7 1, 013191 (2025).

[51] 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).

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

[53] Rishab Dutta, Cameron Cianci, Alexander V. Soudackov, Yuchen Wang, Chuzhi Xu, David A. Mazziotti, Lea F. Santos, and Victor S. Batista, "Qumode-Based Variational Quantum Eigensolver for Molecular Excited States", Journal of Chemical Theory and Computation 22 2, 993 (2026).

[54] Yuya Seki, Yuichiro Matsuzaki, and Shiro Kawabata, "Excited State Search Using Quantum Annealing", Journal of the Physical Society of Japan 90 5, 054002 (2021).

[55] Y. Nishida, "Research on Quantum Circuit Learning Models for Molecular Dynamics Simulation", The Journal of Physical Chemistry A 129 40, 9434 (2025).

[56] Yifeng Xiong, Daryus Chandra, Soon Xin Ng, and Lajos Hanzo, "Sampling Overhead Analysis of Quantum Error Mitigation: Uncoded vs. Coded Systems", IEEE Access 8, 228967 (2020).

[57] Scott E. Smart and Prineha Narang, "Many-body eigenstates from quantum manifold optimization", Physical Review A 110 5, 052430 (2024).

[58] Soichi Shirai, Shih-Yen Tseng, Hokuto Iwakiri, Takahiro Horiba, Hirotoshi Hirai, and Sho Koh, "Enhancing Accuracy of Quantum-Selected Configuration Interaction Calculations Using Multireference Perturbation Theory: Application to Aromatic Molecules", ACS Omega 10 35, 39736 (2025).

[59] Dhritimalya Roy and Somnath Nag, "Towards quantum simulation of rotating nuclei using quantum variational algorithms", The European Physical Journal Plus 141 7, 808 (2026).

[60] Hamza Hasnaoui, Leonardo Limongi, Taira Giordani, Beatrice Polacchi, Alberto Quaranta, Martino Bernard, Fabio Sciarrino, and Mirko Lobino, "Optical quantum computing", Applied Physics Reviews 13 3, 031306 (2026).

[61] Kazuki Ikeda, Zhong-Bo Kang, Dmitri E. Kharzeev, and Wenyang Qian, "Quantum simulation of real-time current correlators and DIS-inspired observables in the Schwinger model", Journal of High Energy Physics 2026 7, 242 (2026).

[62] 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).

[63] Sandip Maiti, Debasish Banerjee, Bipasha Chakraborty, and Emilie Huffman, "Spontaneous symmetry breaking in a SO(3) non-Abelian lattice gauge theory in 2+1D with quantum algorithms", Physical Review Research 7 1, 013283 (2025).

[64] Oleksandr Kyriienko and Vincent E. Elfving, "Generalized quantum circuit differentiation rules", Physical Review A 104 5, 052417 (2021).

[65] Júlia Barberà-Rodríguez, Nicolas Gama, Anand Kumar Narayanan, and David Joseph, "Finding dense sublattices as low energy states of a Hamiltonian", Physical Review Research 6 4, 043279 (2024).

[66] Gregory Boyd and Bálint Koczor, "Training Variational Quantum Circuits with CoVaR: Covariance Root Finding with Classical Shadows", Physical Review X 12 4, 041022 (2022).

[67] R Carobene, S Barison, and A Giachero, "Sequence of penalties method to study excited states using VQE", Quantum Science and Technology 8 3, 035014 (2023).

[68] María Laura Olivera-Atencio, Jesús Casado-Pascual, and Denis Lacroix, "Exploring fixed points and eigenstates of quantum systems with reinforcement learning", Physical Review Research 8 3, 033087 (2026).

[69] Ira Iryanti, Jun Hao Hue, Qidir Maulana Binu Soesanto, M Yusrul Hanna, Choirun Nisaa Rangkuti, Kwek Leong Chuan, and Yanoar P Sarwono, "Connecting ansatz design to density functional observables using weighted subspace search vqe for kohn-sham hamiltonian", Physica Scripta 100 12, 125414 (2025).

[70] Vinit Singh, Amandeep Singh Bhatia, Mandeep Kaur Saggi, Manas Sajjan, and Sabre Kais, "Quantum machine learning for complex systems: paradigms, applications, and challenges", Academia Quantum 3 2(2026).

[71] Piyush Kumar Sinha and R. Marimuthu, "Comparative Study of Quantum Algorithms Based on Input Parameters for Renewable Energy Sources", IEEE Access 14, 64302 (2026).

[72] Daniel Faílde, Juan Santos-Suárez, David A. Herrera-Martí, and Javier Mas, "Hamiltonian forging of a thermofield double", Physical Review A 111 1, 012432 (2025).

[73] Tariq M. Khan and Antonio Robles-Kelly, "Machine Learning: Quantum vs Classical", IEEE Access 8, 219275 (2020).

[74] 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).

[75] Lang-Xing Cheng and Dan-Bo Zhang, "Variational quantum simulation of ground states and thermal states for lattice gauge theory with multi-objective optimization", Physics Letters A 546, 130516 (2025).

[76] Hiroki Kawai and Yuya O. Nakagawa, "Predicting excited states from ground state wavefunction by supervised quantum machine learning", Machine Learning: Science and Technology 1 4, 045027 (2020).

[77] Xin Wang, Mingrui Jing, and Chengkai Zhu, "Computable and Faithful Lower Bound on Entanglement Cost", Physical Review Letters 134 19, 190202 (2025).

[78] Shuo Liu, Shi-Xin Zhang, Chang-Yu Hsieh, Shengyu Zhang, and Hong Yao, "Probing many-body localization by excited-state variational quantum eigensolver", Physical Review B 107 2, 024204 (2023).

[79] Yifu Zhang and Lei Ma, "Fast super robust nonadiabatic geometric quantum computation", Physics Letters A 521, 129794 (2024).

[80] Hongbin Liu, Guang Hao Low, Damian S. Steiger, Thomas Häner, Markus Reiher, and Matthias Troyer, "Prospects of quantum computing for molecular sciences", Materials Theory 6 1, 11 (2022).

[81] Jaehyun Bae, Gwangsu Yoo, Satoshi Nakamura, Shota Ohnishi, and Dae Sin Kim, "Hardware efficient decomposition of the Laplace operator and its application to the Helmholtz and the Poisson equation on quantum computer", Quantum Information Processing 23 7, 270 (2024).

[82] Britant and Abhishek Tiwari, 2025 17th International Conference on COMmunication Systems and NETworks (COMSNETS) 1068 (2025) ISBN:979-8-3315-3119-5.

[83] Dominic Alfonso, Yueh-Lin Lee, Hari P. Paudel, and Yuhua Duan, "Chemical applications of variational quantum eigenvalue-based quantum algorithms: Perspective and survey", Applied Physics Reviews 12 3, 031304 (2025).

[84] Keita Kanno, Masaya Kohda, Ryosuke Imai, Sho Koh, Kosuke Mitarai, Wataru Mizukami, and Yuya O. Nakagawa, "Quantum-selected configuration interaction: Classical diagonalization of Hamiltonians in subspaces selected by quantum computers", Physical Review Research 8 2, 023268 (2026).

[85] Ji-Yao Chen, Bochen Huang, D. L. Zhou, Norbert Schuch, Chenfeng Cao, and Muchun Yang, "Tangent Space Excitation Ansatz for Quantum Circuits", Physical Review Letters 136 15, 150601 (2026).

[86] Theo Juncker von Buchwald, Karl Michael Ziems, Erik Rosendahl Kjellgren, Stephan P. A. Sauer, Jacob Kongsted, and Sonia Coriani, "Reduced Density Matrix Formulation of Quantum Linear Response", Journal of Chemical Theory and Computation acs.jctc.4c00574 (2024).

[87] Kohdai Kuroiwa and Yuya O. Nakagawa, "Penalty methods for a variational quantum eigensolver", Physical Review Research 3 1, 013197 (2021).

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

[89] Nifeeya Singh, Pooja Siwach, and P. Arumugam, "Advancing quantum simulations of the nuclear shell model with Gray-code–based resource-efficient protocols", Physical Review C 112 3, 034320 (2025).

[90] Martin R. Albrecht, Miloš Prokop, Yixin Shen, and Petros Wallden, "Variational quantum solutions to the Shortest Vector Problem", Quantum 7, 933 (2023).

[91] Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko, "Quantum kernel methods for solving regression problems and differential equations", Physical Review A 107 3, 032428 (2023).

[92] Theo Juncker von Buchwald, Erik Rosendahl Kjellgren, Jacob Kongsted, Stephan P. A. Sauer, Sonia Coriani, and Karl Michael Ziems, "Reduced Density Matrix and Cumulant Approximations of Quantum Linear Response", Journal of Chemical Theory and Computation 22 4, 1652 (2026).

[93] Qi Gao, Gavin O. Jones, Mario Motta, Michihiko Sugawara, Hiroshi C. Watanabe, Takao Kobayashi, Eriko Watanabe, Yu-ya Ohnishi, Hajime Nakamura, and Naoki Yamamoto, "Applications of quantum computing for investigations of electronic transitions in phenylsulfonyl-carbazole TADF emitters", npj Computational Materials 7 1, 70 (2021).

[94] Martín A. Mosquera, "Excited-state response theory within the context of the coupled-cluster formalism", Physical Review A 106 5, 052805 (2022).

[95] Bruno Murta, Pedro M. Q. Cruz, and J. Fernández-Rossier, "Preparing valence-bond-solid states on noisy intermediate-scale quantum computers", Physical Review Research 5 1, 013190 (2023).

[96] Philipp Schleich, Joseph Boen, Lukasz Cincio, Abhinav Anand, Jakob S. Kottmann, Sergei Tretiak, Pavel A. Dub, and Alán Aspuru-Guzik, "Partitioning Quantum Chemistry Simulations with Clifford Circuits", Journal of Chemical Theory and Computation 19 15, 4952 (2023).

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

[98] Oleksandr Kyriienko, "Quantum inverse iteration algorithm for programmable quantum simulators", npj Quantum Information 6 1, 7 (2020).

[99] Yasar Y. Atas, Jinglei Zhang, Randy Lewis, Amin Jahanpour, Jan F. Haase, and Christine A. Muschik, "SU(2) hadrons on a quantum computer via a variational approach", Nature Communications 12 1, 6499 (2021).

[100] Dan-Bo Zhang, Bin-Lin Chen, Zhan-Hao Yuan, and Tao Yin, "Variational quantum eigensolvers by variance minimization", Chinese Physics B 31 12, 120301 (2022).

[101] Michele Grossi, Oriel Kiss, Francesco De Luca, Carlo Zollo, Ian Gremese, and Antonio Mandarino, "Finite-size criticality in fully connected spin models on superconducting quantum hardware", Physical Review E 107 2, 024113 (2023).

[102] Harshdeep Singh, Sonjoy Majumder, and Sabyashachi Mishra, "SHARC-VQE: Simplified Hamiltonian approach with refinement and correction enabled variational quantum eigensolver for molecular simulation", The Journal of Chemical Physics 162 11, 114117 (2025).

[103] Thomas E. O’Brien, Bruno Senjean, Ramiro Sagastizabal, Xavier Bonet-Monroig, Alicja Dutkiewicz, Francesco Buda, Leonardo DiCarlo, and Lucas Visscher, "Calculating energy derivatives for quantum chemistry on a quantum computer", npj Quantum Information 5 1, 113 (2019).

[104] Rihito Sakurai, Oliver J. Backhouse, George H. Booth, Wataru Mizukami, and Hiroshi Shinaoka, "Comparative study on compact quantum circuits of hybrid quantum-classical algorithms for quantum impurity models", Physical Review Research 6 2, 023110 (2024).

[105] Alberto B. de Palhares, Joab M. Varela, Diogo H. G. Duarte, and Rafael Chaves, "Exploring Clifford and near-Clifford initialization strategies for quantum variational optimization", International Journal of Modern Physics C 2643005 (2026).

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

[107] Jie Liu, Zhenyu Li, and Jinlong Yang, "An efficient adaptive variational quantum solver of the Schrödinger equation based on reduced density matrices", The Journal of Chemical Physics 154 24, 244112 (2021).

[108] 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).

[109] Yuichiro Yoshida, Nayuta Takemori, and Wataru Mizukami, " Ab initio extended Hubbard model of short polyenes for efficient quantum computing", The Journal of Chemical Physics 161 8, 084303 (2024).

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

[111] Guojian Wu, Dejian Huang, Feng Shuang, and Fang Gao, "Variational quantum eigenvalue solver algorithm utilizing bridge-inspired quantum circuits and a gradient filter module", Computer Physics Communications 298, 109117 (2024).

[112] Nicholas H. Stair, Renke Huang, and Francesco A. Evangelista, "A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons", Journal of Chemical Theory and Computation 16 4, 2236 (2020).

[113] Bin-Lin Chen and Dan-Bo Zhang, "Variational Quantum Eigensolver with Mutual Variance-Hamiltonian Optimization", Chinese Physics Letters 40 1, 010303 (2023).

[114] Ophelia Crawford, Barnaby van Straaten, Daochen Wang, Thomas Parks, Earl Campbell, and Stephen Brierley, "Efficient quantum measurement of Pauli operators in the presence of finite sampling error", Quantum 5, 385 (2021).

[115] 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).

[116] Nicolas PD Sawaya and Joonsuk Huh, "Improved Resource‐Tunable Near‐Term Quantum Algorithms for Transition Probabilities, with Applications in Physics and Variational Quantum Linear Algebra", Advanced Quantum Technologies 6 9, 2300042 (2023).

[117] Piyush Kumar Sinha and R. Marimuthu, "Quantum Algorithms’ Approach: A Case Study on Bladeless Wind Turbine and Solar Panel", IEEE Access 13, 85951 (2025).

[118] 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).

[119] Thomas E. Baker, "Block Lanczos method for excited states on a quantum computer", Physical Review A 110 1, 012420 (2024).

[120] Manh Tien Nguyen, Yueh-Lin Lee, Dominic Alfonso, Qing Shao, and Yuhua Duan, "Description of reaction and vibrational energetics of CO2–NH3 interaction using quantum computing algorithms", AVS Quantum Science 5 1, 013801 (2023).

[121] Ashutosh Kumar, Ayush Asthana, Vibin Abraham, T. Daniel Crawford, Nicholas J. Mayhall, Yu Zhang, Lukasz Cincio, Sergei Tretiak, and Pavel A. Dub, "Quantum Simulation of Molecular Response Properties in the NISQ Era", Journal of Chemical Theory and Computation 19 24, 9136 (2023).

[122] Bauman Nicholas P, Jaroslav Chládek, Libor Veis, Jiří Pittner, and Kowalski Karol, "Variational quantum eigensolver for approximate diagonalization of downfolded Hamiltonians using generalized unitary coupled cluster ansatz", Quantum Science and Technology 6 3, 034008 (2021).

[123] Harshdeep Singh, Sonjoy Majumder, and Sabyashachi Mishra, "Hückel molecular orbital theory on a quantum computer: A scalable system-agnostic variational implementation with compact encoding", The Journal of Chemical Physics 160 19, 194106 (2024).

[124] Salahuddin Abdul Rahman, Özkan Karabacak, and Rafal Wisniewski, "Feedback-based quantum strategies for constrained combinatorial optimization problems", Future Generation Computer Systems 174, 107979 (2026).

[125] Kamal Choudhary, "Quantum computation for predicting electron and phonon properties of solids", Journal of Physics: Condensed Matter 33 38, 385501 (2021).

[126] Jannis Ehrlich, Daniel F Urban, and Christian Elsässer, "Variational quantum-algorithm based self-consistent calculations for the two-site DMFT model on noisy quantum computing hardware", Journal of Physics: Condensed Matter 37 22, 225901 (2025).

[127] Mohannad Ibrahim, Nicholas T. Bronn, and Gregory T. Byrd, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 39 (2023) ISBN:979-8-3503-4323-6.

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

[129] 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).

[130] Yi Liu, Zixuan Hu, Xiao Zheng, and Zhenyu Li, "Variational-quantum-eigensolver solution for Lindblad-driven nonequilibrium electron-transport problems", Physical Review A 112 6, 062438 (2025).

[131] Cameron Cianci, Lea F. Santos, and Victor S. Batista, "Subspace-Search Quantum Imaginary Time Evolution for Excited State Computations", Journal of Chemical Theory and Computation 20 20, 8940 (2024).

[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] Qing-Xing Xie, Zidong Lin, Yun-Long Liu, and Yan Zhao, "Beyond VQE and QPE: a noise- and sampling-error-tolerant quantum algorithm with Heisenberg-limited precision", Quantum Information Processing 25 4, 128 (2026).

[134] Tatsuhiko N. Ikeda, Hideki Kono, and Keisuke Fujii, "Measuring Trotter error and its application to precision-guaranteed Hamiltonian simulations", Physical Review Research 6 3, 033285 (2024).

[135] Aleksei V. Ivanov, Christoph Sünderhauf, Nicole Holzmann, Tom Ellaby, Rachel N. Kerber, Glenn Jones, and Joan Camps, "Quantum computation for periodic solids in second quantization", Physical Review Research 5 1, 013200 (2023).

[136] 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).

[137] Hans Hon Sang Chan, Nathan Fitzpatrick, Javier Segarra-Martí, Michael J. Bearpark, and David P. Tew, "Molecular excited state calculations with adaptive wavefunctions on a quantum eigensolver emulation: reducing circuit depth and separating spin states", Physical Chemistry Chemical Physics 23 46, 26438 (2021).

[138] Xiaokai Hou, Guoqing Zhou, Shan Jin, Yang Li, Wei Huang, Ao Sun, Xiaoting Wang, and Bingjie Xu, "Efficient variational quantum Korkin-Zolotarev algorithm for solving shortest-vector problems", Physical Review A 113 4, 042611 (2026).

[139] Kaixuan Huang, Xiaoxia Cai, Hao Li, Zi-Yong Ge, Ruijuan Hou, Hekang Li, Tong Liu, Yunhao Shi, Chitong Chen, Dongning Zheng, Kai Xu, Zhi-Bo Liu, Zhendong Li, Heng Fan, and Wei-Hai Fang, "Variational Quantum Computation of Molecular Linear Response Properties on a Superconducting Quantum Processor", The Journal of Physical Chemistry Letters 13 39, 9114 (2022).

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

[141] "Band Theory and Beyond: Applications of Quantum Algorithms for Quantum Chemistry", University of North Texas (2022).

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

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

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

[145] Javier Argüello-Luengo, Alejandro González-Tudela, Tao Shi, Peter Zoller, and J. Ignacio Cirac, "Quantum simulation of two-dimensional quantum chemistry in optical lattices", Physical Review Research 2 4, 042013 (2020).

[146] 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).

[147] Nancy Barraza, Gabriel Alvarado Barrios, Jie Peng, Lucas Lamata, Enrique Solano, and Francisco Albarrán-Arriagada, "Analog quantum approximate optimization algorithm", Quantum Science and Technology 7 4, 045035 (2022).

[148] Scott E. Smart, Davis M. Welakuh, and Prineha Narang, "Many-Body Excited States with a Contracted Quantum Eigensolver", Journal of Chemical Theory and Computation 20 9, 3580 (2024).

[149] Yuchen Wang, LeeAnn M Sager-Smith, and David A Mazziotti, "Quantum simulation of bosons with the contracted quantum eigensolver", New Journal of Physics 25 10, 103005 (2023).

[150] Naman Khandelwal, Nidhi Verma, Pooja Jamdagni, and Ashok Kumar, "Quantum computation of the electronic structure of some prototype solids", Scientific Reports 15 1, 44074 (2025).

[151] Mi-Ra Hwang, Eylee Jung, MuSeong Kim, and DaeKil Park, "Euclidean time method in generalized eigenvalue equation", Quantum Information Processing 23 3, 62 (2024).

[152] S. X. Li, W. L. Mu, J. B. You, and X. Q. Shao, "Simulation of a feedback-based algorithm for quantum optimization for a realistic neutral-atom system with an optimized small-angle controlled-phase gate", Physical Review A 109 6, 062603 (2024).

[153] George S. Barron, Bryan T. Gard, Orien J. Altman, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Preserving Symmetries for Variational Quantum Eigensolvers in the Presence of Noise", Physical Review Applied 16 3, 034003 (2021).

[154] 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).

[155] Advait Vagerwal and Zhenhua Jiang, NAECON 2024 - IEEE National Aerospace and Electronics Conference 89 (2024) ISBN:979-8-3503-6762-1.

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

[157] Carlos Bravo-Prieto, "Quantum autoencoders with enhanced data encoding", Machine Learning: Science and Technology 2 3, 035028 (2021).

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

[159] Manuel Hagelueken, Marco F Huber, and Marco Roth, "Data efficient prediction of excited-state properties using quantum neural networks", New Journal of Physics 27 5, 054508 (2025).

[160] Kazuhiro Seki, Tomonori Shirakawa, and Seiji Yunoki, "Symmetry-adapted variational quantum eigensolver", Physical Review A 101 5, 052340 (2020).

[161] Pauline J. Ollitrault, Alexander Miessen, and Ivano Tavernelli, "Molecular Quantum Dynamics: A Quantum Computing Perspective", Accounts of Chemical Research 54 23, 4229 (2021).

[162] Nicholas P. Bauman, Guang Hao Low, and Karol Kowalski, "Quantum simulations of excited states with active-space downfolded Hamiltonians", The Journal of Chemical Physics 151 23, 234114 (2019).

[163] Guoming Wang, Sukin Sim, and Peter D. Johnson, "State Preparation Boosters for Early Fault-Tolerant Quantum Computation", Quantum 6, 829 (2022).

[164] A J Martin and J W Martin, "Studying the hyperfine interaction using the variational quantum eigensolver", European Journal of Physics 47 2, 025404 (2026).

[165] Pingyu Zhu, Chao Wu, Yang Wang, Jiacheng Liu, Gongyu Xia, Yan Wang, Qilin Zheng, Miaomiao Yu, Chang Zhao, Yuxing Du, Kaikai Zhang, Kun Wang, and Ping Xu, "Efficient Quantum Estimation of Hamiltonian Spectra via Shallow Circuits", Journal of Chemical Theory and Computation 21 9, 4417 (2025).

[166] Taichi Kosugi and Yu-ichiro Matsushita, "Linear-response functions of molecules on a quantum computer: Charge and spin responses and optical absorption", Physical Review Research 2 3, 033043 (2020).

[167] Pejman Jouzdani, Stefan Bringuier, and Mark Kostuk, "A method of determining molecular excited-states using quantum computation", MRS Advances 6 22, 558 (2021).

[168] Jonathon P. Misiewicz and Francesco A. Evangelista, "Implementation of the Projective Quantum Eigensolver on a Quantum Computer", The Journal of Physical Chemistry A 128 11, 2220 (2024).

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

[170] 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).

[171] Joel Bierman, Yingzhou Li, and Jianfeng Lu, "Improving the Accuracy of Variational Quantum Eigensolvers with Fewer Qubits Using Orbital Optimization", Journal of Chemical Theory and Computation 19 3, 790 (2023).

[172] Ivana Miháliková, Michal Krejčí, and Martin Friák, "The impact of quantum circuit architecture and hyperparameters on variational quantum algorithms exemplified in the electronic structure of the GaAs crystal", Scientific Reports 15 1, 15746 (2025).

[173] Robert A. Lang, Shashank G. Mehendale, Ilya G. Ryabinkin, and Artur F. Izmaylov, "Multistate Iterative Qubit Coupled Cluster (MS-iQCC): A Quantum-Inspired, State-Averaged Approach to Ground- And Excited-State Energies", Journal of Chemical Theory and Computation 22 4, 1714 (2026).

[174] Shiro Tamiya, Sho Koh, and Yuya O. Nakagawa, "Calculating nonadiabatic couplings and Berry's phase by variational quantum eigensolvers", Physical Review Research 3 2, 023244 (2021).

[175] Sergi Ramos-Calderer, Adrián Pérez-Salinas, Diego García-Martín, Carlos Bravo-Prieto, Jorge Cortada, Jordi Planagumà, and José I. Latorre, "Quantum unary approach to option pricing", Physical Review A 103 3, 032414 (2021).

[176] 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).

[177] Victor Wei, Alev Orfi, Felix Fehse, and William A. Coish, "Finding the Dynamics of an Integrable Quantum Many‐Body System via Machine Learning", Advanced Physics Research 3 1, 2300078 (2024).

[178] Sean Greenaway, Frédéric Sauvage, Kiran E. Khosla, and Florian Mintert, "Efficient assessment of process fidelity", Physical Review Research 3 3, 033031 (2021).

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

[180] Emanuele Mendicelli and David Schaich, "Towards Quantum Simulation of Lower-Dimensional Supersymmetric Lattice Models", (2024).

[181] 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).

[182] Qianjun Yao and He Li, "Efficient excited-state calculations for molecules based on contextual subspace method and symmetry optimizations", New Journal of Physics 27 9, 094508 (2025).

[183] Gabriel Greene‐Diniz and David Muñoz Ramo, "Generalized unitary coupled cluster excitations for multireference molecular states optimized by the variational quantum eigensolver", International Journal of Quantum Chemistry 121 4, e26352 (2021).

[184] Joel Bierman, Yingzhou Li, and Jianfeng Lu, "Qubit Count Reduction by Orthogonally Constrained Orbital Optimization for Variational Quantum Excited-State Solvers", Journal of Chemical Theory and Computation 20 8, 3131 (2024).

[185] Rimika Jaiswal, Izabella Lovas, and Leon Balents, "Simulating a quasiparticle on a quantum device", Physical Review A 111 1, 012629 (2025).

[186] Xi He, "Quantum correlation alignment for unsupervised domain adaptation", Physical Review A 102 3, 032410 (2020).

[187] Luca Nigro, Carlo Barbieri, and Enrico Prati, "Simulation of a Three‐Nucleons System Transition on Quantum Circuits", Advanced Quantum Technologies 8 5, 2400371 (2025).

[188] Hai-Ping Cheng, Erik Deumens, James K. Freericks, Chenglong Li, and Beverly A. Sanders, "Application of Quantum Computing to Biochemical Systems: A Look to the Future", Frontiers in Chemistry 8, 587143 (2020).

[189] Somasundaram R., Jayaharish R., Rohith Ramanan, and Chandra Chowdhury, "Quantum computing for molecular vibrational energies: A comprehensive study", Materials Today Quantum 6, 100031 (2025).

[190] C. D. Pemmaraju and Amol Deshmukh, "Levy-Lieb embedding of density-functional theory and its quantum kernel: Illustration for the Hubbard dimer using near-term quantum algorithms", Physical Review A 106 4, 042807 (2022).

[191] 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).

[192] G. Xu, Y. B. Guo, X. Li, K. Wang, Z. Fan, Z. S. Zhou, H. J. Liao, and T. Xiang, "Concurrent quantum eigensolver for multiple low-energy eigenstates", Physical Review A 107 5, 052423 (2023).

[193] Maria A. Castellanos, Mario Motta, and Julia E. Rice, " Quantum computation of π → π* and n → π* excited states of aromatic heterocycles ", Molecular Physics 122 7-8, e2282736 (2024).

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

[195] Utkarsh Azad and Harjinder Singh, "Quantum chemistry calculations using energy derivatives on quantum computers", Chemical Physics 558, 111506 (2022).

[196] Carlos Bistafa, Norihiko Takahashi, Jumpei Koyama, Shintaro Sato, and Yukihiro Okuno, "Accuracy and Potential of Hardware-Efficient Ansätze for Molecular Ground and Excited State Electronic Structure Calculations: Benchmark and Analysis of High-Depth Quantum Circuits", ACS Omega 10 45, 54585 (2025).

[197] Carlos L Benavides-Riveros, Yuchen Wang, Samuel Warren, and David A Mazziotti, "Quantum simulation of excited states from parallel contracted quantum eigensolvers", New Journal of Physics 26 3, 033020 (2024).

[198] Ruo-Nan Li, Yuan-Hong Tao, Jin-Min Liang, Shu-Hui Wu, and Shao-Ming Fei, "Full quantum eigensolvers based on variance", Physica Scripta 99 9, 095207 (2024).

[199] Tomoya TAKANASHI, Takeshi YOSHIKAWA, and Hiromi NAKAI, "Development of Quantum Algorithm qUCC-LR for Excited-State Calculation Using Dynamic Polarizability", Journal of Computer Chemistry, Japan 20 4, 140 (2021).

[200] Shaobo Zhang, Akib Karim, Harry M. Quiney, and Muhammad Usman, "Full band-structure calculation of semiconducting materials on a noisy quantum processor", Physical Review A 110 6, 062415 (2024).

[201] Pengyu Zhou and Honghui Shang, "Exploring electron-phonon coupling using quantum computing methods", Physica Scripta 99 12, 125105 (2024).

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

[203] Jinfeng Zeng, Chenfeng Cao, Chao Zhang, Pengxiang Xu, and Bei Zeng, "A variational quantum algorithm for Hamiltonian diagonalization", Quantum Science and Technology 6 4, 045009 (2021).

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

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

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

[207] Wei-Bin Ewe, Dax Enshan Koh, Siong Thye Goh, Hong-Son Chu, and Ching Eng Png, "Variational Quantum-Based Simulation of Waveguide Modes", IEEE Transactions on Microwave Theory and Techniques 70 5, 2517 (2022).

[208] Kenji Sugisaki and Yuji Mochizuki, "Quantum Applications", The Journal of The Institute of Image Information and Television Engineers 77 1, 47 (2023).

[209] Ken M. Nakanishi, Kosuke Mitarai, and Keisuke Fujii, "Subspace-search variational quantum eigensolver for excited states", Physical Review Research 1 3, 033062 (2019).

[210] Ansh Jaiswal, Sheetal, Shyam R, Alli. A, Veena S Badiger, and Peer Mohammed Jeelan, 2025 IEEE Pune Section International Conference (PuneCon) 1 (2025) ISBN:979-8-3315-8834-2.

[211] Manuela Weigold, Johanna Barzen, Frank Leymann, and Daniel Vietz, Communications in Computer and Information Science 1429, 34 (2021) ISBN:978-3-030-87567-1.

[212] Yordan S. Yordanov, Crispin H. W. Barnes, and David R. M. Arvidsson-Shukur, "Molecular-excited-state calculations with the qubit-excitation-based adaptive variational quantum eigensolver protocol", Physical Review A 106 3, 032434 (2022).

[213] Irma Avdic and David A. Mazziotti, "Enhanced shadow tomography of molecular excited states via the enforcement of N -representability conditions by semidefinite programming", Physical Review A 110 5, 052407 (2024).

[214] Saad Yalouz, Bruno Senjean, Jakob Günther, Francesco Buda, Thomas E O’Brien, and Lucas Visscher, "A state-averaged orbital-optimized hybrid quantum–classical algorithm for a democratic description of ground and excited states", Quantum Science and Technology 6 2, 024004 (2021).

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

[216] 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).

[217] 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).

[218] Yi Fan, Jie Liu, Zhenyu Li, and Jinlong Yang, "Equation-of-Motion Theory to Calculate Accurate Band Structures with a Quantum Computer", The Journal of Physical Chemistry Letters 12 36, 8833 (2021).

[219] Taichi Kosugi and Yu-ichiro Matsushita, "Construction of Green's functions on a quantum computer: Quasiparticle spectra of molecules", Physical Review A 101 1, 012330 (2020).

[220] Kade Head-Marsden, Johannes Flick, Christopher J. Ciccarino, and Prineha Narang, "Quantum Information and Algorithms for Correlated Quantum Matter", Chemical Reviews 121 5, 3061 (2021).

[221] Lexin Ding, Cheng-Lin Hong, and Christian Schilling, "Ground and Excited States from Ensemble Variational Principles", Quantum 8, 1525 (2024).

[222] Benedikt Fauseweh, "Quantum many-body simulations on digital quantum computers: State-of-the-art and future challenges", Nature Communications 15 1, 2123 (2024).

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

[224] Bálint Koczor and Simon C. Benjamin, "Quantum natural gradient generalized to noisy and nonunitary circuits", Physical Review A 106 6, 062416 (2022).

[225] Joonho Kim and Yaron Oz, "Entanglement diagnostics for efficient VQA optimization", Journal of Statistical Mechanics: Theory and Experiment 2022 7, 073101 (2022).

[226] Keita Omiya, Yuya O. Nakagawa, Sho Koh, Wataru Mizukami, Qi Gao, and Takao Kobayashi, "Analytical Energy Gradient for State-Averaged Orbital-Optimized Variational Quantum Eigensolvers and Its Application to a Photochemical Reaction", Journal of Chemical Theory and Computation 18 2, 741 (2022).

[227] Dileep Singh, Shashank Mehendale, Arvind, and Kavita Dorai, "Accurate excited state energy calculations of the H$$_\textbf{2}$$ molecule using a variational quantum deflation algorithm on an NMR quantum simulator", Pramana 100 2, 89 (2026).

[228] Mikio Morita, Yoshinori Tomita, Junpei Koyama, and Koichi Kimura, "Simulator Demonstration of Large Scale Variational Quantum Algorithm on HPC Cluster", IEEE Access 12, 85219 (2024).

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

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

[231] Emiel Koridon, Saad Yalouz, Bruno Senjean, Francesco Buda, Thomas E. O'Brien, and Lucas Visscher, "Orbital transformations to reduce the 1-norm of the electronic structure Hamiltonian for quantum computing applications", Physical Review Research 3 3, 033127 (2021).

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

[233] Earl Campbell, "Random Compiler for Fast Hamiltonian Simulation", Physical Review Letters 123 7, 070503 (2019).

[234] Takahiro Yoshikura, Seiichiro L. Ten-no, and Takashi Tsuchimochi, "Quantum Inverse Algorithm via Adaptive Variational Quantum Linear Solver: Applications to General Eigenstates", The Journal of Physical Chemistry A 127 31, 6577 (2023).

[235] Martin Plesch, Martin Friák, and Ijaz Ahamed Mohammad, "Efficient implementation of single particle Hamiltonians in exponentially reduced qubit space", Quantum 10, 2099 (2026).

[236] Yi-Qiang Wu, Xuan Liu, Hanlin Li, and Fuqiang Wang, "Energy-embedded neural solvers for one-dimensional quantum systems", Physical Review A 112 6, 062611 (2025).

[237] Francesco Ghisoni, Francesco Scala, Daniele Bajoni, and Dario Gerace, "Resource-efficient quantum algorithm for linear systems of equations", Physical Review A 113 3, 032405 (2026).

[238] Qing‐Xing Xie, Wen‐gang Zhang, Xu‐Sheng Xu, Sheng Liu, and Yan Zhao, "Qubit unitary coupled cluster with generalized single and paired double excitations ansatz for variational quantum eigensolver", International Journal of Quantum Chemistry 122 24, e27001 (2022).

[239] Anton Simen Albino, Rodrigo Bloot, and Raphael F. I. Gomes, "Variable ansatz applied to spectral operator decomposition in a physical superconducting quantum device", Quantum Information Processing 22 6, 233 (2023).

[240] Johannes Selisko, Maximilian Amsler, Christopher Wever, Yukio Kawashima, Georgy Samsonidze, Rukhsan Ul Haq, Francesco Tacchino, Ivano Tavernelli, and Thomas Eckl, "Dynamical mean field theory for real materials on a quantum computer", npj Computational Materials 11 1, 325 (2025).

[241] Soichi Shirai, Takahiro Horiba, and Hirotoshi Hirai, "Calculation of Core-Excited and Core-Ionized States Using Variational Quantum Deflation Method and Applications to Photocatalyst Modeling", ACS Omega 7 12, 10840 (2022).

[242] Peter Reinholdt, Erik Kjellgren, Karl Michael Ziems, Sonia Coriani, Stephan P. A. Sauer, and Jacob Kongsted, "Self-consistent Quantum Linear Response with a Polarizable Embedding Environment", The Journal of Physical Chemistry A 129 5, 1504 (2025).

[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] Bozhi Wang, Jingwei Wen, Jiawei Wu, Haonan Xie, Fan Yang, Dong Ruan, Shijie Wei, and Gui-lu Long, "Improving the full quantum eigensolver with exponentiated operators", Physical Review B 109 24, 245117 (2024).

[245] Takeshi Yoshikawa, Tomoya Takanashi, and Hiromi Nakai, "Scalable Quantum-Classical Hybrid Algorithm for Excited States Based on Divide-and-Conquer Unitary Coupled-Cluster Linear-Response Theory Using Dynamical Polarizability", The Journal of Physical Chemistry A 130 28, 5602 (2026).

[246] Tilas Kabengele, Yash M. Lokare, J. B. Marston, and Brenda M. Rubenstein, "Modeling stochastic chemical kinetics on quantum computers", The Journal of Chemical Physics 164 19, 194117 (2026).

[247] Dibyendu Mondal and Rahul Maitra, "Determination of Molecular Symmetry Adapted Eigenroots in the Variational Quantum Eigensolver Framework", International Journal of Quantum Chemistry 125 7, e70029 (2025).

[248] Max Rossmannek, Panagiotis Kl. Barkoutsos, Pauline J. Ollitrault, and Ivano Tavernelli, "Quantum HF/DFT-embedding algorithms for electronic structure calculations: Scaling up to complex molecular systems", The Journal of Chemical Physics 154 11, 114105 (2021).

[249] Utkarsh Azad and Animesh Sinha, "qLEET: visualizing loss landscapes, expressibility, entangling power and training trajectories for parameterized quantum circuits", Quantum Information Processing 22 6, 256 (2023).

[250] Leopoldo Mejía, Jia Yin, David R. Reichman, Roi Baer, Chao Yang, and Eran Rabani, "Stochastic Real-Time Second-Order Green’s Function Theory for Neutral Excitations in Molecules and Nanostructures", Journal of Chemical Theory and Computation 19 16, 5563 (2023).

[251] William A Wheeler, Kevin G Kleiner, and Lucas K Wagner, "Ensemble variational Monte Carlo for optimization of correlated excited state wave functions", Electronic Structure 6 2, 025001 (2024).

[252] Sumeet, M. Hörmann, and K. P. Schmidt, "Quantum algorithm for one-quasiparticle excitations in the thermodynamic limit via cluster-additive block diagonalization", Physical Review Research 8 2, 023358 (2026).

[253] Andy C. Y. Li, M. Sohaib Alam, Thomas Iadecola, Ammar Jahin, Joshua Job, Doga Murat Kurkcuoglu, Richard Li, Peter P. Orth, A. Barış Özgüler, Gabriel N. Perdue, and Norm M. Tubman, "Benchmarking variational quantum eigensolvers for the square-octagon-lattice Kitaev model", Physical Review Research 5 3, 033071 (2023).

[254] Chao Liu and Yuxin Deng, "Resource-Efficient Simulation of Molecular Ground and Excited States Based on Contextual Subspace and Quantum Subspace Expansion", Journal of Chemical Theory and Computation (2026).

[255] Chufan Lyu, Xiaoyu Tang, Junning Li, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Variational quantum simulation of long-range interacting systems", New Journal of Physics 25 5, 053022 (2023).

[256] Daniel Wines, Ramya Gurunathan, Kevin F. Garrity, Brian DeCost, Adam J. Biacchi, Francesca Tavazza, and Kamal Choudhary, "Recent progress in the JARVIS infrastructure for next-generation data-driven materials design", Applied Physics Reviews 10 4, 041302 (2023).

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

[258] Shoukuan Zhao, Diandong Tang, Xiaoxiao Xiao, Ruixia Wang, Qiming Sun, Zhen Chen, Xiaoxia Cai, Zhendong Li, Haifeng Yu, and Wei-Hai Fang, "Quantum Computation of Conical Intersections on a Programmable Superconducting Quantum Processor", The Journal of Physical Chemistry Letters 15 28, 7244 (2024).

[259] Feng Zhang, Niladri Gomes, Yongxin Yao, Peter P. Orth, and Thomas Iadecola, "Adaptive variational quantum eigensolvers for highly excited states", Physical Review B 104 7, 075159 (2021).

[260] Yuchen Wang and David A. Mazziotti, "Electronic excited states from a variance-based contracted quantum eigensolver", Physical Review A 108 2, 022814 (2023).

[261] Ha-Vu Phan, Quoc-Thang Phan, Yuan-Kang Wu, and Quoc Dung Phan, 2025 IEEE Industry Applications Society Annual Meeting (IAS) 1 (2025) ISBN:978-1-6654-5776-7.

[262] Jaeho Choi, Seunghyeok Oh, and Joongheon Kim, 2020 International Conference on Information Networking (ICOIN) 1 (2020) ISBN:978-1-7281-4199-2.

[263] Alexey Pyrkov, Alex Aliper, Dmitry Bezrukov, Yen-Chu Lin, Daniil Polykovskiy, Petrina Kamya, Feng Ren, and Alex Zhavoronkov, "Quantum computing for near-term applications in generative chemistry and drug discovery", Drug Discovery Today 28 8, 103675 (2023).

[264] Pauline J. Ollitrault, Alberto Baiardi, Markus Reiher, and Ivano Tavernelli, "Hardware efficient quantum algorithms for vibrational structure calculations", Chemical Science 11 26, 6842 (2020).

[265] Junyong Lee and Shiho Kim, Advances in Computers 140, 65 (2026) ISBN:9780443223822.

[266] Yu Zhang, Lukasz Cincio, Christian F. A. Negre, Piotr Czarnik, Patrick J. Coles, Petr M. Anisimov, Susan M. Mniszewski, Sergei Tretiak, and Pavel A. Dub, "Variational quantum eigensolver with reduced circuit complexity", npj Quantum Information 8 1, 96 (2022).

[267] Michele Minervini, Madison Chin, Jacob Kupperman, Nana Liu, Ivy Luo, Meghan Ly, Soorya Rethinasamy, Kathie Wang, and Mark M. Wilde, "Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems", Physical Review A 113 4, 042407 (2026).

[268] Jin-Min Liang, Shi-Jie Wei, and Shao-Ming Fei, "Quantum gradient descent algorithms for nonequilibrium steady states and linear algebraic systems", Science China Physics, Mechanics & Astronomy 65 5, 250313 (2022).

[269] Enhua Xu, Yuma Shimomoto, Seiichiro L. Ten-no, and Takashi Tsuchimochi, "Many-Body-Expansion Based on Variational Quantum Eigensolver and Deflation for Dynamical Correlation", The Journal of Physical Chemistry A 128 12, 2507 (2024).

[270] Yuchen Wang, "Diabatic Potential Energy Matrices at the Interface of Nonadiabatic Dynamics, Machine Learning, and Quantum Computing", Atoms 14 3, 19 (2026).

[271] Kenji Sugisaki, Takumi Kato, Yuichiro Minato, Koji Okuwaki, and Yuji Mochizuki, "Variational quantum eigensolver simulations with the multireference unitary coupled cluster ansatz: a case study of the C2v quasi-reaction pathway of beryllium insertion into a H2 molecule", Physical Chemistry Chemical Physics 24 14, 8439 (2022).

[272] Qianjun Yao, Qun Ji, Xiaopeng Li, Yehui Zhang, Xinyu Chen, Ming-Gang Ju, Jie Liu, and Jinlan Wang, "Machine Learning Accelerates Precise Excited-State Potential Energy Surface Calculations on a Quantum Computer", The Journal of Physical Chemistry Letters 15 27, 7061 (2024).

[273] Bujiao Wu, Jinzhao Sun, Qi Huang, and Xiao Yuan, "Overlapped grouping measurement: A unified framework for measuring quantum states", Quantum 7, 896 (2023).

[274] Sheng-Hsuan Lin, Rohit Dilip, Andrew G. Green, Adam Smith, and Frank Pollmann, "Real- and Imaginary-Time Evolution with Compressed Quantum Circuits", PRX Quantum 2 1, 010342 (2021).

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

[276] Yutaka Shikano, Hiroshi C. Watanabe, Ken M. Nakanishi, and Yu-ya Ohnishi, "Post-Hartree–Fock method in quantum chemistry for quantum computer", The European Physical Journal Special Topics 230 4, 1037 (2021).

[277] Yuchen Wang, Cameron Cianci, Irma Avdic, Rishab Dutta, Samuel Warren, Brandon Allen, Nam P. Vu, Lea F. Santos, Victor S. Batista, and David A. Mazziotti, "Characterizing Conical Intersections of Nucleobases on Quantum Computers", Journal of Chemical Theory and Computation 21 3, 1213 (2025).

[278] Juan Felipe Huan Lew-Yee and Mario Piris, "Efficient Energy Measurement of Chemical Systems via One-Particle Reduced Density Matrix: A NOF-VQE Approach for Optimized Sampling", Journal of Chemical Theory and Computation 21 5, 2402 (2025).

[279] Smriti Sharma, "Quantum algorithms for simulation of quantum chemistry problems by quantum computers: an appraisal", Foundations of Chemistry 24 2, 263 (2022).

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

[281] Alexandre Drinko, Guilherme Ilário Correr, Ivan Medina, Pedro Coutinho Azado, Askery Canabarro, and Diogo Oliveira Soares-Pinto, "Benchmarking Variational Quantum Eigensolvers for Entanglement Detection in Many-Body Hamiltonian Ground States", (2024).

[282] Waleed Sherif, "Simultaneous approximation of multiple degenerate states using a single neural network quantum state", Machine Learning: Science and Technology 7 1, 015029 (2026).

[283] Saverio Monaco, Oriel Kiss, Antonio Mandarino, Sofia Vallecorsa, and Michele Grossi, "Quantum phase detection generalization from marginal quantum neural network models", Physical Review B 107 8, L081105 (2023).

[284] Gennaro De Luca, Andrew Vlasic, Michael Vitz, and Anh Pham, "Empirical power of quantum encoding methods for binary classification", Quantum Machine Intelligence 7 2, 72 (2025).

[285] C. De Lucia, A. Martone, F. A. D'Aniello, A. Mastroianni, G. Nunziata, G. De Gregorio, R. Folprecht, F. Knapp, N. Lo Iudice, and P. Veselý, "Solution of the equation-of-motion phonon method eigenvalue problems on the D-wave quantum annealer", Physical Review C 113 6, 064325 (2026).

[286] Jiaxin Li, Zhaobing Fan, Hongmei Yao, Chunlin Yang, Shao-Ming Fei, Zi-Tong Zhou, Meng-Han Dou, and Teng-Yang Ma, "Variational quantum algorithm for generalized eigenvalue problems of non-Hermitian systems", Physical Review A 113 1, 012406 (2026).

[287] Zhijie Sun, Xiaopeng Li, Jie Liu, Zhenyu Li, and Jinlong Yang, "Circuit-Efficient Qubit Excitation-Based Variational Quantum Eigensolver", Journal of Chemical Theory and Computation 21 10, 5071 (2025).

[288] Salahuddin Abdul Rahman, Özkan Karabacak, and Rafal Wisniewski, "Feedback-Based Quantum Algorithm for Excited States Calculation", IEEE Transactions on Quantum Engineering 7, 1 (2026).

[289] Maria-Andreea Filip, "Fighting Noise with Noise: A Stochastic Projective Quantum Eigensolver", Journal of Chemical Theory and Computation 20 14, 5964 (2024).

[290] Giuseppe Clemente, Arianna Crippa, Karl Jansen, and Cenk Tüysüz, Quantum Computer Music 433 (2022) ISBN:978-3-031-13908-6.

[291] Eoin Carolan, Nathan Keenan, Gabriele Cenedese, and Giuliano Benenti, "Variational preparation of quantum many-body scars with shallow circuits", Physical Review A 114 1, 012613 (2026).

[292] Adrián Pérez-Salinas, Juan Cruz-Martinez, Abdulla A. Alhajri, and Stefano Carrazza, "Determining the proton content with a quantum computer", Physical Review D 103 3, 034027 (2021).

[293] Mafalda Ramôa, Panagiotis G. Anastasiou, Luis Paulo Santos, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Reducing the resources required by ADAPT-VQE using coupled exchange operators and improved subroutines", npj Quantum Information 11 1, 86 (2025).

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

[295] Carlos Bistafa, Josh J. M. Kirsopp, Antonio Márquez Romero, Jumpei Koyama, and Michal Krompiec, "A Hybrid Quantum Computing Method for UV–Vis Spectroscopy of Solvated Molecules at Room Temperature", The Journal of Physical Chemistry A 129 43, 9991 (2025).

[296] Kübra Yeter-Aydeniz, Raphael C. Pooser, and George Siopsis, "Practical quantum computation of chemical and nuclear energy levels using quantum imaginary time evolution and Lanczos algorithms", npj Quantum Information 6 1, 63 (2020).

[297] Yongbin Kim and Anna I. Krylov, "Two Algorithms for Excited-State Quantum Solvers: Theory and Application to EOM-UCCSD", The Journal of Physical Chemistry A 127 31, 6552 (2023).

[298] Dan-Bo Zhang, Hongxi Xing, Hui Yan, Enke Wang, and Shi-Liang Zhu, "Selected topics of quantum computing for nuclear physics* ", Chinese Physics B 30 2, 020306 (2021).

[299] Anthony W. Schlimgen, Kade Head-Marsden, LeeAnn M. Sager, Prineha Narang, and David A. Mazziotti, "Quantum simulation of the Lindblad equation using a unitary decomposition of operators", Physical Review Research 4 2, 023216 (2022).

[300] Shreya Verma, Abhishek Mitra, Qiaohong Wang, Ruhee D’Cunha, Bhavnesh Jangid, Matthew R. Hennefarth, Valay Agarawal, Leon Otis, Soumi Haldar, Matthew R. Hermes, and Laura Gagliardi, "Multireference Embedding and Fragmentation Methods for Classical and Quantum Computers: From Model Systems to Realistic Applications", Chemical Reviews 126 1, 184 (2026).

[301] Chongji Jiang and Junchen Pei, "Quantum computing of the pairing Hamiltonian at finite temperature", Physical Review C 107 4, 044308 (2023).

[302] Joel Bierman, Yingzhou Li, and Jianfeng Lu, "Quantum Orbital Minimization Method for Excited States Calculation on a Quantum Computer", Journal of Chemical Theory and Computation 18 8, 4674 (2022).

[303] Juntao Lai, Qiang Fu, Zhenyu Li, and Jinlong Yang, "Accurate Vibrational Frequency Calculations for Quantum Computing via an Analytic Second-Order Energy Derivative Framework", Journal of Chemical Theory and Computation 22 9, 4304 (2026).

[304] Nikolay V Tkachenko, Lukasz Cincio, Alexander I Boldyrev, Sergei Tretiak, Pavel A Dub, and Yu Zhang, "Quantum Davidson algorithm for excited states", Quantum Science and Technology 9 3, 035012 (2024).

[305] Alan Bidart, Prateek Vaish, Tilas Kabengele, Yaoqi Pang, Yuan Liu, and Brenda M. Rubenstein, "Quantum Computing Beyond Ground-State Electronic Structure: A Review of Progress Toward Quantum Chemistry Out of the Ground State", Annual Review of Physical Chemistry 77 1, 417 (2026).

[306] Xi He, Li Sun, Chufan Lyu, and Xiaoting Wang, "Quantum locally linear embedding for nonlinear dimensionality reduction", Quantum Information Processing 19 9, 309 (2020).

[307] Jingcheng Dai, Atharva Vidwans, Eric H. Wan, Alexander X. Miller, and Micheline B. Soley, "Molecular Resonance Identification in Complex Absorbing Potentials via Integrated Quantum Computing and High-Throughput Computing", Journal of Chemical Theory and Computation 22 6, 2687 (2026).

[308] 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).

[309] Qi Gao, Michihiko Sugawara, Paul D. Nation, Takao Kobayashi, Yu-ya Ohnishi, Hiroyuki Tezuka, and Naoki Yamamoto, "A Quantum-Classical Method Applied to Material Design: Photochromic Materials Optimization for Photopharmacology Applications", Intelligent Computing 3, 0108 (2024).

[310] Kosuke Mitarai and Keisuke Fujii, "Methodology for replacing indirect measurements with direct measurements", Physical Review Research 1 1, 013006 (2019).

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

[312] Bálint Koczor, "Exponential Error Suppression for Near-Term Quantum Devices", Physical Review X 11 3, 031057 (2021).

[313] Lingxia Zhang, Xiaodie Lin, Peidong Wang, Kaiyan Yang, Xiao Zeng, Zhaohui Wei, and Zizhu Wang, "Variational optimization for quantum problems using deep generative networks", Communications Physics 8 1, 334 (2025).

[314] Michal Krejčí, Lucie Krejčí, Ijaz Ahamed Mohammad, Martin Plesch, and Martin Friák, "Minimum measurements quantum protocol for band structure calculations", Quantum Science and Technology 11 3, 035059 (2026).

[315] Manas Sajjan, Shree Hari Sureshbabu, and Sabre Kais, "Quantum Machine-Learning for Eigenstate Filtration in Two-Dimensional Materials", Journal of the American Chemical Society 143 44, 18426 (2021).

[316] Jan Lukas Bosse and Ashley Montanaro, "Probing ground-state properties of the kagome antiferromagnetic Heisenberg model using the variational quantum eigensolver", Physical Review B 105 9, 094409 (2022).

[317] Eduarda Sangiogo Gil, Markus Oppel, Jakob S. Kottmann, and Leticia González, "SHARC meets TEQUILA: mixed quantum-classical dynamics on a quantum computer using a hybrid quantum-classical algorithm", Chemical Science 16 2, 596 (2025).

[318] Nannan Ma, Heng Dai, and Jiangbin Gong, "Penalty-free quantum algorithm to find energy eigenstates", Physical Review A 113 4, 042625 (2026).

[319] Hakon Volkmann, Raamamurthy Sathyanarayanan, Alejandro Saenz, Karl Jansen, and Stefan Kühn, "Chemically Accurate Potential Curves for H2 Molecules Using Explicitly Correlated Qubit-ADAPT", Journal of Chemical Theory and Computation 20 3, 1244 (2024).

[320] Qing Guo and Ping-Xing Chen, "Optimization of VQE-UCC Algorithm Based on Spin State Symmetry", Frontiers in Physics 9, 735321 (2021).

[321] Chris N. Self, Kiran E. Khosla, Alistair W. R. Smith, Frédéric Sauvage, Peter D. Haynes, Johannes Knolle, Florian Mintert, and M. S. Kim, "Variational quantum algorithm with information sharing", npj Quantum Information 7 1, 116 (2021).

[322] Piyush Kumar Sinha and Marimuthu R, "Comparative study of quantum and classical algorithms for renewable energy sources", Results in Engineering 27, 107062 (2025).

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

[324] Nicolas PD Sawaya, Daniel Marti-Dafcik, Yang Ho, Daniel P Tabor, David E Bernal Neira, Alicia B Magann, Shavindra Premaratne, Pradeep Dubey, Anne Matsuura, Nathan Bishop, Wibe A de Jong, Simon Benjamin, Ojas Parekh, Norm Tubman, Katherine Klymko, and Daan Camps, "HamLib: A library of Hamiltonians for benchmarking quantum algorithms and hardware", Quantum 8, 1559 (2024).

[325] Kaoru Mizuta, Mikiya Fujii, Shigeki Fujii, Kazuhide Ichikawa, Yutaka Imamura, Yukihiro Okuno, and Yuya O. Nakagawa, "Deep variational quantum eigensolver for excited states and its application to quantum chemistry calculation of periodic materials", Physical Review Research 3 4, 043121 (2021).

[326] Dibyajyoti Chakravarti and Leonardo Guidoni, Handbook of Electronic Structure Theory 263 (2026) ISBN:9780443265969.

[327] 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).

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

[329] Saad Yalouz, Emiel Koridon, Bruno Senjean, Benjamin Lasorne, Francesco Buda, and Lucas Visscher, "Analytical Nonadiabatic Couplings and Gradients within the State-Averaged Orbital-Optimized Variational Quantum Eigensolver", Journal of Chemical Theory and Computation 18 2, 776 (2022).

[330] Nicolas P. D. Sawaya, Francesco Paesani, and Daniel P. Tabor, "Near- and long-term quantum algorithmic approaches for vibrational spectroscopy", Physical Review A 104 6, 062419 (2021).

[331] "Computational Techniques for Accelerated Materials Discovery", University of North Texas (2021).

[332] Kamal Choudhary, Daniel Wines, Kangming Li, Kevin F. Garrity, Vishu Gupta, Aldo H. Romero, Jaron T. Krogel, Kayahan Saritas, Addis Fuhr, Panchapakesan Ganesh, Paul R. C. Kent, Keqiang Yan, Yuchao Lin, Shuiwang Ji, Ben Blaiszik, Patrick Reiser, Pascal Friederich, Ankit Agrawal, Pratyush Tiwary, Eric Beyerle, Peter Minch, Trevor David Rhone, Ichiro Takeuchi, Robert B. Wexler, Arun Mannodi-Kanakkithodi, Elif Ertekin, Avanish Mishra, Nithin Mathew, Mitchell Wood, Andrew Dale Rohskopf, Jason Hattrick-Simpers, Shih-Han Wang, Luke E. K. Achenie, Hongliang Xin, Maureen Williams, Adam J. Biacchi, and Francesca Tavazza, "JARVIS-Leaderboard: a large scale benchmark of materials design methods", npj Computational Materials 10 1, 93 (2024).

[333] Qingyu Li, Yuhan Huang, Xiaokai Hou, Ying Li, Xiaoting Wang, and Abolfazl Bayat, "Ensemble-learning error mitigation for variational quantum shallow-circuit classifiers", Physical Review Research 6 1, 013027 (2024).

[334] Diego García-Vega, Fernando Plou Llorente, Alejandro Leal Castaño, Elías F. Combarro, and José Ranilla, "pLazyQML: A parallel package for efficient execution of QML models on classical computers", (2025).

[335] Kübra Yeter‐Aydeniz, Bryan T. Gard, Jacek Jakowski, Swarnadeep Majumder, George S. Barron, George Siopsis, Travis S. Humble, and Raphael C. Pooser, "Benchmarking Quantum Chemistry Computations with Variational, Imaginary Time Evolution, and Krylov Space Solver Algorithms", Advanced Quantum Technologies 4 7, 2100012 (2021).

[336] Huanfeng Zhao, Peng Zhang, and Tzu-Chieh Wei, "A universal variational quantum eigensolver for non-Hermitian systems", Scientific Reports 13 1, 22313 (2023).

[337] Christian Vorwerk, Nan Sheng, Marco Govoni, Benchen Huang, and Giulia Galli, "Quantum embedding theories to simulate condensed systems on quantum computers", Nature Computational Science 2 7, 424 (2022).

[338] S. Padmakala, 2023 International Conference on Sustainable Communication Networks and Application (ICSCNA) 1716 (2023) ISBN:979-8-3503-1398-7.

[339] Thi Ha Kyaw, Tim Menke, Sukin Sim, Abhinav Anand, Nicolas P.D. Sawaya, William D. Oliver, Gian Giacomo Guerreschi, and Alán Aspuru-Guzik, "Quantum Computer-Aided Design: Digital Quantum Simulation of Quantum Processors", Physical Review Applied 16 4, 044042 (2021).

[340] Pauline J. Ollitrault, Abhinav Kandala, Chun-Fu Chen, Panagiotis Kl. Barkoutsos, Antonio Mezzacapo, Marco Pistoia, Sarah Sheldon, Stefan Woerner, Jay M. Gambetta, and Ivano Tavernelli, "Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor", Physical Review Research 2 4, 043140 (2020).

[341] Yohei Ibe, Yuya O. Nakagawa, Nathan Earnest, Takahiro Yamamoto, Kosuke Mitarai, Qi Gao, and Takao Kobayashi, "Calculating transition amplitudes by variational quantum deflation", Physical Review Research 4 1, 013173 (2022).

[342] Yi Fan, Jie Liu, Zhenyu Li, and Jinlong Yang, "Quantum Circuit Matrix Product State Ansatz for Large-Scale Simulations of Molecules", Journal of Chemical Theory and Computation 19 16, 5407 (2023).

[343] Ana Clara das Neves Silva, Lucas Queiroz Galvão, and Clebson Cruz, "Simulating thermodynamic properties of dinuclear metal complexes using Variational Quantum Algorithms", Physica Scripta 99 9, 095131 (2024).

[344] Yanick S. Kind and Benedikt Fauseweh, "Exploring Variational Entanglement Hamiltonians", Quantum 10, 2133 (2026).

[345] Carla Lupo, François Jamet, Wai Hei Terence Tse, Ivan Rungger, and Cedric Weber, "Maximally localized dynamical quantum embedding for solving many-body correlated systems", Nature Computational Science 1 6, 410 (2021).

[346] Salahuddin Abdul Rahman, Özkan Karabacak, and Rafal Wisniewski, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 169 (2024) ISBN:979-8-3315-4137-8.

[347] Hirotoshi Hirai, "Excited-state molecular dynamics simulation based on variational quantum algorithms", Chemical Physics Letters 816, 140404 (2023).

[348] José A. S. Laranjeira, Mateus M. Ferrer, Anderson R. Albuquerque, Carlos A. Paskocimas, Julio R. Sambrano, and Guilherme S. L. Fabris, Engineering Materials 267 (2022) ISBN:978-3-031-07621-3.

[349] Chisomo Daka and Somnath Bhattacharyya, "NISQ in practice: navigating noise, scale, and hardware constraints of near-term devices in quantum machine learning workflows", New Journal of Physics 28 8, 081201 (2026).

[350] Sk Mujaffar Hossain, Seung‐Cheol Lee, and Satadeep Bhattacharjee, "Quantum Simulations of Battery Electrolytes Using Variational Quantum Eigensolver, Equation‐of‐Motion, and Sample‐Based Diagonalization Methods: Active‐Space Design, Dissociation, and Excited States of LiPF6, NaPF6, and FSI Salts", Advanced Quantum Technologies 9 2, e00871 (2026).

[351] 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).

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

[353] Yuki Sato, Ruho Kondo, Satoshi Koide, Hideki Takamatsu, and Nobuyuki Imoto, "Variational quantum algorithm based on the minimum potential energy for solving the Poisson equation", Physical Review A 104 5, 052409 (2021).

[354] Phillip W K Jensen, Lasse Bjørn Kristensen, Jakob S Kottmann, and Alán Aspuru-Guzik, "Quantum computation of eigenvalues within target intervals", Quantum Science and Technology 6 1, 015004 (2021).

[355] Yi Fan, Jie Liu, Xiongzhi Zeng, Zhiqian Xu, Honghui Shang, Zhenyu Li, and Jinlong Yang, "Q²Chemistry: A quantum computation platform for quantum chemistry", JUSTC 52 12, 2 (2022).

[356] 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 Optimizer Methods", IEEE Transactions on Quantum Engineering 7, 3104115 (2026).

[357] Zhikun Han, Chufan Lyu, Yuxuan Zhou, Jiahao Yuan, Ji Chu, Wuerkaixi Nuerbolati, Hao Jia, Lifu Nie, Weiwei Wei, Zusheng Yang, Libo Zhang, Ziyan Zhang, Chang-Kang Hu, Ling Hu, Jian Li, Dian Tan, Abolfazl Bayat, Song Liu, Fei Yan, and Dapeng Yu, "Multilevel variational spectroscopy using a programmable quantum simulator", Physical Review Research 6 1, 013015 (2024).

[358] Yuheng Guo, Feixiang Guo, Bozitao Zhong, Xingyu Chen, Xijun Yuan, Xian‐Min Jin, and Hao Tang, "Quantum‐Enhanced Computing for the Antiferromagnetic J1−J2$J_1-J_2$ Heisenberg Model", Advanced Quantum Technologies 8 7, 2300240 (2025).

[359] Juan Carlos Garcia-Escartin, "Finding eigenvectors with a quantum variational algorithm", Quantum Information Processing 23 7, 254 (2024).

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

[361] O. V. Borzenkova, G. I. Struchalin, A. S. Kardashin, V. V. Krasnikov, N. N. Skryabin, S. S. Straupe, S. P. Kulik, and J. D. Biamonte, "Variational simulation of Schwinger's Hamiltonian with polarization qubits", Applied Physics Letters 118 14, 144002 (2021).

[362] Yun-Hsuan Chen, Jen-Yu Chang, Tsung-Wei Huang, and En-Jui Kuo, 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC) 937 (2026) ISBN:979-8-3315-6110-9.

[363] Frank T. Cerasoli, Kyle Sherbert, Jagoda Sławińska, and Marco Buongiorno Nardelli, "Quantum computation of silicon electronic band structure", Physical Chemistry Chemical Physics 22 38, 21816 (2020).

[364] Hai-Ling Liu, Yu-Sen Wu, Lin-Chun Wan, Shi-Jie Pan, Su-Juan Qin, Fei Gao, and Qiao-Yan Wen, "Variational quantum algorithm for the Poisson equation", Physical Review A 104 2, 022418 (2021).

[365] Yuhan Zheng, Zhijie Sun, Jie Liu, Yi Fan, Zhenyu Li, and Jinlong Yang, "Quantum Equation-of-Motion Method with Single, Double, and Triple Excitations", Journal of Chemical Theory and Computation 20 20, 9032 (2024).

[366] Shashank Kumar Ranu and Daniel D. Stancil, "Single-magnon excited states of a Heisenberg spin chain using a quantum computer", Physical Review B 106 18, 184402 (2022).

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

[368] Alain Delgado, Juan Miguel Arrazola, Soran Jahangiri, Zeyue Niu, Josh Izaac, Chase Roberts, and Nathan Killoran, "Variational quantum algorithm for molecular geometry optimization", Physical Review A 104 5, 052402 (2021).

[369] Chufan Lyu, Victor Montenegro, and Abolfazl Bayat, "Accelerated variational algorithms for digital quantum simulation of many-body ground states", Quantum 4, 324 (2020).

[370] 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).

[371] Matteo Bruschi, Federico Gallina, and Barbara Fresch, "A Quantum Algorithm from Response Theory: Digital Quantum Simulation of Two-Dimensional Electronic Spectroscopy", The Journal of Physical Chemistry Letters 15 5, 1484 (2024).

[372] Kevin Lively, Tim Bode, Jochen Szangolies, Jian-Xin Zhu, and Benedikt Fauseweh, "Noise robust detection of quantum phase transitions", Physical Review Research 6 4, 043254 (2024).

[373] Paul Carpentier, Nathalie Pillet, Denis Lacroix, Noël Dubray, and David Regnier, "Construction of Continuous Collective Energy Landscapes for Large Amplitude Nuclear Many-Body Problems", Physical Review Letters 133 15, 152501 (2024).

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

[375] Phillip W. K. Jensen, Erik Rosendahl Kjellgren, Peter Reinholdt, Karl Michael Ziems, Sonia Coriani, Jacob Kongsted, and Stephan P. A. Sauer, "Quantum Equation of Motion with Orbital Optimization for Computing Molecular Properties in Near-Term Quantum Computing", Journal of Chemical Theory and Computation 20 9, 3613 (2024).

[376] Nick S. Blunt, Joan Camps, Ophelia Crawford, Róbert Izsák, Sebastian Leontica, Arjun Mirani, Alexandra E. Moylett, Sam A. Scivier, Christoph Sünderhauf, Patrick Schopf, Jacob M. Taylor, and Nicole Holzmann, "Perspective on the Current State-of-the-Art of Quantum Computing for Drug Discovery Applications", Journal of Chemical Theory and Computation 18 12, 7001 (2022).

[377] Manas Sajjan, Hadiseh Alaeian, and Sabre Kais, "Magnetic phases of spatially modulated spin-1 chains in Rydberg excitons: Classical and quantum simulations", The Journal of Chemical Physics 157 22, 224111 (2022).

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

[379] Michal Ďuriška, Ivana Miháliková, and Martin Friák, "Quantum computing of the electronic structure of crystals by the Variational Quantum Deflation algorithm", Physica Scripta 100 4, 045105 (2025).

[380] Hayun Park and Hunpyo Lee, "Quantum eigensolver on extension of optimized binary configurations", Physical Review B 110 20, 205142 (2024).

[381] Samir Belaloui, Nacer Eddine Belaloui, and Achour Benslama, "On the simulation of conical intersections in water and methanimine molecules via variational quantum algorithms", International Journal of Modern Physics B 40 20, 2650191 (2026).

[382] Jin-Min Liang, Shu-Qian Shen, Ming Li, and Lei Li, "Variational quantum algorithms for dimensionality reduction and classification", Physical Review A 101 3, 032323 (2020).

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

[384] Erik Lötstedt, Kaoru Yamanouchi, and Yutaka Tachikawa, "Evaluation of vibrational energies and wave functions of CO2 on a quantum computer", AVS Quantum Science 4 3, 036801 (2022).

[385] Pavel P. Popov, Michael Meth, Maciej Lewestein, Philipp Hauke, Martin Ringbauer, Erez Zohar, and Valentin Kasper, "Variational quantum simulation of U(1) lattice gauge theories with qudit systems", Physical Review Research 6 1, 013202 (2024).

[386] Jakob S. Kottmann, Abhinav Anand, and Alán Aspuru-Guzik, "A feasible approach for automatically differentiable unitary coupled-cluster on quantum computers", Chemical Science 12 10, 3497 (2021).

[387] Nobuyuki Yoshioka, Hideaki Hakoshima, Yuichiro Matsuzaki, Yuuki Tokunaga, Yasunari Suzuki, and Suguru Endo, "Generalized Quantum Subspace Expansion", Physical Review Letters 129 2, 020502 (2022).

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

[389] 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).

[390] Rebecca Erbanni, Kishor Bharti, Leong-Chuan Kwek, and Dario Poletti, "NISQ algorithm for the matrix elements of a generic observable", SciPost Physics 15 4, 180 (2023).

[391] Jingwei Wen, Dingshun Lv, Man‐Hong Yung, and Gui‐Lu Long, "Variational quantum packaged deflation for arbitrary excited states", Quantum Engineering 3 4(2021).

[392] Takeshi Yoshikawa, Tomoya Takanashi, and Hiromi Nakai, "Quantum Algorithm of the Divide-and-Conquer Unitary Coupled Cluster Method with a Variational Quantum Eigensolver", Journal of Chemical Theory and Computation 18 9, 5360 (2022).

[393] Cheng Chu, Lei Jiang, and Fan Chen, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 700 (2025) ISBN:979-8-3315-5736-2.

[394] Zhan-Hao Yuan, Tao Yin, and Dan-Bo Zhang, "Hybrid quantum-classical algorithms for solving quantum chemistry in Hamiltonian–wave-function space", Physical Review A 103 1, 012413 (2021).

[395] Chufan Lyu, Xusheng Xu, Man-Hong Yung, and Abolfazl Bayat, "Symmetry enhanced variational quantum spin eigensolver", Quantum 7, 899 (2023).

[396] Ernesto Campos, Aly Nasrallah, and Jacob Biamonte, "Abrupt transitions in variational quantum circuit training", Physical Review A 103 3, 032607 (2021).

[397] Kyle Sherbert, Frank Cerasoli, and Marco Buongiorno Nardelli, "A systematic variational approach to band theory in a quantum computer", RSC Advances 11 62, 39438 (2021).

[398] Tieyu Zhao and Yingying Chi, "Quantum Weighted Fractional-Order Transform", Fractal and Fractional 7 3, 269 (2023).

[399] Sebastian Grieninger, Kazuki Ikeda, and Ismail Zahed, "Quasiparton distributions in massive QED2: Toward quantum computation", Physical Review D 110 7, 076008 (2024).

[400] Smik Patel, Praveen Jayakumar, Tzu-Ching Yen, and Artur F. Izmaylov, "Quantum Measurement for Quantum Chemistry on a Quantum Computer", Chemical Reviews 125 16, 7490 (2025).

[401] Ryan LaRose, Arkin Tikku, Étude O’Neel-Judy, Lukasz Cincio, and Patrick J. Coles, "Variational quantum state diagonalization", npj Quantum Information 5 1, 57 (2019).

[402] Anthony Gandon, Alberto Baiardi, Pauline Ollitrault, and Ivano Tavernelli, "Nonadiabatic Molecular Dynamics with Fermionic Subspace-Expansion Algorithms on Quantum Computers", Journal of Chemical Theory and Computation 20 14, 5951 (2024).

[403] Peter Reinholdt, Erik Rosendahl Kjellgren, Juliane Holst Fuglsbjerg, Karl Michael Ziems, Sonia Coriani, Stephan P. A. Sauer, and Jacob Kongsted, "Subspace Methods for the Simulation of Molecular Response Properties on a Quantum Computer", Journal of Chemical Theory and Computation 20 9, 3729 (2024).

[404] Shu Kanno, Suguru Endo, Yasunari Suzuki, and Yuuki Tokunaga, "Quantum algorithm for the calculation of transition amplitudes in hybrid tensor networks", Physical Review A 104 4, 042424 (2021).

[405] Johannes Selisko, Maximilian Amsler, Thomas Hammerschmidt, Ralf Drautz, and Thomas Eckl, "Extending the variational quantum eigensolver to finite temperatures", Quantum Science and Technology 9 1, 015026 (2024).

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

[407] Long Chen, Xinyu Li, Jianlin Su, Siqi Huang, Zixuan Cai, Zhicai Yu, Yuming Ning, Qiang Xiao, Jianan Zhang, Qian Ma, Zhihao Lan, Jianwei You, and Tie-Jun Cui, "Variational Quantum Algorithm for Photonic Crystals(Invited Paper)", Progress In Electromagnetics Research 184, 1 (2025).

[408] Michael Kreshchuk, Shaoyang Jia, William Kirby, Gary Goldstein, James Vary, and Peter Love, "Light-Front Field Theory on Current Quantum Computers", Entropy 23 5, 597 (2021).

[409] Daming Li, "Variational Methods for Solving High-Dimensional Quantum Systems", Journal of Modern Physics 16 05, 686 (2025).

[410] Gwangsu Yoo, Jaehyun Bae, and Dae Sin Kim, "Symmetry-protecting ansatz for the variational quantum eigensolver of mirror- and occupation-number-inversion-symmetrical systems", Physical Review A 111 3, 032615 (2025).

[411] Andrey Kardashin, Alexey Uvarov, Dmitry Yudin, and Jacob Biamonte, "Certified variational quantum algorithms for eigenstate preparation", Physical Review A 102 5, 052610 (2020).

[412] Takumi Kobori, Taichi Kosugi, Hirofumi Nishi, Synge Todo, and Yu-ichiro Matsushita, "Accelerated spin-adapted ground-state preparation with nonvariational quantum algorithms", Physical Review A 113 6, 062454 (2026).

[413] Nobuyuki Yoshioka, Takeshi Sato, Yuya O. Nakagawa, Yu-ya Ohnishi, and Wataru Mizukami, "Variational quantum simulation for periodic materials", Physical Review Research 4 1, 013052 (2022).

[414] Ayush Asthana, Ashutosh Kumar, Vibin Abraham, Harper Grimsley, Yu Zhang, Lukasz Cincio, Sergei Tretiak, Pavel A. Dub, Sophia E. Economou, Edwin Barnes, and Nicholas J. Mayhall, "Quantum self-consistent equation-of-motion method for computing molecular excitation energies, ionization potentials, and electron affinities on a quantum computer", Chemical Science 14 9, 2405 (2023).

[415] Carlos Bravo-Prieto, Josep Lumbreras-Zarapico, Luca Tagliacozzo, and José I. Latorre, "Scaling of variational quantum circuit depth for condensed matter systems", Quantum 4, 272 (2020).

[416] Qing-Xing Xie, Sheng Liu, and Yan Zhao, "Orthogonal State Reduction Variational Eigensolver for the Excited-State Calculations on Quantum Computers", Journal of Chemical Theory and Computation 18 6, 3737 (2022).

[417] Mario Motta and Julia E. Rice, "Emerging quantum computing algorithms for quantum chemistry", WIREs Computational Molecular Science 12 3, e1580 (2022).

[418] Keisuke Fujii, Kaoru Mizuta, Hiroshi Ueda, Kosuke Mitarai, Wataru Mizukami, and Yuya O. Nakagawa, "Deep Variational Quantum Eigensolver: A Divide-And-Conquer Method for Solving a Larger Problem with Smaller Size Quantum Computers", PRX Quantum 3 1, 010346 (2022).

[419] Yann Beaujeault-Taudière and Denis Lacroix, "Solving the Lipkin model using quantum computers with two qubits only with a hybrid quantum-classical technique based on the generator coordinate method", Physical Review C 109 2, 024327 (2024).

[420] Tian-Yin Li, Hong-Xi Xing, and Dan-Bo Zhang, "Quantum computing based high-energy nuclear physics", Acta Physica Sinica 72 20, 200303 (2023).

[421] Ning Yang and Jing Guo, "A Quantum-Computing-Based Method for Solving Quantum Confinement Problem in Semiconductor", IEEE Transactions on Electron Devices 70 3, 1366 (2023).

[422] Da Bean Han, Kang-Min Hu, Hyang-Tag Lim, and Hyun Woo Kim, "Exploring Hückel Molecular Orbital Energies through Variational and Phase Estimation Quantum Algorithms", The Journal of Physical Chemistry Letters 17 8, 2205 (2026).

[423] 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).

[424] Robert M. Parrish, Edward G. Hohenstein, Peter L. McMahon, and Todd J. Martínez, "Quantum Computation of Electronic Transitions Using a Variational Quantum Eigensolver", Physical Review Letters 122 23, 230401 (2019).

[425] Joonho Lee, William J. Huggins, Martin Head-Gordon, and K. Birgitta Whaley, "Generalized Unitary Coupled Cluster Wave functions for Quantum Computation", Journal of Chemical Theory and Computation 15 1, 311 (2019).

[426] Robert M. Parrish and Peter L. McMahon, "Quantum Filter Diagonalization: Quantum Eigendecomposition without Full Quantum Phase Estimation", arXiv:1909.08925, (2019).

[427] Sam McArdle, Suguru Endo, Alan Aspuru-Guzik, Simon Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402, (2018).

[428] Ilya G. Ryabinkin, Scott N. Genin, and Artur F. Izmaylov, "Constrained Variational Quantum Eigensolver: Quantum Computer Search Engine in the Fock Space", Journal of Chemical Theory and Computation 15 1, 249 (2019).

[429] Robert M. Parrish, Joseph T. Iosue, Asier Ozaeta, and Peter L. McMahon, "A Jacobi Diagonalization and Anderson Acceleration Algorithm For Variational Quantum Algorithm Parameter Optimization", arXiv:1904.03206, (2019).

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

[431] Robert M. Parrish, Edward G. Hohenstein, Peter L. McMahon, and Todd J. Martinez, "Hybrid Quantum/Classical Derivative Theory: Analytical Gradients and Excited-State Dynamics for the Multistate Contracted Variational Quantum Eigensolver", arXiv:1906.08728, (2019).

[432] Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, and Alán Aspuru-Guzik, "Quantum Chemistry in the Age of Quantum Computing", arXiv:1812.09976, (2018).

[433] Joonho Lee, William J. Huggins, Martin Head-Gordon, and K. Birgitta Whaley, "Generalized Unitary Coupled Cluster Wavefunctions for Quantum Computation", arXiv:1810.02327, (2018).

[434] Suguru Endo, Tyson Jones, Sam McArdle, Xiao Yuan, and Simon Benjamin, "Variational quantum algorithms for discovering Hamiltonian spectra", arXiv:1806.05707, (2018).

[435] Hikaru Wakaura and Takao Tomono, "Genetic-Multi-initial Generalized VQE: Advanced VQE method using Genetic Algorithms then Local Search", arXiv:2109.02009, (2021).

[436] Hikaru Wakaura and Takao Tomono, "Effect of constraint and Tabu Search term on Variational Quantum Eigensolver and Subspace-Search Variational Quantum Eigensolver", arXiv:2103.12574, (2021).

[437] Rafael I. Nepomechie, "Bethe ansatz on a quantum computer?", arXiv:2010.01609, (2020).

[438] Hikaru Wakaura and Andriyan Bayu Suksmono, "Tangent Vector Variational Quantum Eigensolver: A Robust Variational Quantum Eigensolver against the inaccuracy of derivative", arXiv:2105.01141, (2021).

[439] Taichi Kosugi and Yu-ichiro Matsushita, "Charge and spin response functions on a quantum computer: applications to molecules", arXiv:1911.00293, (2019).

[440] Hikaru Wakaura and Takao Tomono, "Frame Superposition Cluster: The method to derive the transition matrices in high accuracy", arXiv:2107.02979, (2021).

[441] Kyle Sherbert and Marco Buongiorno Nardelli, "Orthogonal-ansatz VQE: Locating excited states without modifying a cost-function", arXiv:2204.04361, (2022).

[442] Alexander Gresch, Lennart Bittel, and Martin Kliesch, "Scalable approach to many-body localization via quantum data", arXiv:2202.08853, (2022).

[443] Scott E. Smart, Davis M. Welakuh, and Prineha Narang, "Many-Body Excited States with a Contracted Quantum Eigensolver", arXiv:2305.09653, (2023).

[444] Guanglei Xu, Yi-Bin Guo, Xuan Li, Zong-Sheng Zhou, Hai-Jun Liao, and T. Xiang, "Variational determination of arbitrarily many eigenpairs in one quantum circuit", arXiv:2206.11036, (2022).

[445] Tilas Kabengele, Yash M. Lokare, J. B. Marston, and Brenda M. Rubenstein, "Modeling Stochastic Chemical Kinetics on Quantum Computers", arXiv:2404.08770, (2024).

[446] Julien Gacon, "Scalable Quantum Algorithms for Noisy Quantum Computers", arXiv:2403.00940, (2024).

[447] I-Chi Chen, Nouhaila Innan, Suman Kumar Roy, and Jason Saroni, "Crossing The Gap Using Variational Quantum Eigensolver: A Comparative Study", arXiv:2405.11687, (2024).

[448] Sutapa Samanta, Jian-Xin Zhu, and Armin Rahmani, "Realizing supersymmetry in a digitized quantum device", arXiv:2504.18703, (2025).

[449] David Schaich and Christopher Culver, "Exploring lattice supersymmetry with variational quantum deflation", arXiv:2410.11514, (2024).

[450] Christian Soize, Loïc Joubert-Doriol, and Artur F. Izmaylov, "Quantum computer formulation of the FKP-operator eigenvalue problem for probabilistic learning on manifolds", arXiv:2502.14580, (2025).

[451] David Quiroga, Jason Han, and Anastasios Kyrillidis, "Quantum EigenGame for excited state calculation", arXiv:2503.13644, (2025).

[452] John Kerfoot, Emanuele Mendicelli, and David Schaich, "Quantum Variational Methods for Supersymmetric Quantum Mechanics", arXiv:2510.26506, (2025).

[453] Guolong Zhong, Yi Fan, and Zhenyu Li, "Scalable parallel simulation of quantum circuits on CPU and GPU systems", arXiv:2509.04955, (2025).

[454] Martin Plesch, Martin Friák, and Ijaz Ahamed Mohammad, "Efficient implementation of single particle Hamiltonians in exponentially reduced qubit space", arXiv:2601.00247, (2026).

[455] Sandip Maiti, "Frustration-Induced Expressibility Limitations in Variational Quantum Algorithms", arXiv:2604.11688, (2026).

[456] Stefano Scali, Brian Coyle, Giuseppe Buonaiuto, and Michal Krompiec, "Quantum Randomized Subspace Iteration", arXiv:2604.09483, (2026).

[457] Xian Lu, Xinying Li, Fei Wang, Shuai Hou, Chengkang Pan, Xin Yi, and Yongmei Li, "Aicir: A Full-Stack Quantum Circuit Simulator with AscendNPU Support", arXiv:2608.09733, (2026).

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

[459] Sumit Tapas Chongder, "QR-SPPS: Quantum-Native Retail Shock Propagation and Policy Stress Simulator", arXiv:2607.16275, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-18 00:58:04) and SAO/NASA ADS (last updated successfully 2026-08-17 12:17:53). The list may be incomplete as not all publishers provide suitable and complete citation data.

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