Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization

Dominic W. Berry; Craig Gidney; Mario Motta; Jarrod R. McClean; Ryan Babbush

doi:10.22331/q-2019-12-02-208

Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization

Dominic W. Berry¹, Craig Gidney², Mario Motta³, Jarrod R. McClean², and Ryan Babbush²

¹Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2109, Australia
²Google Research, Venice, CA 90291, United States
³Division of Chemistry, California Institute of Technology, Pasadena, CA 91125, United States

Published:	2019-12-02, volume 3, page 208
Eprint:	arXiv:1902.02134v4
Doi:	https://doi.org/10.22331/q-2019-12-02-208
Citation:	Quantum 3, 208 (2019).

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Abstract

Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with $\widetilde{\cal O}(N^{3/2} \lambda)$ T complexity, where $N$ is number of orbitals and $\lambda$ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring about seven hundred times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.

Popular summary

Simulation of quantum chemistry is one of the most important potential applications of quantum computers, because it could be used to design new molecules for a wide range of applications. For example, it could be used to gain understanding of biological Nitrogen fixation by simulation of the FeMoco molecule. We show how to greatly accelerate the simulation of quantum chemistry by taking advantage of the structure of the system, together with several other new advances in quantum algorithms. Our most efficient approach takes advantage of the fact that the description of the energy has many terms that are close to zero and can be ignored. Together with a more efficient way of inputting information about the nonzero terms into the quantum algorithm, for the example of FeMoco we achieve a speedup of about a factor of 700.

► BibTeX data

@article{Berry2019qubitizationof,
  doi = {10.22331/q-2019-12-02-208},
  url = {https://doi.org/10.22331/q-2019-12-02-208},
  title = {Qubitization of {A}rbitrary {B}asis {Q}uantum {C}hemistry {L}everaging {S}parsity and {L}ow {R}ank {F}actorization},
  author = {Berry, Dominic W. and Gidney, Craig and Motta, Mario and McClean, Jarrod R. and Babbush, Ryan},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {208},
  month = dec,
  year = {2019}
}

► References

[1] R. P. Feynman, International Journal of Theoretical Physics 21, 467 (1982).
https://doi.org/10.1007/BF02650179

[2] S. Lloyd, Science 273, 1073 (1996).
https://doi.org/10.1126/science.273.5278.1073

[3] M. Mohseni, P. Read, H. Neven, S. Boixo, V. Denchev, R. Babbush, A. Fowler, V. Smelyanskiy, and J. Martinis, Nature 543, 171 (2017).
https://doi.org/10.1038/543171a

[4] A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Science 309, 1704 (2005).
https://doi.org/10.1126/science.1113479

[5] A. Y. Kitaev, arXiv:quant-ph/9511026 (1995).
arXiv:quant-ph/9511026

[6] D. S. Abrams and S. Lloyd, Physical Review Letters 79, 2586 (1997).
https://doi.org/10.1103/PhysRevLett.79.2586

[7] R. Babbush, N. Wiebe, J. McClean, J. McClain, H. Neven, and G. K.-L. Chan, Physical Review X 8, 011044 (2018a).
https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.011044

[8] R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Physical Review X 8, 041015 (2018b).
https://doi.org/10.1103/PhysRevX.8.041015

[9] I. D. Kivlichan, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, W. Sun, Z. Jiang, N. Rubin, A. Fowler, A. Aspuru-Guzik, R. Babbush, and H. Neven, arXiv:1902.10673 (2019).
arXiv:1902.10673

[10] G. H. Low and N. Wiebe, arXiv:1805.00675 (2018).
arXiv:1805.00675

[11] R. Babbush, D. W. Berry, J. R. McClean, and H. Neven, npj Quantum Information 5, 92 (2019a).
https://doi.org/10.1038/s41534-019-0199-y

[12] J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Molecular Physics 109, 735 (2011).
https://doi.org/10.1080/00268976.2011.552441

[13] D. Wecker, B. Bauer, B. K. Clark, M. B. Hastings, and M. Troyer, Physical Review A 90, 022305 (2014).
https://doi.org/10.1103/PhysRevA.90.022305

[14] R. Babbush, J. McClean, D. Wecker, A. Aspuru-Guzik, and N. Wiebe, Physical Review A 91, 022311 (2015).
https://doi.org/10.1103/PhysRevA.91.022311

[15] D. Poulin, M. B. Hastings, D. Wecker, N. Wiebe, A. C. Doherty, and M. Troyer, Quantum Information and Computation 15, 361 (2015).
https://doi.org/10.26421/QIC15.5-6

[16] J. T. Seeley, M. J. Richard, and P. J. Love, Journal of Chemical Physics 137, 224109 (2012).
https://doi.org/10.1063/1.4768229

[17] K. Setia and J. D. Whitfield, The Journal of Chemical Physics 148, 164104 (2018).
https://doi.org/10.1063/1.5019371

[18] S. Bravyi, J. M. Gambetta, A. Mezzacapo, and K. Temme, arXiv:1701.08213 (2017).
arXiv:1701.08213

[19] M. Steudtner and S. Wehner, New Journal of Physics 20, 063010 (2018).
https://doi.org/10.1088/1367-2630/aac54f

[20] Z. Jiang, J. McClean, R. Babbush, and H. Neven, arXiv:1812.08190 (2018).
arXiv:1812.08190

[21] L. Veis and J. Pittner, Journal of Chemical Physics 140, 214111 (2014).
https://doi.org/10.1063/1.4880755

[22] D. W. Berry, M. Kieferová, A. Scherer, Y. R. Sanders, G. H. Low, N. Wiebe, C. Gidney, and R. Babbush, npj Quantum Information 4, 22 (2018).
https://doi.org/10.1038/s41534-018-0071-5

[23] D. Poulin, A. Y. Kitaev, D. Steiger, M. Hastings, and M. Troyer, Physical Review Letters 121, 010501 (2017).
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.010501

[24] N. M. Tubman, C. Mejuto-Zaera, J. M. Epstein, D. Hait, D. S. Levine, W. Huggins, Z. Jiang, J. R. McClean, R. Babbush, M. Head-Gordon, and K. B. Whaley, arXiv:1809.05523 (2018).
arXiv:1809.05523

[25] G. H. Low and I. L. Chuang, Quantum 3, 163 (2019).
https://doi.org/10.22331/q-2019-07-12-163

[26] R. Babbush, D. W. Berry, I. D. Kivlichan, A. Y. Wei, P. J. Love, and A. Aspuru-Guzik, New Journal of Physics 18, 33032 (2016).
https://doi.org/10.1088/1367-2630/18/3/033032

[27] E. Campbell, Physical Review Letters 123, 070503 (2019).
https://doi.org/10.1103/PhysRevLett.123.070503

[28] N. Cody Jones, J. D. Whitfield, P. L. McMahon, M.-H. Yung, R. V. Meter, A. Aspuru-Guzik, and Y. Yamamoto, New Journal of Physics 14, 115023 (2012).
https://doi.org/10.1088/1367-2630/14/11/115023

[29] M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Proceedings of the National Academy of Sciences 114, 7555 (2017).
http://www.pnas.org/content/114/29/7555.abstract

[30] D. Litinski, Quantum 3, 128 (2019a).
https://quantum-journal.org/papers/q-2019-03-05-128/

[31] I. Kassal, S. P. Jordan, P. J. Love, M. Mohseni, and A. Aspuru-Guzik, Proceedings of the National Academy of Sciences 105, 18681 (2008).
http://www.pnas.org/content/105/48/18681.abstract

[32] B. Toloui and P. J. Love, arXiv:1312.2579 (2013).
arXiv:1312.2579

[33] M. B. Hastings, D. Wecker, B. Bauer, and M. Troyer, Quantum Information and Computation 15, 1 (2015).
https://doi.org/10.26421/QIC15.1-2

[34] K. Sugisaki, S. Yamamoto, S. Nakazawa, K. Toyota, K. Sato, D. Shiomi, and T. Takui, The Journal of Physical Chemistry A 120, 6459 (2016).
https://doi.org/10.1021/acs.jpca.6b04932

[35] F. Motzoi, M. P. Kaicher, and F. K. Wilhelm, Physical Review Letters 119, 160503 (2017).
https://doi.org/10.1103/PhysRevLett.119.160503

[36] M. Motta, E. Ye, J. R. McClean, Z. Li, A. J. Minnich, R. Babbush, and G. K.-L. Chan, arXiv:1808.02625 (2018).
arXiv:1808.02625

[37] C. Gidney and A. G. Fowler, Quantum 3, 135 (2019).
https://doi.org/10.22331/q-2019-04-30-135

[38] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Physical Review A 86, 032324 (2012).
https://doi.org/10.1103/PhysRevA.86.032324

[39] A. G. Fowler and C. Gidney, arXiv:1808.06709 (2018).
arXiv:1808.06709

[40] H. Beinert, R. Holm, and E. Munck, Science 277, 653 (1997).

[41] M. Szegedy, in 45th Annual IEEE Symposium on Foundations of Computer Science (IEEE, 2004) pp. 32–41.
https://doi.org/10.1109/FOCS.2004.53

[42] A. M. Childs and N. Wiebe, Quantum Information and Computation 12, 901 (2012).
https://doi.org/10.26421/QIC12.11-12

[43] A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su, Proceedings of the National Academy of Sciences 115, 9456 (2018).
https://doi.org/10.1073/pnas.1801723115

[44] G. H. Low, V. Kliuchnikov, and L. Schaeffer, arXiv:1812.00954 (2018).
arXiv:1812.00954

[45] Z. Li, J. Li, N. S. Dattani, C. J. Umrigar, and G. K.-L. Chan, The Journal of Chemical Physics 150, 024302 (2019).
https://doi.org/10.1063/1.5063376

[46] E. G. Hohenstein, S. I. L. Kokkila, R. M. Parrish, and T. J. Martinez, The Journal of Physical Chemistry B 117, 12972 (2013).
https://doi.org/10.1021/jp4021905

[47] J. L. Whitten, The Journal of Chemical Physics 58, 4496 (1973).
https://doi.org/10.1063/1.1679012

[48] E. G. Hohenstein and C. D. Sherrill, The Journal of Chemical Physics 132, 184111 (2010).
https://doi.org/10.1063/1.3426316

[49] N. H. F. Beebe and J. Linderberg, International Journal of Quantum Chemistry 12, 683 (1977).
https://doi.org/10.1002/qua.560120408

[50] H. Koch, A. S. de Meras, and T. B. Pedersen, The Journal of Chemical Physics 118, 9481 (2003).
https://doi.org/10.1063/1.1578621

[51] F. Aquilante, L. De Vico, N. Ferré, G. Ghigo, P.-Å. Malmqvist, P. Neogrády, T. B. Pedersen, M. Pitoňák, M. Reiher, B. O. Roos, L. Serrano-Andrés, M. Urban, V. Veryazov, and R. Lindh, Journal of Computational Chemistry 31, 224 (2010).
https://doi.org/10.1002/jcc.21318

[52] T. Helgaker, P. Jorgensen, and J. Olsen, Molecular Electronic Structure Theory (Wiley, 2002).
https://doi.org/10.1002/9781119019572

[53] B. Peng and K. Kowalski, Journal of Chemical Theory and Computation 13, 4179 (2017).
https://doi.org/10.1021/acs.jctc.7b00605

[54] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, in STOC '14 Proceedings of the 46th Annual ACM Symposium on Theory of Computing (2014) pp. 283–292.
https://doi.org/10.1145/2591796.2591854

[55] G. H. Low and I. L. Chuang, Physical Review Letters 118, 010501 (2017).
https://doi.org/10.1103/PhysRevLett.118.010501

[56] I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, G. K.-L. Chan, and R. Babbush, Physical Review Letters 120, 110501 (2018).
https://doi.org/10.1103/PhysRevLett.120.110501

[57] D. A. Mazziotti, Physical Review Letters 108, 263002 (2012).
https://doi.org/10.1103/PhysRevLett.108.263002

[58] N. Rubin, R. Babbush, and J. McClean, New Journal of Physics 20, 053020 (2018).
http://iopscience.iop.org/article/10.1088/1367-2630/aab919/meta

[59] W. A. Al-Saidi, S. Zhang, and H. Krakauer, Journal of Chemical Physics 124, 224101 (2006).
https://doi.org/10.1063/1.2200885

[60] D. Vanderbilt, Physical Review B 41, 7892 (1990).
https://doi.org/10.1103/PhysRevB.41.7892

[61] V. Giovannetti, S. Lloyd, and L. Maccone, Physical Review Letters 100, 160501 (2008).
https://doi.org/10.1103/PhysRevLett.100.160501

[62] C. Gidney, Quantum 2, 74 (2018).
https://doi.org/10.22331/q-2018-06-18-74

[63] R. Babbush, D. W. Berry, and H. Neven, Physical Review A 99, 040301 (2019b).
https://doi.org/10.1103/PhysRevA.99.040301

[64] J. R. McClean, R. Babbush, P. J. Love, and A. Aspuru-Guzik, The Journal of Physical Chemistry Letters 5, 4368 (2014).
https://doi.org/10.1021/jz501649m

[65] D. Litinski, arXiv:1905.06903 (2019b).
arXiv:1905.06903

[66] S. A. Cuccaro, T. G. Draper, S. A. Kutin, and D. P. Moulton, arXiv:quant-ph/0410184 (2004).
arXiv:quant-ph/0410184

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[15] Pei Zeng, Jinzhao Sun, Liang Jiang, and Qi Zhao, "Simple and High-Precision Hamiltonian Simulation by Compensating Trotter Error with Linear Combination of Unitary Operations", PRX Quantum 6 1, 010359 (2025).

[16] Andreas Juul Bay-Smidt, Frederik Ravn Klausen, Christoph Sünderhauf, Róbert Izsák, Gemma C. Solomon, and Nick S. Blunt, "Fault-Tolerant Quantum Simulation of Generalized Hubbard Models", PRX Quantum 6 3, 030348 (2025).

[17] Samuel Jaques and Thomas Häner, "Leveraging State Sparsity for More Efficient Quantum Simulations", ACM Transactions on Quantum Computing 3 3, 1 (2022).

[18] Michael Garn and Angus Kan, "Quantum Resource Estimates for Computing Binary Elliptic Curve Discrete Logarithms", IEEE Transactions on Quantum Engineering 6, 1 (2025).

[19] Matthew Otten, Byeol Kang, Dmitry Fedorov, Joo-Hyoung Lee, Anouar Benali, Salman Habib, Stephen K. Gray, and Yuri Alexeev, "QREChem: quantum resource estimation software for chemistry applications", Frontiers in Quantum Science and Technology 2, 1232624 (2023).

[20] Calvin Ku, Yu-Cheng Chen, Alice Hu, and Min-Hsiu Hsieh, "Optimizing quantum chemistry simulations with a hybrid quantization scheme", Communications Physics 9 1, 148 (2026).

[21] Weijie Du and James P. Vary, "Systematic many-fermion Hamiltonian input scheme and spectral calculations on quantum computers", Physics Letters B 866, 139548 (2025).

[22] Pauline J. Ollitrault, Jérôme F. Gonthier, Dario Rocca, Gian-Luca Anselmetti, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, and Michael Streif, "Improving the Runtime of Quantum Phase Estimation for Chemistry through Basis Set Optimization", Journal of Chemical Theory and Computation 21 23, 12034 (2025).

[23] Chuan-Dong Song, Jian Li, Yan-Yan Hou, Qing-Hui Liu, and Zhuo Wang, "Quantum canonical correlation analysis algorithm", Laser Physics Letters 20 10, 105203 (2023).

[24] William J. Huggins, Jarrod R. McClean, Nicholas C. Rubin, Zhang Jiang, Nathan Wiebe, K. Birgitta Whaley, and Ryan Babbush, "Efficient and noise resilient measurements for quantum chemistry on near-term quantum computers", npj Quantum Information 7 1, 23 (2021).

[25] Qiaohong Wang, Ruhee D’Cunha, Abhishek Mitra, Stephen K. Gray, Matthew Otten, and Laura Gagliardi, "Nonunitary Variational Quantum Eigensolver with the Localized Active Space Method and Cost Mitigation", The Journal of Physical Chemistry A 129 34, 7999 (2025).

[26] 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).

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[48] John M. Martinis, "Saving superconducting quantum processors from decay and correlated errors generated by gamma and cosmic rays", npj Quantum Information 7 1, 90 (2021).

[49] Craig Gidney and Martin Ekerå, "How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits", Quantum 5, 433 (2021).

[50] Samuel Jaques and Arthur G. Rattew, "QRAM: A Survey and Critique", Quantum 9, 1922 (2025).

[51] Edward G. Hohenstein, Oumarou Oumarou, Rachael Al-Saadon, Gian-Luca R. Anselmetti, Maximilian Scheurer, Christian Gogolin, and Robert M. Parrish, "Efficient quantum analytic nuclear gradients with double factorization", The Journal of Chemical Physics 158 11, 114119 (2023).

[52] Clémence Chevignard, Pierre-Alain Fouque, and André Schrottenloher, Lecture Notes in Computer Science 16541, 371 (2026) ISBN:978-3-032-25290-6.

[53] Alexey Pyrkov, Alex Aliper, Dmitry Bezrukov, and Alex Zhavoronkov, Applied Artificial Intelligence for Drug Discovery 289 (2026) ISBN:978-3-031-98021-3.

[54] Luka Skoric, Dan E. Browne, Kenton M. Barnes, Neil I. Gillespie, and Earl T. Campbell, "Parallel window decoding enables scalable fault tolerant quantum computation", Nature Communications 14 1, 7040 (2023).

[55] Smik Patel, Aritra Sankar Brahmachari, Joshua T. Cantin, Linjun Wang, and Artur F. Izmaylov, "Global Minimization of Electronic Hamiltonian 1-Norm via Linear Programming in the Block Invariant Symmetry Shift (BLISS) Method", Journal of Chemical Theory and Computation 21 2, 703 (2025).

[56] Kenji Sugisaki, Chikako Sakai, Kazuo Toyota, Kazunobu Sato, Daisuke Shiomi, and Takeji Takui, "Quantum Algorithm for Full Configuration Interaction Calculations without Controlled Time Evolutions", The Journal of Physical Chemistry Letters 12 45, 11085 (2021).

[57] Mario Motta, Kevin J. Sung, and James Shee, "Quantum Algorithms for the Variational Optimization of Correlated Electronic States with Stochastic Reconfiguration and the Linear Method", The Journal of Physical Chemistry A 128 40, 8762 (2024).

[58] Ryan Babbush, William J. Huggins, Dominic W. Berry, Shu Fay Ung, Andrew Zhao, David R. Reichman, Hartmut Neven, Andrew D. Baczewski, and Joonho Lee, "Quantum simulation of exact electron dynamics can be more efficient than classical mean-field methods", Nature Communications 14 1, 4058 (2023).

[59] Keli Huang and Jens Palsberg, "Toffoli Requires Six Quantum Neighbor Gates", ACM Transactions on Quantum Computing 7 2, 1 (2026).

[60] Yulong Dong, Lin Lin, and Yu Tong, "Ground-State Preparation and Energy Estimation on Early Fault-Tolerant Quantum Computers via Quantum Eigenvalue Transformation of Unitary Matrices", PRX Quantum 3 4, 040305 (2022).

[61] Stepan Fomichev, Kasra Hejazi, Modjtaba Shokrian Zini, Matthew Kiser, Joana Fraxanet, Pablo Antonio Moreno Casares, Alain Delgado, Joonsuk Huh, Arne-Christian Voigt, Jonathan E. Mueller, and Juan Miguel Arrazola, "Initial State Preparation for Quantum Chemistry on Quantum Computers", PRX Quantum 5 4, 040339 (2024).

[62] Élie Gouzien and Nicolas Sangouard, "Factoring 2048-bit RSA Integers in 177 Days with 13 436 Qubits and a Multimode Memory", Physical Review Letters 127 14, 140503 (2021).

[63] Calvin Ku, Yu-Cheng Chen, Alice Hu, and Min-Hsiu Hsieh, "Benchmarking quantum simulation of chemical Hamiltonians using the sorted-list encoding", Physical Review Research 8 3, 033091 (2026).

[64] Ignacio Loaiza and Artur F. Izmaylov, "Block-Invariant Symmetry Shift: Preprocessing Technique for Second-Quantized Hamiltonians to Improve Their Decompositions to Linear Combination of Unitaries", Journal of Chemical Theory and Computation 19 22, 8201 (2023).

[65] 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).

[66] Hans Gundlach, Keeper Sharkey, Jayson Lynch, Victoria Hazoglou, Kung-Chuan Hsu, Carl Dukatz, Eleanor Crane, Karin Walczyk, Marcin Bodziak, Johannes Galatsanos-Dueck, and Neil Thompson, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 2289 (2025) ISBN:979-8-3315-5736-2.

[67] Maxine Luo and J. Ignacio Cirac, "Efficient Simulation of Quantum Chemistry Problems in an Enlarged Basis Set", PRX Quantum 6 1, 010355 (2025).

[68] Mathias Soeken and Martin Roetteler, 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) 366 (2020) ISBN:978-1-7281-8969-7.

[69] Seonghoon Choi, Tzu-Ching Yen, and Artur F. Izmaylov, "Improving Quantum Measurements by Introducing “Ghost” Pauli Products", Journal of Chemical Theory and Computation 18 12, 7394 (2022).

[70] Mihael Erakovic, Freek Witteveen, Dylan Harley, Jakob Günther, Moritz Bensberg, Oinam Romesh Meitei, Minsik Cho, Troy Van Voorhis, Markus Reiher, and Matthias Christandl, "High Ground State Overlap via Quantum Embedding Methods", PRX Life 3 1, 013003 (2025).

[71] Ryan Babbush, Robbie King, Sergio Boixo, William Huggins, Tanuj Khattar, Guang Hao Low, Jarrod R. McClean, Thomas O’Brien, and Nicholas C. Rubin, "Grand Challenge of Quantum Applications", PRX Quantum 7 2, 020101 (2026).

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

[73] Yuan Su, Dominic W. Berry, Nathan Wiebe, Nicholas Rubin, and Ryan Babbush, "Fault-Tolerant Quantum Simulations of Chemistry in First Quantization", PRX Quantum 2 4, 040332 (2021).

[74] Yuan Su, Hsin-Yuan Huang, and Earl T. Campbell, "Nearly tight Trotterization of interacting electrons", Quantum 5, 495 (2021).

[75] Muhammad AbuGhanem and Hichem Eleuch, "NISQ Computers: A Path to Quantum Supremacy", IEEE Access 12, 102941 (2024).

[76] Kenji Sugisaki, "Does the full configuration interaction method based on quantum phase estimation with Trotter decomposition satisfy the size consistency condition?", AIP Advances 14 9, 095021 (2024).

[77] Pauline J. Ollitrault, Cristian L. Cortes, Jérôme F. Gonthier, Robert M. Parrish, Dario Rocca, Gian-Luca Anselmetti, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, and Michael Streif, "Enhancing Initial State Overlap through Orbital Optimization for Faster Molecular Electronic Ground-State Energy Estimation", Physical Review Letters 133 25, 250601 (2024).

[78] Kianna Wan, Mario Berta, and Earl T. Campbell, "Randomized Quantum Algorithm for Statistical Phase Estimation", Physical Review Letters 129 3, 030503 (2022).

[79] Thomas E. O’Brien, Lev B. Ioffe, Yuan Su, David Fushman, Hartmut Neven, Ryan Babbush, and Vadim Smelyanskiy, "Quantum Computation of Molecular Structure Using Data from Challenging-To-Classically-Simulate Nuclear Magnetic Resonance Experiments", PRX Quantum 3 3, 030345 (2022).

[80] Yasunori Lee and Keita Kanno, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 2170 (2025) ISBN:979-8-3315-5736-2.

[81] 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).

[82] Xianzhi Huang, Lili Niu, Jie Chen, Lichao Li, Kashif Hayat, and Weiping Liu, "Quantum and classical computational synergy for emerging contaminants management: Advanced insights into cytochrome P450 metabolic mechanisms", Critical Reviews in Environmental Science and Technology 54 24, 1827 (2024).

[83] Mark Webber, Vincent Elfving, Sebastian Weidt, and Winfried K. Hensinger, "The impact of hardware specifications on reaching quantum advantage in the fault tolerant regime", AVS Quantum Science 4 1, 013801 (2022).

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

[85] Alexander Miessen, Pauline J. Ollitrault, Francesco Tacchino, and Ivano Tavernelli, "Quantum algorithms for quantum dynamics", Nature Computational Science 3 1, 25 (2022).

[86] Kohdai Kuroiwa and Yuya O. Nakagawa, "Averaging gate approximation error and performance of Unitary Coupled Cluster ansatz in Pre-FTQC Era", Quantum 9, 1800 (2025).

[87] Jack S. Baker, Pablo A. M. Casares, Modjtaba Shokrian Zini, Jaydeep Thik, Debasish Banerjee, Chen Ling, Alain Delgado, and Juan Miguel Arrazola, "Simulating optically active spin defects with a quantum computer", Physical Review A 110 3, 032606 (2024).

[88] Yukun Zhang, Xiaoming Zhang, Jinzhao Sun, Heng Lin, Yifei Huang, Dingshun Lv, and Xiao Yuan, "Quantum Algorithms for Quantum Molecular Systems: A Survey", WIREs Computational Molecular Science 15 3, e70020 (2025).

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

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

[91] Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer, "Trading T gates for dirty qubits in state preparation and unitary synthesis", Quantum 8, 1375 (2024).

[92] Yuta Matsuzawa and Yuki Kurashige, "Jastrow-type Decomposition in Quantum Chemistry for Low-Depth Quantum Circuits", Journal of Chemical Theory and Computation 16 2, 944 (2020).

[93] Nicholas P. Bauman, Hongbin Liu, Eric J. Bylaska, Sriram Krishnamoorthy, Guang Hao Low, Christopher E. Granade, Nathan Wiebe, Nathan A. Baker, Bo Peng, Martin Roetteler, Matthias Troyer, and Karol Kowalski, "Toward Quantum Computing for High-Energy Excited States in Molecular Systems: Quantum Phase Estimations of Core-Level States", Journal of Chemical Theory and Computation 17 1, 201 (2021).

[94] Jarrod R. McClean, Nicholas C. Rubin, Joonho Lee, Matthew P. Harrigan, Thomas E. O’Brien, Ryan Babbush, William J. Huggins, and Hsin-Yuan Huang, "What the foundations of quantum computer science teach us about chemistry", The Journal of Chemical Physics 155 15, 150901 (2021).

[95] Carlos Outeiral, Martin Strahm, Jiye Shi, Garrett M. Morris, Simon C. Benjamin, and Charlotte M. Deane, "The prospects of quantum computing in computational molecular biology", WIREs Computational Molecular Science 11 1, e1481 (2021).

[96] Yuri Alexeev, Maximilian Amsler, Marco Antonio Barroca, Sanzio Bassini, Torey Battelle, Daan Camps, David Casanova, Young Jay Choi, Frederic T. Chong, Charles Chung, Christopher Codella, Antonio D. Córcoles, James Cruise, Alberto Di Meglio, Ivan Duran, Thomas Eckl, Sophia Economou, Stephan Eidenbenz, Bruce Elmegreen, Clyde Fare, Ismael Faro, Cristina Sanz Fernández, Rodrigo Neumann Barros Ferreira, Keisuke Fuji, Bryce Fuller, Laura Gagliardi, Giulia Galli, Jennifer R. Glick, Isacco Gobbi, Pranav Gokhale, Salvador de la Puente Gonzalez, Johannes Greiner, Bill Gropp, Michele Grossi, Emanuel Gull, Burns Healy, Matthew R. Hermes, Benchen Huang, Travis S. Humble, Nobuyasu Ito, Artur F. Izmaylov, Ali Javadi-Abhari, Douglas Jennewein, Shantenu Jha, Liang Jiang, Barbara Jones, Wibe Albert de Jong, Petar Jurcevic, William Kirby, Stefan Kister, Masahiro Kitagawa, Joel Klassen, Katherine Klymko, Kwangwon Koh, Masaaki Kondo, Dog̃a Murat Kürkçüog̃lu, Krzysztof Kurowski, Teodoro Laino, Ryan Landfield, Matt Leininger, Vicente Leyton-Ortega, Ang Li, Meifeng Lin, Junyu Liu, Nicolas Lorente, Andre Luckow, Simon Martiel, Francisco Martin-Fernandez, Margaret Martonosi, Claire Marvinney, Arcesio Castaneda Medina, Dirk Merten, Antonio Mezzacapo, Kristel Michielsen, Abhishek Mitra, Tushar Mittal, Kyungsun Moon, Joel Moore, Sarah Mostame, Mario Motta, Young-Hye Na, Yunseong Nam, Prineha Narang, Yu-ya Ohnishi, Daniele Ottaviani, Matthew Otten, Scott Pakin, Vincent R. Pascuzzi, Edwin Pednault, Tomasz Piontek, Jed Pitera, Patrick Rall, Gokul Subramanian Ravi, Niall Robertson, Matteo A.C. Rossi, Piotr Rydlichowski, Hoon Ryu, Georgy Samsonidze, Mitsuhisa Sato, Nishant Saurabh, Vidushi Sharma, Kunal Sharma, Soyoung Shin, George Slessman, Mathias Steiner, Iskandar Sitdikov, In-Saeng Suh, Eric D. Switzer, Wei Tang, Joel Thompson, Synge Todo, Minh C. Tran, Dimitar Trenev, Christian Trott, Huan-Hsin Tseng, Norm M. Tubman, Esin Tureci, David García Valiñas, Sofia Vallecorsa, Christopher Wever, Konrad Wojciechowski, Xiaodi Wu, Shinjae Yoo, Nobuyuki Yoshioka, Victor Wen-zhe Yu, Seiji Yunoki, Sergiy Zhuk, and Dmitry Zubarev, "Quantum-centric supercomputing for materials science: A perspective on challenges and future directions", Future Generation Computer Systems 160, 666 (2024).

[97] Kazuki Sakamoto, Hayata Morisaki, Junichi Haruna, Etsuko Itou, Keisuke Fujii, and Kosuke Mitarai, "End-to-end complexity for simulating the Schwinger model on quantum computers", Quantum 8, 1474 (2024).

[98] Raffaele Santagati, Alan Aspuru-Guzik, Ryan Babbush, Matthias Degroote, Leticia González, Elica Kyoseva, Nikolaj Moll, Markus Oppel, Robert M. Parrish, Nicholas C. Rubin, Michael Streif, Christofer S. Tautermann, Horst Weiss, Nathan Wiebe, and Clemens Utschig-Utschig, "Drug design on quantum computers", Nature Physics 20 4, 549 (2024).

[99] Hayata Morisaki, Kosuke Mitarai, Keisuke Fujii, and Yuya O. Nakagawa, "Classical variational optimization of a PREPARE circuit for quantum phase estimation of quantum chemistry Hamiltonians", Physical Review Research 6 4, 043186 (2024).

[100] Nicholas H. Stair and Francesco A. Evangelista, "Simulating Many-Body Systems with a Projective Quantum Eigensolver", PRX Quantum 2 3, 030301 (2021).

[101] Sam McArdle, Earl Campbell, and Yuan Su, "Exploiting fermion number in factorized decompositions of the electronic structure Hamiltonian", Physical Review A 105 1, 012403 (2022).

[102] Bruno Senjean, Saad Yalouz, and Matthieu Saubanère, "Toward density functional theory on quantum computers?", SciPost Physics 14 3, 055 (2023).

[103] Timothy N. Georges, Marius Bothe, Christoph Sünderhauf, Bjorn K. Berntson, Róbert Izsák, and Aleksei V. Ivanov, "Quantum simulations of chemistry in first quantization with any basis set", npj Quantum Information 11 1, 55 (2025).

[104] Richard Meister, Simon C. Benjamin, and Earl T. Campbell, "Tailoring Term Truncations for Electronic Structure Calculations Using a Linear Combination of Unitaries", Quantum 6, 637 (2022).

[105] Konrad Deka and Emil Zak, "Simultaneously Optimizing Symmetry Shifts and Tensor Factorizations for Cost-Efficient Fault-Tolerant Quantum Simulations of Electronic Hamiltonians", Journal of Chemical Theory and Computation 21 9, 4458 (2025).

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

[107] Javier Argüello-Luengo, Tao Shi, and Alejandro González-Tudela, "Engineering analog quantum chemistry Hamiltonians using cold atoms in optical lattices", Physical Review A 103 4, 043318 (2021).

[108] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush, "Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction", PRX Quantum 2 3, 030305 (2021).

[109] Tyler Takeshita, Nicholas C. Rubin, Zhang Jiang, Eunseok Lee, Ryan Babbush, and Jarrod R. McClean, "Increasing the Representation Accuracy of Quantum Simulations of Chemistry without Extra Quantum Resources", Physical Review X 10 1, 011004 (2020).

[110] Seonghoon Choi and Artur F. Izmaylov, "Measurement Optimization Techniques for Excited Electronic States in Near-Term Quantum Computing Algorithms", Journal of Chemical Theory and Computation 19 11, 3184 (2023).

[111] Joshua J. Goings, Alec White, Joonho Lee, Christofer S. Tautermann, Matthias Degroote, Craig Gidney, Toru Shiozaki, Ryan Babbush, and Nicholas C. Rubin, "Reliably assessing the electronic structure of cytochrome P450 on today’s classical computers and tomorrow’s quantum computers", Proceedings of the National Academy of Sciences 119 38, e2203533119 (2022).

[112] Nicholas H. Stair, Cristian L. Cortes, Robert M. Parrish, Jeffrey Cohn, and Mario Motta, "Stochastic quantum Krylov protocol with double-factorized Hamiltonians", Physical Review A 107 3, 032414 (2023).

[113] William J. Huggins and Jarrod R. McClean, "Accelerating Quantum Algorithms with Precomputation", Quantum 8, 1264 (2024).

[114] BinBin Cai, Fei Gao, and Gregor Leander, "Quantum attacks on two-round even-mansour", Frontiers in Physics 10, 1028014 (2022).

[115] Athena Caesura, Cristian L. Cortes, William Pol, Sukin Sim, Mark Steudtner, Gian-Luca R. Anselmetti, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, Michael Streif, and Christofer S. Tautermann, "Faster Quantum Chemistry Simulations on a Quantum Computer with Improved Tensor Factorization and Active Volume Compilation", PRX Quantum 6 3, 030337 (2025).

[116] G. Wendin, Encyclopedia of Condensed Matter Physics 246 (2024) ISBN:9780323914086.

[117] Dario Rocca, Cristian L. Cortes, Jérôme F. Gonthier, Pauline J. Ollitrault, Robert M. Parrish, Gian-Luca Anselmetti, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, and Michael Streif, "Reducing the Runtime of Fault-Tolerant Quantum Simulations in Chemistry through Symmetry-Compressed Double Factorization", Journal of Chemical Theory and Computation 20 11, 4639 (2024).

[118] Thomas E. O’Brien, Stefano Polla, Nicholas C. Rubin, William J. Huggins, Sam McArdle, Sergio Boixo, Jarrod R. McClean, and Ryan Babbush, "Error Mitigation via Verified Phase Estimation", PRX Quantum 2 2, 020317 (2021).

[119] Lewis W. Anderson, Martin Kiffner, Tom O'Leary, Jason Crain, and Dieter Jaksch, "Solving lattice gauge theories using the quantum Krylov algorithm and qubitization", Quantum 9, 1669 (2025).

[120] Mario Motta, Erika Ye, Jarrod R. McClean, Zhendong Li, Austin J. Minnich, Ryan Babbush, and Garnet Kin-Lic Chan, "Low rank representations for quantum simulation of electronic structure", npj Quantum Information 7 1, 83 (2021).

[121] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B. Buckley, David A. Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Zijun Chen, Benjamin Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Edward Farhi, Austin Fowler, Brooks Foxen, Craig Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Matthew P. Harrigan, Alan Ho, Sabrina Hong, Trent Huang, William J. Huggins, Lev Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul V. Klimov, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Pavel Laptev, Mike Lindmark, Erik Lucero, Orion Martin, John M. Martinis, Jarrod R. McClean, Matt McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojciech Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Hartmut Neven, Murphy Yuezhen Niu, Thomas E. O’Brien, Eric Ostby, Andre Petukhov, Harald Putterman, Chris Quintana, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Doug Strain, Kevin J. Sung, Marco Szalay, Tyler Y. Takeshita, Amit Vainsencher, Theodore White, Nathan Wiebe, Z. Jamie Yao, Ping Yeh, and Adam Zalcman, "Hartree-Fock on a superconducting qubit quantum computer", Science 369 6507, 1084 (2020).

[122] Pablo A. M. Casares, Roberto Campos, and M. A. Martin-Delgado, "TFermion: A non-Clifford gate cost assessment library of quantum phase estimation algorithms for quantum chemistry", Quantum 6, 768 (2022).

[123] Yingkai Ouyang, David R. White, and Earl T. Campbell, "Compilation by stochastic Hamiltonian sparsification", Quantum 4, 235 (2020).

[124] Baptiste Claudon, Julien Zylberman, César Feniou, Fabrice Debbasch, Alberto Peruzzo, and Jean-Philip Piquemal, "Polylogarithmic-depth controlled-NOT gates without ancilla qubits", Nature Communications 15 1, 5886 (2024).

[125] Ignacio Loaiza, Aritra Sankar Brahmachari, and Artur F Izmaylov, "Majorana tensor decomposition: a unifying framework for decompositions of fermionic Hamiltonians to linear combination of unitaries", Quantum Science and Technology 10 3, 035035 (2025).

[126] Shuchen Zhu, Aarthi Sundaram, and Guang Hao Low, "Unified architecture for quantum lookup tables", Physical Review Research 7 4, 043230 (2025).

[127] A. K. Fedorov and M. S. Gelfand, "Towards practical applications in quantum computational biology", Nature Computational Science 1 2, 114 (2021).

[128] Zhicheng Zhang, Qisheng Wang, and Mingsheng Ying, "Parallel Quantum Algorithm for Hamiltonian Simulation", Quantum 8, 1228 (2024).

[129] Connor T. Hann, Gideon Lee, S.M. Girvin, and Liang Jiang, "Resilience of Quantum Random Access Memory to Generic Noise", PRX Quantum 2 2, 020311 (2021).

[130] Ignacio Loaiza, Alireza Marefat Khah, Nathan Wiebe, and Artur F Izmaylov, "Reducing molecular electronic Hamiltonian simulation cost for linear combination of unitaries approaches", Quantum Science and Technology 8 3, 035019 (2023).

[131] Cristian L. Cortes, Matthias Loipersberger, Robert M. Parrish, Sam Morley-Short, William Pol, Sukin Sim, Mark Steudtner, Christofer S. Tautermann, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, and Michael Streif, "Fault-Tolerant Quantum Algorithm for Symmetry-Adapted Perturbation Theory", PRX Quantum 5 1, 010336 (2024).

[132] Christopher Chamberland and Earl T. Campbell, "Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery", PRX Quantum 3 1, 010331 (2022).

[133] Zhu Sun, Gregory Boyd, Zhenyu Cai, Hamza Jnane, Bálint Koczor, Richard Meister, Romy Minko, Benjamin Pring, Simon C. Benjamin, and Nikitas Stamatopoulos, "Low-depth phase oracle using a parallel piecewise circuit", Physical Review A 111 6, 062420 (2025).

[134] Bill Poirier and Jonathan Jerke, "Full-dimensional Schrödinger wavefunction calculations using tensors and quantum computers: the Cartesian component-separated approach", Physical Chemistry Chemical Physics 24 7, 4437 (2022).

[135] Aashna Anil Zade, Kenji Sugisaki, Matthias Werner, Ana Palacios, Jordi Riu, Jan Nogué, Artur Garcia-Saez, Arnau Riera, and V. S. Prasannaa, "Quantum annealing eigensolver as a NISQ era tool for probing strong correlation effects in quantum chemistry", The European Physical Journal Plus 140 9, 930 (2025).

[136] Zhendong Li, "Expressibility of comb tensor network states (CTNS) for the P-cluster and the FeMo-cofactor of nitrogenase", Electronic Structure 3 1, 014001 (2021).

[137] Fanhao Shen, Yujie Ji, Debin Xiang, Yanzhe Wang, Ke Wang, Chuanyu Zhang, Aosai Zhang, Yiren Zou, Yu Gao, Zhengyi Cui, Gongyu Liu, Jianan Yang, Yihang Han, Jinfeng Deng, Anbang Wang, Zhihong Zhang, Hekang Li, Qiujiang Guo, Pengfei Zhang, Chao Song, Liqiang Lu, Zhen Wang, and Jianwei Yin, "A bucket-brigade quantum random access memory", Nature Physics 22 5, 745 (2026).

[138] Marco Majland, Rasmus Berg Jensen, Patrick Ettenhuber, Irfansha Shaik, Nikolaj Thomas Zinner, and Ove Christiansen, "Fault-Tolerant Quantum Computations of Vibrational Wave Functions", Journal of Chemical Theory and Computation 22 1, 30 (2026).

[139] Kasra Rajabzadeh Dizaji, Ariq Haqq, Alicia B Magann, and Christian Arenz, "Hamiltonian simulation in Zeno subspaces", Physica Scripta 101 25, 255106 (2026).

[140] Oumarou Oumarou, Pauline J. Ollitrault, Cristian L. Cortes, Maximilian Scheurer, Robert M. Parrish, and Christian Gogolin, "Molecular Properties from Quantum Krylov Subspace Diagonalization", Journal of Chemical Theory and Computation 21 9, 4543 (2025).

[141] 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).

[142] Oumarou Oumarou, Maximilian Scheurer, Robert M. Parrish, Edward G. Hohenstein, and Christian Gogolin, "Accelerating Quantum Computations of Chemistry Through Regularized Compressed Double Factorization", Quantum 8, 1371 (2024).

[143] Christoph Sünderhauf, Earl Campbell, and Joan Camps, "Block-encoding structured matrices for data input in quantum computing", Quantum 8, 1226 (2024).

[144] Mohsen Bagherimehrab, Yuval R. Sanders, Dominic W. Berry, Gavin K. Brennen, and Barry C. Sanders, "Nearly Optimal Quantum Algorithm for Generating the Ground State of a Free Quantum Field Theory", PRX Quantum 3 2, 020364 (2022).

[145] Alessandro Luongo, Antonio Michele Miti, Varun Narasimhachar, and Adithya Sireesh, 2025 62nd ACM/IEEE Design Automation Conference (DAC) 1 (2025) ISBN:979-8-3315-0304-8.

[146] William J. Huggins, Oskar Leimkuhler, Torin F. Stetina, and K. Birgitta Whaley, "Efficient State Preparation for the Quantum Simulation of Molecules in First Quantization", PRX Quantum 6 2, 020319 (2025).

[147] Guang Hao Low, Robbie King, Dominic W. Berry, Qiushi Han, A. Eugene DePrince, Alec F. White, Ryan Babbush, Rolando D. Somma, and Nicholas C. Rubin, "Fast Quantum Simulation of Electronic Structure by Spectral Amplification", Physical Review X 15 4, 041016 (2025).

[148] Samson Wang, Sam McArdle, and Mario Berta, "Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra", PRX Quantum 5 2, 020324 (2024).

[149] Isaac H. Kim, Ye-Hua Liu, Sam Pallister, William Pol, Sam Roberts, and Eunseok Lee, "Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules", Physical Review Research 4 2, 023019 (2022).

[150] Nicholas C. Rubin, Dominic W. Berry, Fionn D. Malone, Alec F. White, Tanuj Khattar, A. Eugene DePrince, Sabrina Sicolo, Michael Küehn, Michael Kaicher, Joonho Lee, and Ryan Babbush, "Fault-Tolerant Quantum Simulation of Materials Using Bloch Orbitals", PRX Quantum 4 4, 040303 (2023).

[151] Yuval R. Sanders, Dominic W. Berry, Pedro C.S. Costa, Louis W. Tessler, Nathan Wiebe, Craig Gidney, Hartmut Neven, and Ryan Babbush, "Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization", PRX Quantum 1 2, 020312 (2020).

[152] Nicholas C. Rubin, Dominic W. Berry, Alina Kononov, Fionn D. Malone, Tanuj Khattar, Alec White, Joonho Lee, Hartmut Neven, Ryan Babbush, and Andrew D. Baczewski, "Quantum computation of stopping power for inertial fusion target design", Proceedings of the National Academy of Sciences 121 23, e2317772121 (2024).

[153] Srinjoy Ganguly, Vaishnavi Chandilkar, Prateek Jain, and Luis Gerardo Ayala Bertel, Communications in Computer and Information Science 2127, 146 (2024) ISBN:978-3-031-68616-0.

[154] Aleksei V. Ivanov, Andrew Patterson, Marius Bothe, Christoph Sünderhauf, Bjorn K. Berntson, Jens Jørgen Mortensen, Mikael Kuisma, Earl Campbell, and Róbert Izsák, "Quantum Computation of Electronic Structure with Projector Augmented-Wave Method and Plane Wave Basis Set", Journal of Chemical Theory and Computation 21 15, 7360 (2025).

[155] Vera von Burg, Guang Hao Low, Thomas Häner, Damian S. Steiger, Markus Reiher, Martin Roetteler, and Matthias Troyer, "Quantum computing enhanced computational catalysis", Physical Review Research 3 3, 033055 (2021).

[156] Guang Hao Low, Yuan Su, Yu Tong, and Minh C. Tran, "Complexity of Implementing Trotter Steps", PRX Quantum 4 2, 020323 (2023).

[157] Allan Tosta, Thais de Lima Silva, Giancarlo Camilo, and Leandro Aolita, "Randomized semi-quantum matrix processing", npj Quantum Information 10 1, 93 (2024).

[158] 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).

[159] Mark Steudtner, Sam Morley-Short, William Pol, Sukin Sim, Cristian L. Cortes, Matthias Loipersberger, Robert M. Parrish, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, and Michael Streif, "Fault-tolerant quantum computation of molecular observables", Quantum 7, 1164 (2023).

[160] 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).

[161] Shifan Xu, Alvin Lu, and Yongshan Ding, Proceedings of the 30th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 2 390 (2025) ISBN:9798400710797.

[162] Mason L. Rhodes, Michael Kreshchuk, and Shivesh Pathak, "Exponential Improvements in the Simulation of Lattice Gauge Theories Using Near-Optimal Techniques", PRX Quantum 5 4, 040347 (2024).

[163] Ryan Babbush, Dominic W. Berry, Jarrod R. McClean, and Hartmut Neven, "Quantum simulation of chemistry with sublinear scaling in basis size", npj Quantum Information 5 1, 92 (2019).

[164] Modjtaba Shokrian Zini, Alain Delgado, Roberto dos Reis, Pablo Antonio Moreno Casares, Jonathan E. Mueller, Arne-Christian Voigt, and Juan Miguel Arrazola, "Quantum simulation of battery materials using ionic pseudopotentials", Quantum 7, 1049 (2023).

[165] Jorge Chávez-Saab, Jesús-Javier Chi-Domínguez, Samuel Jaques, and Francisco Rodríguez-Henríquez, "The SQALE of CSIDH: sublinear Vélu quantum-resistant isogeny action with low exponents", Journal of Cryptographic Engineering 12 3, 349 (2022).

[166] Jarrod R. McClean, Nicholas C. Rubin, Kevin J. Sung, Ian D. Kivlichan, Xavier Bonet-Monroig, Yudong Cao, Chengyu Dai, E. Schuyler Fried, Craig Gidney, Brendan Gimby, Pranav Gokhale, Thomas Häner, Tarini Hardikar, Vojtěch Havlíček, Oscar Higgott, Cupjin Huang, Josh Izaac, Zhang Jiang, Xinle Liu, Sam McArdle, Matthew Neeley, Thomas O'Brien, Bryan O'Gorman, Isil Ozfidan, Maxwell D. Radin, Jhonathan Romero, Nicolas P. D. Sawaya, Bruno Senjean, Kanav Setia, Sukin Sim, Damian S. Steiger, Mark Steudtner, Qiming Sun, Wei Sun, Daochen Wang, Fang Zhang, and Ryan Babbush, "OpenFermion: the electronic structure package for quantum computers", Quantum Science and Technology 5 3, 034014 (2020).

[167] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[168] Jarrod R. McClean, Kevin J. Sung, Ian D. Kivlichan, Yudong Cao, Chengyu Dai, E. Schuyler Fried, Craig Gidney, Brendan Gimby, Pranav Gokhale, Thomas Häner, Tarini Hardikar, Vojtěch Havlíček, Oscar Higgott, Cupjin Huang, Josh Izaac, Zhang Jiang, Xinle Liu, Sam McArdle, Matthew Neeley, Thomas O'Brien, Bryan O'Gorman, Isil Ozfidan, Maxwell D. Radin, Jhonathan Romero, Nicholas Rubin, Nicolas P. D. Sawaya, Kanav Setia, Sukin Sim, Damian S. Steiger, Mark Steudtner, Qiming Sun, Wei Sun, Daochen Wang, Fang Zhang, and Ryan Babbush, "OpenFermion: The Electronic Structure Package for Quantum Computers", arXiv:1710.07629, (2017).

[169] Lewis W. Anderson, Martin Kiffner, Tom O'Leary, Jason Crain, and Dieter Jaksch, "Solving lattice gauge theories using the quantum Krylov algorithm and qubitization", arXiv:2403.08859, (2024).

[170] Ryan Babbush, Dominic W. Berry, and Hartmut Neven, "Quantum simulation of the Sachdev-Ye-Kitaev model by asymmetric qubitization", Physical Review A 99 4, 040301 (2019).

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

[172] Craig Gidney and Austin G. Fowler, "Flexible layout of surface code computations using AutoCCZ states", arXiv:1905.08916, (2019).

[173] Travis L. Scholten, Carl J. Williams, Dustin Moody, Michele Mosca, William Hurley, William J. Zeng, Matthias Troyer, and Jay M. Gambetta, "Assessing the Benefits and Risks of Quantum Computers", arXiv:2401.16317, (2024).

[174] Francisco J. R. Ruiz, Tuomas Laakkonen, Johannes Bausch, Matej Balog, Mohammadamin Barekatain, Francisco J. H. Heras, Alexander Novikov, Nathan Fitzpatrick, Bernardino Romera-Paredes, John van de Wetering, Alhussein Fawzi, Konstantinos Meichanetzidis, and Pushmeet Kohli, "Quantum Circuit Optimization with AlphaTensor", arXiv:2402.14396, (2024).

[175] Craig Gidney, "Windowed quantum arithmetic", arXiv:1905.07682, (2019).

[176] William M. Kirby and Peter J. Love, "Contextuality Test of the Nonclassicality of Variational Quantum Eigensolvers", Physical Review Letters 123 20, 200501 (2019).

[177] Yuta Matsuzawa and Yuki Kurashige, "A Jastrow-type decomposition in quantum chemistry for low-depth quantum circuits", arXiv:1909.12410, (2019).

[178] Jarrod R. McClean, Fabian M. Faulstich, Qinyi Zhu, Bryan O'Gorman, Yiheng Qiu, Steven R. White, Ryan Babbush, and Lin Lin, "Discontinuous Galerkin discretization for quantum simulation of chemistry", New Journal of Physics 22 9, 093015 (2020).

[179] Pei Zeng, Jinzhao Sun, Liang Jiang, and Qi Zhao, "Simple and high-precision Hamiltonian simulation by compensating Trotter error with linear combination of unitary operations", arXiv:2212.04566, (2022).

[180] G. Wendin, "Quantum information processing with superconducting circuits: a perspective", arXiv:2302.04558, (2023).

[181] Aleksei V. Ivanov, Andrew Patterson, Marius Bothe, Christoph Sünderhauf, Bjorn K. Berntson, Jens Jørgen Mortensen, Mikael Kuisma, Earl Campbell, and Róbert Izsák, "Quantum Computation of Electronic Structure with Projector Augmented-Wave Method and Plane Wave Basis Set", arXiv:2408.03159, (2024).

[182] Michal Krompiec and David Muñoz Ramo, "Strongly Contracted N-Electron Valence State Perturbation Theory Using Reduced Density Matrices from a Quantum Computer", arXiv:2210.05702, (2022).

[183] Guang Hao Low, "Halving the cost of quantum multiplexed rotations", arXiv:2110.13439, (2021).

[184] Roberto Campos, "Hybrid Quantum-Classical Algorithms", arXiv:2406.12371, (2024).

[185] Guang Hao Low, Yuan Su, Yu Tong, and Minh C. Tran, "On the complexity of implementing Trotter steps", arXiv:2211.09133, (2022).

[186] Charles Yuan, Agnes Villanyi, and Michael Carbin, "Quantum Control Machine: The Limits of Control Flow in Quantum Programming", arXiv:2304.15000, (2023).

[187] Kenji Sugisaki, Shigeaki Nakazawa, Kazuo Toyota, Kazunobu Sato, Daisuke Shiomi, and Takeji Takui, "Quantum chemistry on quantum computers: quantum simulations of the time evolution of wave functions under the S2 operator and determination of the spin quantum number S", Physical Chemistry Chemical Physics (Incorporating Faraday Transactions) 21 28, 15356 (2019).

[188] Sam McArdle, "Learning from Physics Experiments with Quantum Computers: Applications in Muon Spectroscopy", PRX Quantum 2 2, 020349 (2021).

[189] Craig Gidney, "How to factor 2048 bit RSA integers with less than a million noisy qubits", arXiv:2505.15917, (2025).

[190] Kentaro Yamamoto, Yuta Kikuchi, David Amaro, Ben Criger, Silas Dilkes, Ciarán Ryan-Anderson, Andrew Tranter, Joan M. Dreiling, Dan Gresh, Cameron Foltz, Michael Mills, Steven A. Moses, Peter E. Siegfried, Maxwell D. Urmey, Justin J. Burau, Aaron Hankin, Dominic Lucchetti, John P. Gaebler, Natalie C. Brown, Brian Neyenhuis, and David Muñoz Ramo, "Quantum Error-Corrected Computation of Molecular Energies", PRX Quantum 7 2, 020319 (2026).

[191] Sam McArdle, Alexander M. Dalzell, Aleksander Kubica, and Fernando G. S. L. Brandão, "The Fast for the Curious: How to accelerate fault-tolerant quantum applications", arXiv:2510.26078, (2025).

[192] Alexander M. Dalzell, András Gilyén, Connor T. Hann, Sam McArdle, Grant Salton, Quynh T. Nguyen, Aleksander Kubica, and Fernando G. S. L. Brandão, "A distillation-teleportation protocol for fault-tolerant QRAM", arXiv:2505.20265, (2025).

[193] Yasunari Suzuki, Takanori Sugiyama, Tomochika Arai, Wang Liao, Koji Inoue, and Teruo Tanimoto, "Q3DE: A fault-tolerant quantum computer architecture for multi-bit burst errors by cosmic rays", arXiv:2501.00331, (2024).

[194] Hans Gundlach, Keeper Sharkey, Jayson Lynch, Victoria Hazoglou, Kung-Chuan Hsu, Carl Dukatz, Eleanor Crane, Karin Walczyk, Marcin Bodziak, Johannes Galatsanos-Dueck, and Neil Thompson, "Quantum Advantage in Computational Chemistry?", arXiv:2508.20972, (2025).

[195] Nathan A. Baker, Brian Bilodeau, Chi Chen, Yingrong Chen, Marco Eckhoff, Alexandra Efimovskaya, Piero Gasparotto, Puck van Gerwen, Rushi Gong, Kevin Hoang, Zahra Hooshmand, Andrew J. Jenkins, Conrad S. N. Johnston, Run R. Li, Jiashu Liang, Hongbin Liu, Alexis Mills, Maximilian Mörchen, George Nishibuchi, Chong Sun, Bill Ticehurst, Matthias Troyer, Jan P. Unsleber, Stefan Wernli, David B. Williams-Young, and Boqin Zhang, "QDK/Chemistry: A Modular Toolkit for Quantum Chemistry Applications", arXiv:2601.15253, (2026).

[196] Marco Majland, Rasmus Berg Jensen, Patrick Ettenhuber, Irfansha Shaik, Nikolaj Thomas Zinner, and Ove Christiansen, "Fault-tolerant quantum computations of vibrational wave functions", arXiv:2508.16253, (2025).

[197] Alexey Uvarov and Artur F. Izmaylov, "Accuracy and resource advantages of quantum eigenvalue estimation with non-Hermitian transcorrelated electronic Hamiltonians", arXiv:2511.21867, (2025).

[198] Alessandro Berti and Francesco Ghisoni, "Efficient Quantum State Preparation with Bucket Brigade QRAM", arXiv:2510.16149, (2025).

[199] Armando Bellante, Reinis Irmejs, Marta Florido-Llinàs, María Cea Fernández, Marianna Crupi, Matthew Kiser, and J. Ignacio Cirac, "Compiling Quantum Regular Language States", arXiv:2602.02698, (2026).

[200] Alessandro Berti and Francesco Ghisoni, "Efficient Complex-Valued State Preparation on Bucket Brigade QRAM", arXiv:2604.25644, (2026).

[201] Mousumi Kundu, Ashish Kumar Patra, V. Anurag K. S., Ruchika Bhat, Sai Shankar P., Alok Shukla, and Jaiganesh G, "An Analytically Trained Variational Surrogate for Quantum Phase Estimation on NISQ Hardware", arXiv:2607.20943, (2026).

[202] Srushti Patil and Nina Glaser, "Efficient targeting of arbitrary excited states with quantum inverse power iteration through filtering polynomials", arXiv:2606.28255, (2026).

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