Theory of variational quantum simulation

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

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

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

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

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

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

[18] 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).

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[20] 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).

[21] Petar Marević, David Regnier, and Denis Lacroix, "Multiconfigurational time-dependent density functional theory for atomic nuclei: technical and numerical aspects", The European Physical Journal A 60 1, 10 (2024).

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

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

[24] 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).

[25] Kentaro Heya, Ken M. Nakanishi, Kosuke Mitarai, Zhiguang Yan, Kun Zuo, Yasunari Suzuki, Takanori Sugiyama, Shuhei Tamate, Yutaka Tabuchi, Keisuke Fujii, and Yasunobu Nakamura, "Subspace variational quantum simulator", Physical Review Research 5 2, 023078 (2023).

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

[27] Kouhei Nakaji, Suguru Endo, Yuichiro Matsuzaki, and Hideaki Hakoshima, "Measurement optimization of variational quantum simulation by classical shadow and derandomization", Quantum 7, 995 (2023).

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

[29] 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).

[30] Samuel Stein, Nathan Wiebe, Yufei Ding, Peng Bo, Karol Kowalski, Nathan Baker, James Ang, and Ang Li, Proceedings of the 49th Annual International Symposium on Computer Architecture 59 (2022) ISBN:9781450386104.

[31] 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).

[32] David Amaro, Matthias Rosenkranz, Nathan Fitzpatrick, Koji Hirano, and Mattia Fiorentini, "A case study of variational quantum algorithms for a job shop scheduling problem", EPJ Quantum Technology 9 1, 5 (2022).

[33] Filippo Vicentini, Damian Hofmann, Attila Szabó, Dian Wu, Christopher Roth, Clemens Giuliani, Gabriel Pescia, Jannes Nys, Vladimir Vargas-Calderón, Nikita Astrakhantsev, and Giuseppe Carleo, "NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems", SciPost Physics Codebases 7 (2022).

[34] Xiaosi Xu, Simon C. Benjamin, and Xiao Yuan, "Variational Circuit Compiler for Quantum Error Correction", Physical Review Applied 15 3, 034068 (2021).

[35] Patrick Selig, Niall Murphy, Ashwin Sundareswaran R, David Redmond, and Simon Caton, 2021 International Conference on Rebooting Computing (ICRC) 24 (2021) ISBN:978-1-6654-2332-8.

[36] Takanori Nishi, Erik Lötstedt, and Kaoru Yamanouchi, "Simulation of a laser-driven three-level system by a noisy quantum computer", AVS Quantum Science 4 4, 043801 (2022).

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

[38] Lucas Slattery, Benjamin Villalonga, and Bryan K. Clark, "Unitary block optimization for variational quantum algorithms", Physical Review Research 4 2, 023072 (2022).

[39] Marco Pistoia, Syed Farhan Ahmad, Akshay Ajagekar, Alexander Buts, Shouvanik Chakrabarti, Dylan Herman, Shaohan Hu, Andrew Jena, Pierre Minssen, Pradeep Niroula, Arthur Rattew, Yue Sun, and Romina Yalovetzky, 2021 IEEE/ACM International Conference On Computer Aided Design (ICCAD) 1 (2021) ISBN:978-1-6654-4507-8.

[40] Manikandan Kondappan, Manish Chaudhary, Ebubechukwu O. Ilo-Okeke, Valentin Ivannikov, and Tim Byrnes, "Imaginary-time evolution with quantum nondemolition measurements: Multiqubit interactions via measurement nonlinearities", Physical Review A 107 4, 042616 (2023).

[41] Nathaniel Wrobel, Anshumitra Baul, Ka-Ming Tam, and Juana Moreno, "Detecting Quantum Critical Points of Correlated Systems by Quantum Convolutional Neural Network Using Data from Variational Quantum Eigensolver", Quantum Reports 4 4, 574 (2022).

[42] Xiao Yuan, Jinzhao Sun, Junyu Liu, Qi Zhao, and You Zhou, "Quantum Simulation with Hybrid Tensor Networks", Physical Review Letters 127 4, 040501 (2021).

[43] Alexander Miessen, Pauline J. Ollitrault, and Ivano Tavernelli, "Quantum algorithms for quantum dynamics: A performance study on the spin-boson model", Physical Review Research 3 4, 043212 (2021).

[44] Zi-Jian Zhang, Jinzhao Sun, Xiao Yuan, and Man-Hong Yung, "Low-Depth Hamiltonian Simulation by an Adaptive Product Formula", Physical Review Letters 130 4, 040601 (2023).

[45] 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).

[46] Tomasz Szołdra, Piotr Sierant, Maciej Lewenstein, and Jakub Zakrzewski, "Unsupervised detection of decoupled subspaces: Many-body scars and beyond", Physical Review B 105 22, 224205 (2022).

[47] Pablo Rivas and Liang Zhao, 2022 International Conference on Computational Science and Computational Intelligence (CSCI) 85 (2022) ISBN:979-8-3503-2028-2.

[48] Thi Ha Kyaw, Micheline B Soley, Brandon Allen, Paul Bergold, Chong Sun, Victor S Batista, and Alán Aspuru-Guzik, "Boosting quantum amplitude exponentially in variational quantum algorithms", Quantum Science and Technology 9 1, 01LT01 (2024).

[49] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles, "An Adaptive Optimizer for Measurement-Frugal Variational Algorithms", Quantum 4, 263 (2020).

[50] Jacob L. Beckey, M. Cerezo, Akira Sone, and Patrick J. Coles, "Variational quantum algorithm for estimating the quantum Fisher information", Physical Review Research 4 1, 013083 (2022).

[51] Sebastian Brandhofer, Simon Devitt, and Ilia Polian, 2021 IEEE/ACM International Symposium on Nanoscale Architectures (NANOARCH) 1 (2021) ISBN:978-1-6654-0959-9.

[52] Bo Peng and Karol Kowalski, "Variational quantum solver employing the PDS energy functional", Quantum 5, 473 (2021).

[53] 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).

[54] Yuxuan Du, Yibo Yang, Dacheng Tao, and Min-Hsiu Hsieh, "Problem-Dependent Power of Quantum Neural Networks on Multiclass Classification", Physical Review Letters 131 14, 140601 (2023).

[55] 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).

[56] Almudena Carrera Vazquez, Daniel J. Egger, David Ochsner, and Stefan Woerner, "Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation", Quantum 7, 1067 (2023).

[57] Julien Gacon, Jannes Nys, Riccardo Rossi, Stefan Woerner, and Giuseppe Carleo, "Variational quantum time evolution without the quantum geometric tensor", Physical Review Research 6 1, 013143 (2024).

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

[59] Taichi Kosugi, Yusuke Nishiya, Hirofumi Nishi, and Yu-ichiro Matsushita, "Imaginary-time evolution using forward and backward real-time evolution with a single ancilla: First-quantized eigensolver algorithm for quantum chemistry", Physical Review Research 4 3, 033121 (2022).

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

[61] Chenfeng Cao and Xin Wang, "Noise-Assisted Quantum Autoencoder", Physical Review Applied 15 5, 054012 (2021).

[62] Christa Zoufal, David Sutter, and Stefan Woerner, "Error bounds for variational quantum time evolution", Physical Review Applied 20 4, 044059 (2023).

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

[64] Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, and Lukasz Cincio, "Error mitigation with Clifford quantum-circuit data", Quantum 5, 592 (2021).

[65] Alexander M. Czajka, Zhong-Bo Kang, Henry Ma, and Fanyi Zhao, "Quantum simulation of chiral phase transitions", Journal of High Energy Physics 2022 8, 209 (2022).

[66] Jin-Min Liang, Qiao-Qiao Lv, Zhi-Xi Wang, and Shao-Ming Fei, "Assisted quantum simulation of open quantum systems", iScience 26 4, 106306 (2023).

[67] João C. Getelina, Cai-Zhuang Wang, Thomas Iadecola, Yong-Xin Yao, and Peter P. Orth, "Adaptive variational ground state preparation for spin-1 models on qubit-based architectures", Physical Review B 109 8, 085128 (2024).

[68] Markus Hauru, Maarten Van Damme, and Jutho Haegeman, "Riemannian optimization of isometric tensor networks", SciPost Physics 10 2, 040 (2021).

[69] Huo Chen, Niladri Gomes, Siyuan Niu, and Wibe Albert de Jong, "Adaptive variational simulation for open quantum systems", Quantum 8, 1252 (2024).

[70] Richard Meister, Cica Gustiani, and Simon C Benjamin, "Exploring ab initio machine synthesis of quantum circuits", New Journal of Physics 25 7, 073018 (2023).

[71] 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).

[72] Hedayat Alghassi, Amol Deshmukh, Noelle Ibrahim, Nicolas Robles, Stefan Woerner, and Christa Zoufal, "A variational quantum algorithm for the Feynman-Kac formula", Quantum 6, 730 (2022).

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

[74] 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).

[75] Conor Mc Keever and Michael Lubasch, "Classically optimized Hamiltonian simulation", Physical Review Research 5 2, 023146 (2023).

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

[77] Dylan Herman, Cody Googin, Xiaoyuan Liu, Yue Sun, Alexey Galda, Ilya Safro, Marco Pistoia, and Yuri Alexeev, "Quantum computing for finance", Nature Reviews Physics 5 8, 450 (2023).

[78] 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).

[79] Yuchen Guo and Shuo Yang, "Quantum Error Mitigation via Matrix Product Operators", PRX Quantum 3 4, 040313 (2022).

[80] Rishabh Gupta, Manas Sajjan, Raphael D. Levine, and Sabre Kais, "Variational approach to quantum state tomography based on maximal entropy formalism", Physical Chemistry Chemical Physics 24 47, 28870 (2022).

[81] Hirofumi Nishi, Koki Hamada, Yusuke Nishiya, Taichi Kosugi, and Yu-ichiro Matsushita, "Optimal scheduling in probabilistic imaginary-time evolution on a quantum computer", Physical Review Research 5 4, 043048 (2023).

[82] Hao Luo, Qianli Zhou, Zhen Li, and Yong Deng, "Variational Quantum Linear Solver-based Combination Rules in Dempster–Shafer Theory", Information Fusion 102, 102070 (2024).

[83] 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).

[84] João C. Getelina, Niladri Gomes, Thomas Iadecola, Peter P. Orth, and Yong-Xin Yao, "Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations", SciPost Physics 15 3, 102 (2023).

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

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

[87] Zhenhuan Liu, Pei Zeng, You Zhou, and Mile Gu, "Characterizing correlation within multipartite quantum systems via local randomized measurements", Physical Review A 105 2, 022407 (2022).

[88] Francesco Libbi, Jacopo Rizzo, Francesco Tacchino, Nicola Marzari, and Ivano Tavernelli, "Effective calculation of the Green's function in the time domain on near-term quantum processors", Physical Review Research 4 4, 043038 (2022).

[89] Weitang Li, Jiajun Ren, Sainan Huai, Tianqi Cai, Zhigang Shuai, and Shengyu Zhang, "Efficient quantum simulation of electron-phonon systems by variational basis state encoder", Physical Review Research 5 2, 023046 (2023).

[90] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo, "Quantum Natural Gradient", Quantum 4, 269 (2020).

[91] D. Zeuch and N. E. Bonesteel, "Efficient two-qubit pulse sequences beyond CNOT", Physical Review B 102 7, 075311 (2020).

[92] Francisco Escudero, David Fernández-Fernández, Gabriel Jaumà, Guillermo F. Peñas, and Luciano Pereira, "Hardware-Efficient Entangled Measurements for Variational Quantum Algorithms", Physical Review Applied 20 3, 034044 (2023).

[93] Anton Nykänen, Aaron Miller, Walter Talarico, Stefan Knecht, Arseny Kovyrshin, Mårten Skogh, Lars Tornberg, Anders Broo, Stefano Mensa, Benjamin C. B. Symons, Emre Sahin, Jason Crain, Ivano Tavernelli, and Fabijan Pavošević, "Toward Accurate Post-Born–Oppenheimer Molecular Simulations on Quantum Computers: An Adaptive Variational Eigensolver with Nuclear-Electronic Frozen Natural Orbitals", Journal of Chemical Theory and Computation 19 24, 9269 (2023).

[94] Mårten Skogh, Oskar Leinonen, Phalgun Lolur, and Martin Rahm, "Accelerating variational quantum eigensolver convergence using parameter transfer", Electronic Structure 5 3, 035002 (2023).

[95] Jonathan Wei Zhong Lau, Tobias Haug, Leong Chuan Kwek, and Kishor Bharti, "NISQ Algorithm for Hamiltonian simulation via truncated Taylor series", SciPost Physics 12 4, 122 (2022).

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

[97] Sebastian Leontica and David Amaro, "Exploring the neighborhood of 1-layer QAOA with instantaneous quantum polynomial circuits", Physical Review Research 6 1, 013071 (2024).

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

[99] G. Paradezhenko, A. Pervishko, and D. Yudin, "Tensor Train Optimization of Parameterized Quantum Circuits", JETP Letters 118 12, 938 (2023).

[100] Tasneem M Watad and Netanel H Lindner, "Variational quantum algorithms for simulation of Lindblad dynamics", Quantum Science and Technology 9 2, 025015 (2024).

[101] Leonardo Banchi and Gavin E. Crooks, "Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule", Quantum 5, 386 (2021).

[102] I-Chi Chen, Benjamin Burdick, Yongxin Yao, Peter P. Orth, and Thomas Iadecola, "Error-mitigated simulation of quantum many-body scars on quantum computers with pulse-level control", Physical Review Research 4 4, 043027 (2022).

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

[105] 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).

[106] Lento Nagano, Aniruddha Bapat, and Christian W. Bauer, "Quench dynamics of the Schwinger model via variational quantum algorithms", Physical Review D 108 3, 034501 (2023).

[107] Enrique Cervero Martín, Kirill Plekhanov, and Michael Lubasch, "Barren plateaus in quantum tensor network optimization", Quantum 7, 974 (2023).

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

[109] Chee-Kong Lee, Chang-Yu Hsieh, Shengyu Zhang, and Liang Shi, "Simulation of Condensed-Phase Spectroscopy with Near-Term Digital Quantum Computers", Journal of Chemical Theory and Computation 17 11, 7178 (2021).

[110] M. Mahdian and H. Davoodi Yeganeh, "Toward a quantum computing algorithm to quantify classical and quantum correlation of system states", Quantum Information Processing 20 12, 393 (2021).

[111] Kishor Bharti and Tobias Haug, "Quantum-assisted simulator", Physical Review A 104 4, 042418 (2021).

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

[113] 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).

[114] Alessandro Sinibaldi, Clemens Giuliani, Giuseppe Carleo, and Filippo Vicentini, "Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution", Quantum 7, 1131 (2023).

[115] Gaurav Gyawali, Mabrur Ahmed, Eric W. Aspling, Luke Ellert-Beck, and Michael J. Lawler, "Revealing microcanonical phases and phase transitions of strongly correlated systems via time-averaged classical shadows", Physical Review B 108 23, 235141 (2023).

[116] Shi-Ning Sun, Mario Motta, Ruslan N. Tazhigulov, Adrian T.K. Tan, Garnet Kin-Lic Chan, and Austin J. Minnich, "Quantum Computation of Finite-Temperature Static and Dynamical Properties of Spin Systems Using Quantum Imaginary Time Evolution", PRX Quantum 2 1, 010317 (2021).

[117] Stefano Barison, Filippo Vicentini, Ignacio Cirac, and Giuseppe Carleo, "Variational dynamics as a ground-state problem on a quantum computer", Physical Review Research 4 4, 043161 (2022).

[118] James Stokes, Brian Chen, and Shravan Veerapaneni, "Numerical and geometrical aspects of flow-based variational quantum Monte Carlo", Machine Learning: Science and Technology 4 2, 021001 (2023).

[119] Erik Lötstedt, Lidong Wang, Ryuhei Yoshida, Youyuan Zhang, and Kaoru Yamanouchi, "Error-mitigated quantum computing of Heisenberg spin chain dynamics", Physica Scripta 98 3, 035111 (2023).

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

[121] Christa Zoufal, Aurélien Lucchi, and Stefan Woerner, "Variational quantum Boltzmann machines", Quantum Machine Intelligence 3 1, 7 (2021).

[122] Yuri Alexeev, Dave Bacon, Kenneth R. Brown, Robert Calderbank, Lincoln D. Carr, Frederic T. Chong, Brian DeMarco, Dirk Englund, Edward Farhi, Bill Fefferman, Alexey V. Gorshkov, Andrew Houck, Jungsang Kim, Shelby Kimmel, Michael Lange, Seth Lloyd, Mikhail D. Lukin, Dmitri Maslov, Peter Maunz, Christopher Monroe, John Preskill, Martin Roetteler, Martin J. Savage, and Jeff Thompson, "Quantum Computer Systems for Scientific Discovery", PRX Quantum 2 1, 017001 (2021).

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

[124] Katherine Klymko, Carlos Mejuto-Zaera, Stephen J. Cotton, Filip Wudarski, Miroslav Urbanek, Diptarka Hait, Martin Head-Gordon, K. Birgitta Whaley, Jonathan Moussa, Nathan Wiebe, Wibe A. de Jong, and Norm M. Tubman, "Real-Time Evolution for Ultracompact Hamiltonian Eigenstates on Quantum Hardware", PRX Quantum 3 2, 020323 (2022).

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

[126] 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).

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

[128] Matija Medvidović and Dries Sels, "Variational Quantum Dynamics of Two-Dimensional Rotor Models", PRX Quantum 4 4, 040302 (2023).

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

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

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

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

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

[134] Ming-Cheng Chen, Ming Gong, Xiaosi Xu, Xiao Yuan, Jian-Wen Wang, Can Wang, Chong Ying, Jin Lin, Yu Xu, Yulin Wu, Shiyu Wang, Hui Deng, Futian Liang, Cheng-Zhi Peng, Simon C. Benjamin, Xiaobo Zhu, Chao-Yang Lu, and Jian-Wei Pan, "Demonstration of Adiabatic Variational Quantum Computing with a Superconducting Quantum Coprocessor", Physical Review Letters 125 18, 180501 (2020).

[135] Weitang Li, Zigeng Huang, Changsu Cao, Yifei Huang, Zhigang Shuai, Xiaoming Sun, Jinzhao Sun, Xiao Yuan, and Dingshun Lv, "Toward practical quantum embedding simulation of realistic chemical systems on near-term quantum computers", Chemical Science 13 31, 8953 (2022).

[136] Kunal Sharma, Sumeet Khatri, M Cerezo, and Patrick J Coles, "Noise resilience of variational quantum compiling", New Journal of Physics 22 4, 043006 (2020).

[137] Tyler Volkoff and Patrick J Coles, "Large gradients via correlation in random parameterized quantum circuits", Quantum Science and Technology 6 2, 025008 (2021).

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

[139] 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).

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

[141] Manuel G. Algaba, Mario Ponce-Martinez, Carlos Munuera-Javaloy, Vicente Pina-Canelles, Manish J. Thapa, Bruno G. Taketani, Martin Leib, Inés de Vega, Jorge Casanova, and Hermanni Heimonen, "Co-Design quantum simulation of nanoscale NMR", Physical Review Research 4 4, 043089 (2022).

[142] Benjamin A. Cordier, Nicolas P. D. Sawaya, Gian Giacomo Guerreschi, and Shannon K. McWeeney, "Biology and medicine in the landscape of quantum advantages", Journal of The Royal Society Interface 19 196, 20220541 (2022).

[143] 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).

[144] Weitang Li, Jonathan Allcock, Lixue Cheng, Shi-Xin Zhang, Yu-Qin Chen, Jonathan P. Mailoa, Zhigang Shuai, and Shengyu Zhang, "TenCirChem: An Efficient Quantum Computational Chemistry Package for the NISQ Era", Journal of Chemical Theory and Computation 19 13, 3966 (2023).

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

[146] Keisuke Matsumoto, Yuta Shingu, Suguru Endo, Shiro Kawabata, Shohei Watabe, Tetsuro Nikuni, Hideaki Hakoshima, and Yuichiro Matsuzaki, "Calculation of Gibbs partition function with imaginary time evolution on near-term quantum computers", Japanese Journal of Applied Physics 61 4, 042002 (2022).

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

[148] R. Sagastizabal, S. P. Premaratne, B. A. Klaver, M. A. Rol, V. Negîrneac, M. S. Moreira, X. Zou, S. Johri, N. Muthusubramanian, M. Beekman, C. Zachariadis, V. P. Ostroukh, N. Haider, A. Bruno, A. Y. Matsuura, and L. DiCarlo, "Variational preparation of finite-temperature states on a quantum computer", npj Quantum Information 7 1, 130 (2021).

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

[150] Hideyuki Miyahara and Vwani Roychowdhury, "Ansatz-Independent Variational Quantum Classifiers and the Price of Ansatz", Scientific Reports 12 1, 19520 (2022).

[151] 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).

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

[153] Yang Zhao, "The hierarchy of Davydov’s Ansätze: From guesswork to numerically “exact” many-body wave functions", The Journal of Chemical Physics 158 8, 080901 (2023).

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

[155] Yuping Mao, Manish Chaudhary, Manikandan Kondappan, Junheng Shi, Ebubechukwu O. Ilo-Okeke, Valentin Ivannikov, and Tim Byrnes, "Measurement-Based Deterministic Imaginary Time Evolution", Physical Review Letters 131 11, 110602 (2023).

[156] Yongcheng Ding, Yue Ban, and Xi Chen, "Towards Quantum Control with Advanced Quantum Computing: A Perspective", Entropy 24 12, 1743 (2022).

[157] Jason Saroni, Henry Lamm, Peter P. Orth, and Thomas Iadecola, "Reconstructing thermal quantum quench dynamics from pure states", Physical Review B 108 13, 134301 (2023).

[158] Jingwei Wen, Chao Zheng, Zhiguo Huang, and Ling Qian, "Iteration-free digital quantum simulation of imaginary-time evolution based on the approximate unitary expansion", Europhysics Letters 141 6, 68001 (2023).

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

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

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

[162] 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).

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

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

[165] Shichuan Xue, Yizhi Wang, Junwei Zhan, Yaxuan Wang, Ru Zeng, Jiangfang Ding, Weixu Shi, Yong Liu, Yingwen Liu, Anqi Huang, Guangyao Huang, Chunlin Yu, Dongyang Wang, Xiang Fu, Xiaogang Qiang, Ping Xu, Mingtang Deng, Xuejun Yang, and Junjie Wu, "Variational Entanglement-Assisted Quantum Process Tomography with Arbitrary Ancillary Qubits", Physical Review Letters 129 13, 133601 (2022).

[166] 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).

[167] Brian Doolittle, R. Thomas Bromley, Nathan Killoran, and Eric Chitambar, "Variational Quantum Optimization of Nonlocality in Noisy Quantum Networks", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[168] 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).

[169] 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).

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[173] Yuan Yao, Pierre Cussenot, Richard A. Wolf, and Filippo Miatto, "Complex natural gradient optimization for optical quantum circuit design", Physical Review A 105 5, 052402 (2022).

[174] José D. Guimarães, Mikhail I. Vasilevskiy, and Luís S. Barbosa, "Digital quantum simulation of non-perturbative dynamics of open systems with orthogonal polynomials", Quantum 8, 1242 (2024).

[175] Trevor Keen, Thomas Maier, Steven Johnston, and Pavel Lougovski, "Quantum-classical simulation of two-site dynamical mean-field theory on noisy quantum hardware", Quantum Science and Technology 5 3, 035001 (2020).

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

[177] 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).

[178] Kaoru Mizuta, Yuya O. Nakagawa, Kosuke Mitarai, and Keisuke Fujii, "Local Variational Quantum Compilation of Large-Scale Hamiltonian Dynamics", PRX Quantum 3 4, 040302 (2022).

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

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

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

[182] 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).

[183] 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).

[184] Tobias Haug and M. S. Kim, "Natural parametrized quantum circuit", Physical Review A 106 5, 052611 (2022).

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

[186] Sergey Bravyi, Oliver Dial, Jay M. Gambetta, Darío Gil, and Zaira Nazario, "The future of quantum computing with superconducting qubits", Journal of Applied Physics 132 16, 160902 (2022).

[187] Kenji Kubo, Koichi Miyamoto, Kosuke Mitarai, and Keisuke Fujii, "Pricing Multiasset Derivatives by Variational Quantum Algorithms", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[188] Kunal Sharma, M. Cerezo, Zoë Holmes, Lukasz Cincio, Andrew Sornborger, and Patrick J. Coles, "Reformulation of the No-Free-Lunch Theorem for Entangled Datasets", Physical Review Letters 128 7, 070501 (2022).

[189] Kishor Bharti, "Fisher Information: A Crucial Tool for NISQ Research", Quantum Views 5, 61 (2021).

[190] Kian Hwee Lim, Tobias Haug, Leong Chuan Kwek, and Kishor Bharti, "Fast-forwarding with NISQ processors without feedback loop", Quantum Science and Technology 7 1, 015001 (2022).

[191] 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).

[192] Kunal Sharma, M. Cerezo, Lukasz Cincio, and Patrick J. Coles, "Trainability of Dissipative Perceptron-Based Quantum Neural Networks", Physical Review Letters 128 18, 180505 (2022).

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

[194] Igor O. Sokolov, Werner Dobrautz, Hongjun Luo, Ali Alavi, and Ivano Tavernelli, "Orders of magnitude increased accuracy for quantum many-body problems on quantum computers via an exact transcorrelated method", Physical Review Research 5 2, 023174 (2023).

[195] Kaelan Donatella, Zakari Denis, Alexandre Le Boité, and Cristiano Ciuti, "Dynamics with autoregressive neural quantum states: Application to critical quench dynamics", Physical Review A 108 2, 022210 (2023).

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

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

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

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

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

[202] 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).

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

[204] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 129 (2023) ISBN:979-8-3503-4323-6.

[205] Nannan Ma, Wenhao Chu, P. Z. Zhao, and Jiangbin Gong, "Adiabatic quantum learning", Physical Review A 108 4, 042420 (2023).

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

[207] Laura Gentini, Alessandro Cuccoli, Stefano Pirandola, Paola Verrucchi, and Leonardo Banchi, "Noise-resilient variational hybrid quantum-classical optimization", Physical Review A 102 5, 052414 (2020).

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

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

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

[211] Kenji Kubo, Yuya O. Nakagawa, Suguru Endo, and Shota Nagayama, "Variational quantum simulations of stochastic differential equations", Physical Review A 103 5, 052425 (2021).

[212] 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).

[213] Utkarsh Azad and Helena Zhang, 2022 IEEE/ACM 7th Symposium on Edge Computing (SEC) 362 (2022) ISBN:978-1-6654-8611-8.

[214] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, "Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information", Quantum 5, 567 (2021).

[215] Xiaoyang Wang, Xu Feng, Tobias Hartung, Karl Jansen, and Paolo Stornati, "Critical behavior of the Ising model by preparing the thermal state on a quantum computer", Physical Review A 108 2, 022612 (2023).

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

[217] Tyson Jones and Simon C. Benjamin, "Robust quantum compilation and circuit optimisation via energy minimisation", Quantum 6, 628 (2022).

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

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

[220] Laszlo Gyongyosi, "Approximation Method for Optimization Problems in Gate-Model Quantum Computers", Chaos, Solitons & Fractals: X 7, 100066 (2021).

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

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

[223] Jie Zhu, Yuya O Nakagawa, Yong-Sheng Zhang, Chuan-Feng Li, and Guang-Can Guo, "Calculating the Green’s function of two-site fermionic Hubbard model in a photonic system", New Journal of Physics 24 4, 043030 (2022).

[224] 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).

[225] Zhong-Xia Shang, Ming-Cheng Chen, Xiao Yuan, Chao-Yang Lu, and Jian-Wei Pan, "Schrödinger-Heisenberg Variational Quantum Algorithms", Physical Review Letters 131 6, 060406 (2023).

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

[227] 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).

[228] Sheng-Jie Li, Jin-Min Liang, Shu-Qian Shen, and Ming Li, "Variational quantum algorithms for trace norms and their applications", Communications in Theoretical Physics 73 10, 105102 (2021).

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

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

[231] Junyu Liu, Zimu Li, Han Zheng, Xiao Yuan, and Jinzhao Sun, "Towards a variational Jordan–Lee–Preskill quantum algorithm", Machine Learning: Science and Technology 3 4, 045030 (2022).

[232] Kübra Yeter-Aydeniz, George Siopsis, and Raphael C Pooser, "Scattering in the Ising model with the quantum Lanczos algorithm * ", New Journal of Physics 23 4, 043033 (2021).

[233] Huan-Yu Liu, Tai-Ping Sun, Yu-Chun Wu, and Guo-Ping Guo, "Variational Quantum Algorithms for the Steady States of Open Quantum Systems ", Chinese Physics Letters 38 8, 080301 (2021).

[234] Anna Dawid, Julian Arnold, Borja Requena, Alexander Gresch, Marcin Płodzień, Kaelan Donatella, Kim A. Nicoli, Paolo Stornati, Rouven Koch, Miriam Büttner, Robert Okuła, Gorka Muñoz-Gil, Rodrigo A. Vargas-Hernández, Alba Cervera-Lierta, Juan Carrasquilla, Vedran Dunjko, Marylou Gabrié, Patrick Huembeli, Evert van Nieuwenburg, Filippo Vicentini, Lei Wang, Sebastian J. Wetzel, Giuseppe Carleo, Eliška Greplová, Roman Krems, Florian Marquardt, Michał Tomza, Maciej Lewenstein, and Alexandre Dauphin, "Modern applications of machine learning in quantum sciences", arXiv:2204.04198, (2022).

[235] Sam McArdle, Tyson Jones, Suguru Endo, Ying Li, Simon C. Benjamin, and Xiao Yuan, "Variational ansatz-based quantum simulation of imaginary time evolution", npj Quantum Information 5, 75 (2019).

[236] Andrew Arrasmith, Lukasz Cincio, Rolando D. Somma, and Patrick J. Coles, "Operator Sampling for Shot-frugal Optimization in Variational Algorithms", arXiv:2004.06252, (2020).

[237] Naoki Yamamoto, "On the natural gradient for variational quantum eigensolver", arXiv:1909.05074, (2019).

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

[239] Ada Warren, Linghua Zhu, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Adaptive variational algorithms for quantum Gibbs state preparation", arXiv:2203.12757, (2022).

[240] Benjamin A. Cordier, Nicolas P. D. Sawaya, Gian G. Guerreschi, and Shannon K. McWeeney, "Biology and medicine in the landscape of quantum advantages", arXiv:2112.00760, (2021).

[241] 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).

[242] Markus Hauru, Maarten Van Damme, and Jutho Haegeman, "Riemannian optimization of isometric tensor networks", arXiv:2007.03638, (2020).

[243] Yusuke Hama, "Quantum Circuits for Collective Amplitude Damping in Two-Qubit Systems", arXiv:2012.02410, (2020).

[244] Chee-Kong Lee, Chang-Yu Hsieh, Shengyu Zhang, and Liang Shi, "Simulation of Condensed-Phase Spectroscopy with Near-term Digital Quantum Computer", arXiv:2106.10767, (2021).

[245] Baptiste Anselme Martin, Thomas Ayral, François Jamet, Marko J. Rančić, and Pascal Simon, "Combining Matrix Product States and Noisy Quantum Computers for Quantum Simulation", arXiv:2305.19231, (2023).

[246] Luca Cappelli, Francesco Tacchino, Giuseppe Murante, Stefano Borgani, and Ivano Tavernelli, "From Vlasov-Poisson to Schrödinger-Poisson: dark matter simulation with a quantum variational time evolution algorithm", arXiv:2307.06032, (2023).

[247] Yuxuan Yan, Zhenyu Du, Junjie Chen, and Xiongfeng Ma, "Limitations of Noisy Quantum Devices in Computational and Entangling Power", arXiv:2306.02836, (2023).

[248] Debbie Eeltink, Filippo Vicentini, and Vincenzo Savona, "Variational dynamics of open quantum systems in phase space", arXiv:2307.07429, (2023).

[249] Julien Gacon, Christa Zoufal, Giuseppe Carleo, and Stefan Woerner, "Stochastic Approximation of Variational Quantum Imaginary Time Evolution", arXiv:2305.07059, (2023).

[250] Mirko Consiglio, "Variational Quantum Algorithms for Gibbs State Preparation", arXiv:2305.17713, (2023).

[251] Mi-Ra Hwang, Eylee Jung, Museong Kim, and DaeKil Park, "Euclidean time method in Generalized Eigenvalue Equation", arXiv:2307.14640, (2023).

[252] Dirk Oliver Theis, ""Proper" Shift Rules for Derivatives of Perturbed-Parametric Quantum Evolutions", Quantum 7, 1052 (2023).

[253] Samantha V. Barron, Daniel J. Egger, Elijah Pelofske, Andreas Bärtschi, Stephan Eidenbenz, Matthis Lehmkuehler, and Stefan Woerner, "Provable bounds for noise-free expectation values computed from noisy samples", arXiv:2312.00733, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-02-27 18:45:21) and SAO/NASA ADS (last updated successfully 2024-02-27 18:45:23). The list may be incomplete as not all publishers provide suitable and complete citation data.

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