Variational Quantum Computation of Excited States

Oscar Higgott; Daochen Wang; Stephen Brierley

doi:10.22331/q-2019-07-01-156

Variational Quantum Computation of Excited States

Oscar Higgott^1,2, Daochen Wang^1,3, and Stephen Brierley¹

¹Riverlane, 3 Charles Babbage Road, Cambridge CB3 0GT
²Department of Physics and Astronomy, University College London, London, WC1E 6BT
³Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742

Published:	2019-07-01, volume 3, page 156
Eprint:	arXiv:1805.08138v5
Doi:	https://doi.org/10.22331/q-2019-07-01-156
Citation:	Quantum 3, 156 (2019).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

The calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited state energies and their degeneracies. We propose an implementation that requires the same number of qubits as VQE and at most twice the circuit depth. Our method is robust to control errors, is compatible with error-mitigation strategies and can be implemented on near-term quantum computers.

Popular summary

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.

► BibTeX data

@article{Higgott2019variationalquantum,
  doi = {10.22331/q-2019-07-01-156},
  url = {https://doi.org/10.22331/q-2019-07-01-156},
  title = {Variational {Q}uantum {C}omputation of {E}xcited {S}tates},
  author = {Higgott, Oscar and Wang, Daochen and Brierley, Stephen},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {156},
  month = jul,
  year = {2019}
}

► References

[1] Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon. Simulated quantum computation of molecular energies. Science, 309 (5741): 1704–1707, 2005. ISSN 0036-8075. 10.1126/science.1113479. URL https://doi.org/10.1126/science.1113479.
https://doi.org/10.1126/science.1113479

[2] Marcello Benedetti, Delfina Garcia-Pintos, Oscar Perdomo, Vicente Leyton-Ortega, Yunseong Nam, and Alejandro Perdomo-Ortiz. A generative modeling approach for benchmarking and training shallow quantum circuits. npj Quantum Information, 5 (1): 45, 2019. 10.1038/s41534-019-0157-8. URL https://doi.org/10.1038/s41534-019-0157-8.
https://doi.org/10.1038/s41534-019-0157-8

[3] X. Bonet-Monroig, R. Sagastizabal, M. Singh, and T. E. O'Brien. Low-cost error mitigation by symmetry verification. Phys. Rev. A, 98: 062339, Dec 2018. 10.1103/PhysRevA.98.062339. URL https://doi.org/10.1103/PhysRevA.98.062339.
https://doi.org/10.1103/PhysRevA.98.062339

[4] Lukasz Cincio, Yiğit Subaşi, Andrew T Sornborger, and Patrick J Coles. Learning the quantum algorithm for state overlap. New Journal of Physics, 20 (11): 113022, nov 2018. 10.1088/1367-2630/aae94a. URL https://doi.org/10.1088.
https://doi.org/10.1088/1367-2630/aae94a

[5] J. I. Colless, V. V. Ramasesh, D. Dahlen, M. S. Blok, M. E. Kimchi-Schwartz, J. R. McClean, J. Carter, W. A. de Jong, and I. Siddiqi. Computation of molecular spectra on a quantum processor with an error-resilient algorithm. Phys. Rev. X, 8: 011021, Feb 2018. 10.1103/PhysRevX.8.011021. URL https://doi.org/10.1103/PhysRevX.8.011021.
https://doi.org/10.1103/PhysRevX.8.011021

[6] B. Efron. Bootstrap methods: Another look at the jackknife. Ann. Statist., 7 (1): 1–26, 01 1979. 10.1214/aos/1176344552. URL https://doi.org/10.1214/aos/1176344552.
https://doi.org/10.1214/aos/1176344552

[7] Suguru Endo, Simon C. Benjamin, and Ying Li. Practical quantum error mitigation for near-future applications. Phys. Rev. X, 8: 031027, Jul 2018. 10.1103/PhysRevX.8.031027. URL https://doi.org/10.1103/PhysRevX.8.031027.
https://doi.org/10.1103/PhysRevX.8.031027

[8] Henry Eyring. The activated complex in chemical reactions. The Journal of Chemical Physics, 3 (2): 107–115, 1935. 10.1063/1.1749604. URL https://doi.org/10.1063/1.1749604.
https://doi.org/10.1063/1.1749604

[9] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028, 2014. URL https://arxiv.org/abs/1411.4028.
arXiv:1411.4028

[10] Juan Carlos Garcia-Escartin and Pedro Chamorro-Posada. The swap test and the Hong-Ou-Mandel effect are equivalent. Phys. Rev. A, 87: 052330, May 2013. 10.1103/PhysRevA.87.052330. URL https://doi.org/10.1103/PhysRevA.87.052330.
https://doi.org/10.1103/PhysRevA.87.052330

[11] Lov K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, STOC '96, pages 212–219, New York, NY, USA, 1996. ACM. ISBN 0-89791-785-5. 10.1145/237814.237866. URL https://doi.org/10.1145/237814.237866.
https://doi.org/10.1145/237814.237866

[12] Vojtěch Havlíček, Antonio D Córcoles, Kristan Temme, Aram W Harrow, Abhinav Kandala, Jerry M Chow, and Jay M Gambetta. Supervised learning with quantum-enhanced feature spaces. Nature, 567 (7747): 209, 2019. 10.1038/s41586-019-0980-2. URL https://doi.org/10.1038/s41586-019-0980-2.
https://doi.org/10.1038/s41586-019-0980-2

[13] Harold Hotelling. Analysis of a complex of statistical variables into principal components. Journal of educational psychology, 24 (6): 417, 1933. 10.1037/h0071325. URL https://doi.org/10.1037/h0071325.
https://doi.org/10.1037/h0071325

[14] Peter D Johnson, Jonathan Romero, Jonathan Olson, Yudong Cao, and Alán Aspuru-Guzik. QVECTOR: an algorithm for device-tailored quantum error correction. arXiv preprint arXiv:1711.02249, 2017. URL https://arxiv.org/abs/1711.02249.
arXiv:1711.02249

[15] Tyson Jones, Suguru Endo, Sam McArdle, Xiao Yuan, and Simon C. Benjamin. Variational quantum algorithms for discovering hamiltonian spectra. Phys. Rev. A, 99: 062304, Jun 2019. 10.1103/PhysRevA.99.062304. URL https://doi.org/10.1103/PhysRevA.99.062304.
https://doi.org/10.1103/PhysRevA.99.062304

[16] Ian D. Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Gidney, Alán Aspuru-Guzik, Garnet Kin-Lic Chan, and Ryan Babbush. Quantum simulation of electronic structure with linear depth and connectivity. Phys. Rev. Lett., 120: 110501, Mar 2018. 10.1103/PhysRevLett.120.110501. URL https://doi.org/10.1103/PhysRevLett.120.110501.
https://doi.org/10.1103/PhysRevLett.120.110501

[17] Emanuel Knill, Gerardo Ortiz, and Rolando D. Somma. Optimal quantum measurements of expectation values of observables. Phys. Rev. A, 75: 012328, Jan 2007. 10.1103/PhysRevA.75.012328. URL https://doi.org/10.1103/PhysRevA.75.012328.
https://doi.org/10.1103/PhysRevA.75.012328

[18] Joonho Lee, William J Huggins, Martin Head-Gordon, and K Birgitta Whaley. Generalized unitary coupled cluster wavefunctions for quantum computation. Journal of chemical theory and computation, 2018. 10.1021/acs.jctc.8b01004. URL https://doi.org/10.1021/acs.jctc.8b01004.
https://doi.org/10.1021/acs.jctc.8b01004

[19] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum principal component analysis. Nature Physics, 10 (9): 631, 2014. 10.1038/nphys3029. URL https://doi.org/10.1038/nphys3029.
https://doi.org/10.1038/nphys3029

[20] Lester W. Mackey. Deflation methods for sparse PCA. pages 1017–1024, 2009. URL http://papers.nips.cc/paper/3575-deflation-methods-for-sparse-pca.pdf.
http://papers.nips.cc/paper/3575-deflation-methods-for-sparse-pca.pdf

[21] Sam McArdle, Suguru Endo, Ying Li, Simon Benjamin, and Xiao Yuan. Variational quantum simulation of imaginary time evolution with applications in chemistry and beyond. arXiv preprint arXiv:1804.03023, 2018. URL https://arxiv.org/abs/1804.03023.
arXiv:1804.03023

[22] Sam McArdle, Xiao Yuan, and Simon Benjamin. Error-mitigated digital quantum simulation. Phys. Rev. Lett., 122: 180501, May 2019. 10.1103/PhysRevLett.122.180501. URL https://doi.org/10.1103/PhysRevLett.122.180501.
https://doi.org/10.1103/PhysRevLett.122.180501

[23] Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics, 18 (2): 023023, feb 2016. 10.1088/1367-2630/18/2/023023. URL https://doi.org/10.1088.
https://doi.org/10.1088/1367-2630/18/2/023023

[24] Jarrod R. McClean, Mollie E. Kimchi-Schwartz, Jonathan Carter, and Wibe A. de Jong. Hybrid quantum-classical hierarchy for mitigation of decoherence and determination of excited states. Phys. Rev. A, 95: 042308, Apr 2017a. 10.1103/PhysRevA.95.042308. URL https://doi.org/10.1103/PhysRevA.95.042308.
https://doi.org/10.1103/PhysRevA.95.042308

[25] Jarrod R McClean, Kevin J Sung, Ian D Kivlichan, Yudong Cao, Chengyu Dai, E Schuyler Fried, Craig Gidney, Brendan Gimby, Pranav Gokhale, et al. OpenFermion: the electronic structure package for quantum computers. arXiv preprint arXiv:1710.07629, 2017b. URL https://arxiv.org/abs/1710.07629.
arXiv:1710.07629

[26] Nikolaj Moll, Panagiotis Barkoutsos, Lev S Bishop, Jerry M Chow, Andrew Cross, Daniel J Egger, Stefan Filipp, Andreas Fuhrer, Jay M Gambetta, Marc Ganzhorn, et al. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Science and Technology, 3 (3): 030503, jun 2018. 10.1088/2058-9565/aab822. URL https://doi.org/10.1088.
https://doi.org/10.1088/2058-9565/aab822

[27] P. J. J. O'Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Y. Mutus, M. Neeley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis. Scalable quantum simulation of molecular energies. Phys. Rev. X, 6: 031007, Jul 2016. 10.1103/PhysRevX.6.031007. URL https://doi.org/10.1103/PhysRevX.6.031007.
https://doi.org/10.1103/PhysRevX.6.031007

[28] Lawrence Page, Sergey Brin, Rajeev Motwani, and Terry Winograd. The PageRank citation ranking: Bringing order to the web. Technical Report 1999-66, Stanford InfoLab, November 1999. URL http://ilpubs.stanford.edu:8090/422/.
http://ilpubs.stanford.edu:8090/422/

[29] Karl Pearson. On lines and planes of closest fit to systems of points in space. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2 (11): 559–572, 1901. 10.1080/14786440109462720. URL https://doi.org/10.1080/14786440109462720.
https://doi.org/10.1080/14786440109462720

[30] Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy L O'brien. A variational eigenvalue solver on a photonic quantum processor. Nature communications, 5, 2014. 10.1038/ncomms5213. URL https://doi.org/10.1038/ncomms5213.
https://doi.org/10.1038/ncomms5213

[31] John Preskill. Quantum computing in the NISQ era and beyond. Quantum, 2: 79, August 2018. ISSN 2521-327X. 10.22331/q-2018-08-06-79. URL https://doi.org/10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79

[32] Jonathan Romero, Ryan Babbush, Jarrod R McClean, Cornelius Hempel, Peter J Love, and Alán Aspuru-Guzik. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Quantum Science and Technology, 4 (1): 014008, Oct 2018. 10.1088/2058-9565/aad3e4. URL https://doi.org/10.1088.
https://doi.org/10.1088/2058-9565/aad3e4

[33] Nicholas C Rubin, Ryan Babbush, and Jarrod McClean. Application of fermionic marginal constraints to hybrid quantum algorithms. New Journal of Physics, 20 (5): 053020, may 2018. 10.1088/1367-2630/aab919. URL https://doi.org/10.1088.
https://doi.org/10.1088/1367-2630/aab919

[34] 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): 249–255, 2018. 10.1021/acs.jctc.8b00943. URL https://doi.org/10.1021/acs.jctc.8b00943.
https://doi.org/10.1021/acs.jctc.8b00943

[35] Raffaele Santagati, Jianwei Wang, Antonio A. Gentile, Stefano Paesani, Nathan Wiebe, Jarrod R. McClean, Sam Morley-Short, Peter J. Shadbolt, Damien Bonneau, Joshua W. Silverstone, David P. Tew, Xiaoqi Zhou, Jeremy L. O’Brien, and Mark G. Thompson. Witnessing eigenstates for quantum simulation of hamiltonian spectra. Science Advances, 4 (1), 2018. 10.1126/sciadv.aap9646. URL https://doi.org/10.1126/sciadv.aap9646.
https://doi.org/10.1126/sciadv.aap9646

[36] Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput., 26 (5): 1484–1509, October 1997. ISSN 0097-5397. 10.1137/S0097539795293172. URL https://doi.org/10.1137/S0097539795293172.
https://doi.org/10.1137/S0097539795293172

[37] Damian S. Steiger, Thomas Häner, and Matthias Troyer. ProjectQ: an open source software framework for quantum computing. Quantum, 2: 49, January 2018. ISSN 2521-327X. 10.22331/q-2018-01-31-49. URL https://doi.org/10.22331/q-2018-01-31-49.
https://doi.org/10.22331/q-2018-01-31-49

[38] Attila Szabo and Neil S. Ostlund. Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory. Dover Publications, Inc., Mineola, first edition, 1996.

[39] Kristan Temme, Sergey Bravyi, and Jay M. Gambetta. Error mitigation for short-depth quantum circuits. Phys. Rev. Lett., 119: 180509, Nov 2017. 10.1103/PhysRevLett.119.180509. URL https://doi.org/10.1103/PhysRevLett.119.180509.
https://doi.org/10.1103/PhysRevLett.119.180509

[40] Daochen Wang, Oscar Higgott, and Stephen Brierley. Accelerated variational quantum eigensolver. Phys. Rev. Lett., 122: 140504, Apr 2019. 10.1103/PhysRevLett.122.140504. URL https://doi.org/10.1103/PhysRevLett.122.140504.
https://doi.org/10.1103/PhysRevLett.122.140504

[41] Dave Wecker, Matthew B. Hastings, and Matthias Troyer. Progress towards practical quantum variational algorithms. Phys. Rev. A, 92: 042303, Oct 2015. 10.1103/PhysRevA.92.042303. URL https://doi.org/10.1103/PhysRevA.92.042303.
https://doi.org/10.1103/PhysRevA.92.042303

Cited by

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

[2] 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).

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

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

[5] 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).

[6] 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).

[7] 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).

[8] 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).

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

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

[11] 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).

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

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

[14] 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).

[15] 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).

[16] 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).

[17] 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).

[18] 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).

[19] 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).

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

[21] 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).

[22] 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).

[23] 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).

[24] 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).

[25] 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).

[26] 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).

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

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

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

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

[31] 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).

[32] 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).

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

[34] 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).

[35] 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).

[36] 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).

[37] 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).

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

[39] 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).

[40] 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).

[41] 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).

[42] 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).

[43] 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).

[44] 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).

[45] 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).

[46] 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).

[47] 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).

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

[49] 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).

[50] 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).

[51] 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).

[52] 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).

[53] 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).

[54] 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).

[55] 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).

[56] 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).

[57] 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).

[58] 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).

[59] 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).

[60] 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).

[61] 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).

[62] 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).

[63] 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).

[64] 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).

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

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

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

[68] 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).

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

[70] 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).

[71] 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.

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

[73] 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).

[74] 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).

[75] 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).

[76] 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.

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

[78] 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).

[79] 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).

[80] 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).

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

[82] 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).

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

[84] 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).

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

[86] 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).

[87] 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).

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

[89] 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).

[90] 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).

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

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

[93] 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).

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

[95] 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).

[96] 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).

[97] 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).

[98] 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).

[99] 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).

[100] 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).

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

[102] 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).

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

[104] 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).

[105] 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).

[106] 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).

[107] 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).

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

[109] 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).

[110] 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).

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

[112] 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).

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

[114] 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).

[115] 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).

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

[117] 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).

[118] 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).

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

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

[121] 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).

[122] 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).

[123] 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).

[124] 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).

[125] 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).

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

[127] 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).

[128] 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).

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

[130] 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).

[131] 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).

[132] 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).

[133] 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).

[134] 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).

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

[136] 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).

[137] 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).

[138] 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).

[139] 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).

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

[141] 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).

[142] 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).

[143] 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).

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

[145] 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).

[146] 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).

[147] 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).

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

[149] 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).

[150] 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).

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

[152] 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).

[153] 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).

[154] 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).

[155] 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).

[156] 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).

[157] 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).

[158] 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).

[159] 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).

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

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

[162] 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.

[163] 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).

[164] 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.

[165] 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).

[166] 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).

[167] Yi Fan, Jie Liu, Xiongzhi Zeng, Zhiqian Xu, Honghui Shang, Zhenyu Li, and Jinlong Yang, "Q<sup>2</sup>Chemistry: A quantum computation platform for quantum chemistry", JUSTC 52 12, 2 (2022).

[168] 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).

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

[170] 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).

[171] 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).

[172] 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).

[173] 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).

[174] 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).

[175] 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).

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

[177] 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).

[178] 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).

[179] 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).

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

[181] 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).

[182] 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).

[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] 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).

[185] 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).

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

[187] 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).

[188] 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 acs.jctc.4c00069 (2024).

[189] 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).

[190] 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).

[191] 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).

[192] 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).

[193] 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).

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

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

[196] 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).

[197] 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).

[198] 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).

[199] 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).

[200] Scott E. Smart, Davis M. Welakuh, and Prineha Narang, "Many-Body Excited States with a Contracted Quantum Eigensolver", Journal of Chemical Theory and Computation acs.jctc.4c00030 (2024).

[201] 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).

[202] 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).

[203] 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).

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

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

[206] 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).

[207] 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).

[208] 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).

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

[210] 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).

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

[212] 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).

[213] 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).

[214] 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).

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

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

[217] 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).

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

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

[220] 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).

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

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

[223] 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).

[224] 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).

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

[226] 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).

[227] 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).

[228] 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).

[229] 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).

[230] 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).

[231] 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).

[232] 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).

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

[234] 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 acs.jctc.4c00211 (2024).

[235] 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).

[236] 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).

[237] 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).

[238] 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).

[239] 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).

[240] 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).

[241] 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).

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

[243] 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).

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

[245] 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).

[246] 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).

[247] 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).

[248] 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).

[249] 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).

[250] 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).

[251] 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).

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

[253] 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).

[254] Maria A. Castellanos, Mario Motta, and Julia E. Rice, " Quantum computation of π → π* and n → π* excited states of aromatic heterocycles ", Molecular Physics e2282736 (2023).

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

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

[257] 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).

[258] 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).

[259] 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).

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

[261] 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).

[262] 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).

[263] 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).

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

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

[266] 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).

[267] 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).

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

[269] 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).

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

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

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

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

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

[275] Ryan LaRose, Arkin Tikku, Étude O'Neel-Judy, Lukasz Cincio, and Patrick J. Coles, "Variational Quantum State Diagonalization", arXiv:1810.10506, (2018).

[276] 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).

[277] 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).

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

[279] 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).

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

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

[282] Scott E. Smart and Prineha Narang, "Many-Body Eigenstates from Quantum Manifold Optimization", arXiv:2402.07100, (2024).

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

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

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

[286] Yifu Zhang and Lei Ma, "Fast Super Robust Nonadiabatic Geometric Quantum Computation", arXiv:2404.06562, (2024).

[287] Sebastian Grieninger, Kazuki Ikeda, and Ismail Zahed, "Quasi-parton distributions in massive QED2: Towards quantum computation", arXiv:2404.05112, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-06 01:50:52) and SAO/NASA ADS (last updated successfully 2024-05-06 01:50:54). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.