Near-optimal ground state preparation

Lin Lin; Yu Tong

doi:10.22331/q-2020-12-14-372

Near-optimal ground state preparation

Lin Lin^1,2 and Yu Tong¹

¹Department of Mathematics, University of California, Berkeley, CA 94720, USA
²Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

Published:	2020-12-14, volume 4, page 372
Eprint:	arXiv:2002.12508v3
Doi:	https://doi.org/10.22331/q-2020-12-14-372
Citation:	Quantum 4, 372 (2020).

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Abstract

Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.

► BibTeX data

@article{Lin2020nearoptimalground,
  doi = {10.22331/q-2020-12-14-372},
  url = {https://doi.org/10.22331/q-2020-12-14-372},
  title = {Near-optimal ground state preparation},
  author = {Lin, Lin and Tong, Yu},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {372},
  month = dec,
  year = {2020}
}

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[75] Shi Jin, Nana Liu, and Yue Yu, "Quantum Simulation of Partial Differential Equations via Schrödingerization", Physical Review Letters 133 23, 230602 (2024).

[76] Yulong Dong, K. Birgitta Whaley, and Lin Lin, "A quantum hamiltonian simulation benchmark", npj Quantum Information 8 1, 131 (2022).

[77] Qisheng Wang and Zhicheng Zhang, "Quantum Lower Bounds by Sample-to-Query Lifting", SIAM Journal on Computing 54 5, 1294 (2025).

[78] Youle Wang, "Near-optimal quantum kernel principal component analysis", Quantum Science and Technology 10 1, 015034 (2025).

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

[80] Marek Gluza, Jeongrak Son, Bi Hong Tiang, René Zander, Raphael Seidel, Yudai Suzuki, Zoë Holmes, and Nelly H. Y. Ng, "Double-Bracket Quantum Algorithms for Quantum Imaginary-Time Evolution", Physical Review Letters 136 2, 020601 (2026).

[81] Jiaqi Leng, Zhiyan Ding, Zherui Chen, and Lin Lin, "Operator-level quantum acceleration of non-logconcave sampling", Proceedings of the National Academy of Sciences 123 8, e2512789123 (2026).

[82] Yongdan Yang, Ying Li, Xiaosi Xu, and Xiao Yuan, "Resource-efficient quantum-classical hybrid algorithm for energy gap evaluation", Physical Review A 109 5, 052416 (2024).

[83] Shantanav Chakraborty, Aditya Morolia, and Anurudh Peduri, "Quantum Regularized Least Squares", Quantum 7, 988 (2023).

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

[85] Sam McArdle, András Gilyén,, and Mario Berta, "A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits", Quantum 10, 2058 (2026).

[86] Rei Sakuma, Kaito Wada, Shu Kanno, Kimberlee Keithley, Kenji Sugisaki, Takashi Abe, Hajime Nakamura, and Naoki Yamamoto, "Quantum-phase-estimation-based filtering: Performance analysis and application to low-energy spectral calculations", Physical Review A 113 1, 012602 (2026).

[87] Sophia Simon, Raffaele Santagati, Matthias Degroote, Nikolaj Moll, Michael Streif, and Nathan Wiebe, "Improved Precision Scaling for Simulating Coupled Quantum-Classical Dynamics", PRX Quantum 5 1, 010343 (2024).

[88] 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).

[89] Kai Li, Ming Zhang, Xiaowen Liu, Yong Liu, Hongyi Dai, Yijun Zhang, and Chen Dong, "Quantum Linear System Algorithm for General Matrices in System Identification", Entropy 24 7, 893 (2022).

[90] Gabriel Waite, Karl Lin, Samuel J. Elman, and Michael J. Bremner, "Physically motivated guiding states for local Hamiltonians", Physical Review A 114 1, 012431 (2026).

[91] Johannes Feldmeier, Yu-Jie Liu, Mikhail D. Lukin, and Soonwon Choi, "Digital Dissipative State Preparation for Frustration-Free Gapless Quantum Systems", Physical Review Letters 137 16, 160404 (2026).

[92] A. E. Russo, K. M. Rudinger, B. C. A. Morrison, and A. D. Baczewski, "Evaluating Energy Differences on a Quantum Computer with Robust Phase Estimation", Physical Review Letters 126 21, 210501 (2021).

[93] Weijie Du, Yangguang Yang, Zixin Liu, Chao Yang, and James P. Vary, "Quantum-classical computational framework for many-fermion response and structure", Physics Letters B 878, 140538 (2026).

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

[95] Jia-Chen Tang, Xu-Yang Hou, Yu-Huan Huang, Hao Guo, and Chih-Chun Chien, "Evolution of quantum geometric tensor of one-dimensional periodic systems after a quench", Physical Review B 113 17, 174301 (2026).

[96] Sirui Lu, Mari Carmen Bañuls, and J. Ignacio Cirac, "Algorithms for Quantum Simulation at Finite Energies", PRX Quantum 2 2, 020321 (2021).

[97] Stefano Polla, Yaroslav Herasymenko, and Thomas E. O'Brien, "Quantum digital cooling", Physical Review A 104 1, 012414 (2021).

[98] Guang Hao Low and Yuan Su, "Quantum linear system algorithm with optimal queries to initial state preparation", Quantum 10, 2041 (2026).

[99] Lin Lin, Proceedings of the International Congress of Mathematicians 2026 - Volume 7: Invited Lectures (Sections 15–20) 57 (2026) ISBN:978-1-61197-872-8.

[100] Patrick Rall and Bryce Fuller, "Amplitude Estimation from Quantum Signal Processing", Quantum 7, 937 (2023).

[101] S. E. Skelton, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 150 (2024) ISBN:979-8-3315-4137-8.

[102] Zhong-Xia Shang, Dong An, and Changpeng Shao, "Exponential Lindbladian fast forwarding and exponential amplification of certain Gibbs state properties", Reports on Progress in Physics 89 5, 057602 (2026).

[103] Zongkang Zhang, Ying Li, and Xiaosi Xu, "Quantum-Enhanced Neural Networks for Quantum Many-Body Simulations", PRX Intelligence 1 1, 013002 (2026).

[104] Xinying Li and Yun Shang, "Faster quantum sampling of Markov chains in nonregular graphs with fewer qubits", Physical Review A 107 2, 022432 (2023).

[105] Tal Schwartzman, Torsten V. Zache, Hannes Pichler, and H. R. Sadeghpour, "Imaginary Time Evolution and Ground State Preparation Using Unitary Multi-Copy Protocols", PRX Quantum 7 3, 033033 (2026).

[106] Alok Shukla and Prakash Vedula, "Toward Practical Quantum Phase Estimation: A Modular, Scalable, and Adaptive Approach", Advanced Quantum Technologies 9 3, e00683 (2026).

[107] Christopher F. Kane, Niladri Gomes, and Michael Kreshchuk, "Nearly optimal state preparation for quantum simulations of lattice gauge theories", Physical Review A 110 1, 012455 (2024).

[108] Wenyang Qian, Meijian Li, Carlos A. Salgado, and Michael Kreshchuk, "Efficient quantum simulation of QCD jets on the light front", Physical Review D 111 9, 096001 (2025).

[109] Yukun Zhang and Xiao Yuan, "Quantum Algorithms for Fidelity Susceptibility: From Quantum Criticality to Metrology", Physical Review Letters 136 8, 080604 (2026).

[110] Arthur Braida, Shantanav Chakraborty, Alapan Chaudhuri, Joseph Cunningham, Rutvij Menavlikar, Leonardo Novo, and Jérémie Roland, "Unstructured Adiabatic Quantum Optimization: Optimality with Limitations", Quantum 9, 1790 (2025).

[111] Zhong-Bo Kang, Noah Moran, Peter Nguyen, and Wenyang Qian, "Partonic distribution functions and amplitudes using tensor network methods", Journal of High Energy Physics 2025 9, 176 (2025).

[112] Lin Lin and Yu Tong, "Heisenberg-Limited Ground-State Energy Estimation for Early Fault-Tolerant Quantum Computers", PRX Quantum 3 1, 010318 (2022).

[113] Erenay Karacan, Conor Mc Keever, Michael Foss-Feig, David Hayes, and Michael Lubasch, "Filter-enhanced adiabatic quantum computing on a digital quantum processor", Physical Review Research 7 3, 033153 (2025).

[114] 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).

[115] James B. Larsen, Matthew D. Grace, Andrew D. Baczewski, and Alicia B. Magann, "Feedback-based quantum algorithms for ground state preparation", Physical Review Research 6 3, 033336 (2024).

[116] Ioannis Kolotouros, David Joseph, and Anand Kumar Narayanan, "Accelerating quantum imaginary-time evolution with random measurements", Physical Review A 111 1, 012424 (2025).

[117] Lin Lin, "Quantum Algorithms for Eigenvalue Problems", (2023).

[118] Cristian Tabares, Christian Kokail, Peter Zoller, Daniel González-Cuadra, and Alejandro González-Tudela, "Programming Optical-Lattice Fermi-Hubbard Quantum Simulators", PRX Quantum 6 3, 030356 (2025).

[119] 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).

[120] Guang Hao Low and Yuan Su, "Quantum Eigenvalue Processing", 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) 1051 (2024) ISBN:979-8-3315-1674-1.

[121] Navya Gupta, Christopher David White, and Zohreh Davoudi, "Euclidean-Monte-Carlo-informed ground-state preparation for quantum simulation of scalar field theory", Physical Review D 113 11, 114515 (2026).

[122] Alicja Dutkiewicz, Barbara M. Terhal, and Thomas E. O'Brien, "Heisenberg-limited quantum phase estimation of multiple eigenvalues with few control qubits", Quantum 6, 830 (2022).

[123] Mark R. Hirsbrunner, J. Wayne Mullinax, Yizhi Shen, David B. Williams-Young, Katherine Klymko, Roel Van Beeumen, and Norm M. Tubman, "A circuit-generated quantum subspace algorithm for the variational quantum eigensolver", The Journal of Chemical Physics 161 16, 164103 (2024).

[124] Youle Wang, "Resource-efficient quantum phase estimation by randomization", Quantum Science and Technology 10 3, 035036 (2025).

[125] Kiran Adhikari, "Krylov polynomials and quantum query complexity", Physics Letters A 584, 131601 (2026).

[126] Daniel Molpeceres, Sirui Lu, J. Ignacio Cirac, and Barbara Kraus, "Quantum algorithms for cooling: A simple case study", Physical Review Research 7 3, 033162 (2025).

[127] Almudena Carrera Vazquez and Aleksandros Sobczyk, "Spectral Gaps with Quantum Counting Queries and Oblivious State Preparation", Quantum 10, 2067 (2026).

[128] Mark R. Hirsbrunner, Diana Chamaki, J. Wayne Mullinax, and Norm M. Tubman, "Beyond MP2 initialization for unitary coupled cluster quantum circuits", Quantum 8, 1538 (2024).

[129] I. Stetcu, A. Baroni, and J. Carlson, "Variational approaches to constructing the many-body nuclear ground state for quantum computing", Physical Review C 105 6, 064308 (2022).

[130] Patrick Rall, "Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation", Quantum 5, 566 (2021).

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

[132] Yukun Zhang, Yifei Huang, Jinzhao Sun, Dingshun Lv, and Xiao Yuan, "Quantum computing quantum Monte Carlo algorithm", Physical Review A 112 2, 022428 (2025).

[133] 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).

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[135] Lucas K. Kovalsky, Fernando A. Calderon-Vargas, Matthew D. Grace, Alicia B. Magann, James B. Larsen, Andrew D. Baczewski, and Mohan Sarovar, "Self-Healing of Trotter Error in Digital Adiabatic State Preparation", Physical Review Letters 131 6, 060602 (2023).

[136] J. Wayne Mullinax and Norm M. Tubman, "Large-scale sparse wave function circuit simulator for applications with the variational quantum eigensolver", The Journal of Chemical Physics 162 7, 074114 (2025).

[137] Cheong Eung Ahn and Gil Young Cho, "Simulation and randomized measurement of topological phase on a trapped-ion quantum computer", Journal of the Korean Physical Society 81 3, 258 (2022).

[138] Jinzhao Sun, Pei Zeng, Tom Gur, and M. S. Kim, "High-precision and low-depth quantum algorithm design for eigenstate problems", Science Advances 12 3, eaeb1622 (2026).

[139] Siddharth Hariprakash, Neel S. Modi, Michael Kreshchuk, Christopher F. Kane, and Christian W. Bauer, "Strategies for simulating the time evolution of Hamiltonian lattice field theories", Physical Review A 111 2, 022419 (2025).

[140] Katerina Gratsea, Chong Sun, and Peter D. Johnson, "Evaluating the efficiency of ground-state-preparation algorithms", Physical Review A 109 4, 042425 (2024).

[141] Rajesh K. Malla, Hiroki Sukeno, Hongye Yu, Tzu-Chieh Wei, Andreas Weichselbaum, and Robert M. Konik, "Feedback-based quantum algorithm inspired by counterdiabatic driving", Physical Review Research 6 4, 043068 (2024).

[142] A. S. Plyashechnik, A. A. Zhukov, A. V. Lebedev, and W. V. Pogosov, "Spectral theory of energy-selective quantum search with Ising Hamiltonian phase oracles", Physical Review A 114 4, 042463 (2026).

[143] Hans Michael Christensen, Johannes Agerskov, and Frederik Nathan, "Ancilla-train quantum algorithm for simulating non-Markovian open quantum systems", Physical Review A 114 4, 042422 (2026).

[144] 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).

[145] Yusuke Nishiya, Hirofumi Nishi, Yannick Couzinié, Taichi Kosugi, and Yu-ichiro Matsushita, "First-quantized adiabatic time evolution for the ground state of a many-electron system and the optimal nuclear configuration", Physical Review A 109 2, 022423 (2024).

[146] Alireza Ataei, Olle Eriksson, and Vahid Azimi-Mousolou, "Ground-state preparation via nonlinear quantum dissipation", Physical Review B 114 20, 204407 (2026).

[147] S. Pathak, A. E. Russo, S. K. Seritan, and A. D. Baczewski, "Quantifying T -gate-count improvements for ground-state-energy estimation with near-optimal state preparation", Physical Review A 107 4, L040601 (2023).

[148] Judah F. Unmuth-Yockey, "Metropolis-style random sampling of quantum gates for the estimation of low-energy observables", Physical Review D 105 3, 034515 (2022).

[149] 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).

[150] Erenay Karacan, Yanbin Chen, and Christian B. Mendl, "Enhancing Scalability of Quantum Eigenvalue Transformation of Unitary Matrices for Ground State Preparation through Adaptive Finer Filtering", Quantum 9, 1624 (2025).

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

[152] William J. Huggins, Kianna Wan, Jarrod McClean, Thomas E. O’Brien, Nathan Wiebe, and Ryan Babbush, "Nearly Optimal Quantum Algorithm for Estimating Multiple Expectation Values", Physical Review Letters 129 24, 240501 (2022).

[153] Stefano Polla, Gian-Luca R. Anselmetti, and Thomas E. O'Brien, "Optimizing the information extracted by a single qubit measurement", Physical Review A 108 1, 012403 (2023).

[154] Nikhil S. Mande and Ronald de Wolf, "Tight Bounds for Quantum Phase Estimation and Related Problems", Quantum 10, 2140 (2026).

[155] Shantanav Chakraborty, "Implementing any Linear Combination of Unitaries on Intermediate-term Quantum Computers", Quantum 8, 1496 (2024).

[156] Mauro Cainelli, Reo Baba, and Yuki Kurashige, "Numerical Investigation of the Quantum Inverse Algorithm on Small Molecules", Journal of Chemical Theory and Computation acs.jctc.4c00483 (2024).

[157] Trevor Keen, Bo Peng, Karol Kowalski, Pavel Lougovski, and Steven Johnston, "Hybrid quantum-classical approach for coupled-cluster Green's function theory", Quantum 6, 675 (2022).

[158] Dhruv Gopalakrishnan and Michele Mosca, 2024 IEEE 6th International Conference on Trust, Privacy and Security in Intelligent Systems, and Applications (TPS-ISA) 471 (2024) ISBN:979-8-3503-8674-5.

[159] Kianna Wan, "Exponentially faster implementations of Select(H) for fermionic Hamiltonians", Quantum 5, 380 (2021).

[160] Natalie Klco, Alessandro Roggero, and Martin J Savage, "Standard model physics and the digital quantum revolution: thoughts about the interface", Reports on Progress in Physics 85 6, 064301 (2022).

[161] Harry Buhrman, Sevag Gharibian, Zeph Landau, François Le Gall, Norbert Schuch, and Suguru Tamaki, "Beating the Natural Grover Bound for Low-Energy Estimation and State Preparation", Physical Review Letters 135 3, 030601 (2025).

[162] G. E. L. Pexe, L. A. M. Rattighieri, A. L. Malvezzi, and F. F. Fanchini, "Using a feedback-based quantum algorithm to analyze the critical properties of the ANNNI model without classical optimization", Physical Review B 110 22, 224422 (2024).

[163] Robbie King, Guang Hao Low, Ryan Babbush, Rolando D. Somma, and Nicholas C. Rubin, "Quantum Simulation with Sum-of-Squares Spectral Amplification", Physical Review Letters 136 11, 110601 (2026).

[164] 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).

[165] Dominic W. Berry, Yu Tong, Tanuj Khattar, Alec White, Tae In Kim, Guang Hao Low, Sergio Boixo, Zhiyan Ding, Lin Lin, Seunghoon Lee, Garnet Kin-Lic Chan, Ryan Babbush, and Nicholas C. Rubin, "Rapid Initial-State Preparation for the Quantum Simulation of Strongly Correlated Molecules", PRX Quantum 6 2, 020327 (2025).

[166] Alain Delgado, Pablo A. M. Casares, Roberto dos Reis, Modjtaba Shokrian Zini, Roberto Campos, Norge Cruz-Hernández, Arne-Christian Voigt, Angus Lowe, Soran Jahangiri, M. A. Martin-Delgado, Jonathan E. Mueller, and Juan Miguel Arrazola, "Simulating key properties of lithium-ion batteries with a fault-tolerant quantum computer", Physical Review A 106 3, 032428 (2022).

[167] Shi Jin and Nana Liu, "Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 480 2292, 20230370 (2024).

[168] Matthew Thibodeau and Bryan K. Clark, "Nearly-frustration-free ground state preparation", Quantum 7, 1084 (2023).

[169] Ryan V. Mishmash, Tanvi P. Gujarati, Mario Motta, Huanchen Zhai, Garnet Kin-Lic Chan, and Antonio Mezzacapo, "Hierarchical Clifford Transformations to Reduce Entanglement in Quantum Chemistry Wave Functions", Journal of Chemical Theory and Computation 19 11, 3194 (2023).

[170] Koichi Miyamoto and Yuichiro Tada, "Improved quantum algorithm for calculating eigenvalues of differential operators and its application to estimating the decay rate of the perturbation distribution tail in stochastic inflation", Physical Review Research 7 2, 023251 (2025).

[171] Hirofumi Nishi, Taichi Kosugi, Satoshi Hirose, Tatsuya Okayama, and Yu-ichiro Matsushita, "Logical quantum phase estimation for x-ray absorption spectra", Physical Review Applied 25 3, 034026 (2026).

[172] Guanzhong Li, Jingquan Luo, and Lvzhou Li, "Universal quantum spatial search via controlled intermittent quantum walks", Physical Review A 114 3, 032440 (2026).

[173] Alicja Dutkiewicz, Stefano Polla, Maximilian Scheurer, Christian Gogolin, William J. Huggins, and Thomas E. O’Brien, "Error Mitigation and Circuit Division for Early Fault-Tolerant Quantum Phase Estimation", PRX Quantum 6 4, 040318 (2025).

[174] Yongtao Zhan, Zhiyan Ding, Jakob Huhn, Johnnie Gray, John Preskill, Garnet Kin-Lic Chan, and Lin Lin, "Rapid Quantum Ground State Preparation via Dissipative Dynamics", Physical Review X 16 1, 011004 (2026).

[175] L. Wright, F. Barratt, J. Dborin, G. H. Booth, and A. G. Green, "Automatic post-selection by ancillae thermalization", Physical Review Research 3 3, 033151 (2021).

[176] Zhiyan Ding, Chi-Fang Chen, and Lin Lin, "Single-ancilla ground state preparation via Lindbladians", Physical Review Research 6 3, 033147 (2024).

[177] Xiao-Ming Zhang, Yukun Zhang, Wenhao He, and Xiao Yuan, "Exponential Quantum Advantages for Practical Non-Hermitian Eigenproblems", Physical Review Letters 135 14, 140601 (2025).

[178] Yan Wang, Sarah Chehade, and Eugene Dumitrescu, "Semicoherent symmetric quantum processes: Theory and applications", AVS Quantum Science 6 3, 033805 (2024).

[179] Ashish Joshi and Takahiko Koyama, "Quantum Algorithm for Metabolic Network Analysis", (2025).

[180] Hans Hon Sang Chan, Richard Meister, Tyson Jones, David P. Tew, and Simon C. Benjamin, "Grid-based methods for chemistry simulations on a quantum computer", Science Advances 9 9, eabo7484 (2023).

[181] Jordi Weggemans, "Lower Bounds for Unitary Property Testing with Proofs and Advice", Quantum 9, 1717 (2025).

[182] Takeru Utsumi and Yoshifumi Nakata, "Explicit decoders using fixed-point amplitude amplification based on QSVT", Quantum 10, 2024 (2026).

[183] Garnet Kin-Lic Chan, "Spiers Memorial Lecture: Quantum chemistry, classical heuristics, and quantum advantage", Faraday Discussions 254, 11 (2024).

[184] Gwonhak Lee, Minhyeok Kang, Jungsoo Hong, Stepan Fomichev, and Joonsuk Huh, "Filtered quantum phase estimation", Quantum Science and Technology 11 4, 045053 (2026).

[185] Jiasu Wang, Yulong Dong, and Lin Lin, "On the energy landscape of symmetric quantum signal processing", Quantum 6, 850 (2022).

[186] Kaoru Mizuta and Keisuke Fujii, "Recursive quantum eigenvalue and singular-value transformation: Analytic construction of matrix sign function by Newton iteration", Physical Review Research 6 1, L012007 (2024).

[187] Arnaud Rémi, François Damanet, and Christophe Geuzaine, 2026 International Conference on Quantum Control, Computing and Learning (qCCL) 1 (2026) ISBN:979-8-3195-1865-1.

[188] Thomas E. O'Brien, Michael Streif, Nicholas C. Rubin, Raffaele Santagati, Yuan Su, William J. Huggins, Joshua J. Goings, Nikolaj Moll, Elica Kyoseva, Matthias Degroote, Christofer S. Tautermann, Joonho Lee, Dominic W. Berry, Nathan Wiebe, and Ryan Babbush, "Efficient quantum computation of molecular forces and other energy gradients", Physical Review Research 4 4, 043210 (2022).

[189] Gabriel Greene-Diniz, David Zsolt Manrique, Kentaro Yamamoto, Evgeny Plekhanov, Nathan Fitzpatrick, Michal Krompiec, Rei Sakuma, and David Muñoz Ramo, "Quantum Computed Green's Functions using a Cumulant Expansion of the Lanczos Method", Quantum 8, 1383 (2024).

[190] Hirofumi Nishi, Taichi Kosugi, Yusuke Nishiya, and Yu-ichiro Matsushita, "Quadratic acceleration of multistep probabilistic algorithms for state preparation", Physical Review Research 6 2, L022041 (2024).

[191] S E Skelton, "von Neumann measurement and quantum phase estimation of block-encoded Hamiltonians", Quantum Science and Technology 11 3, 035014 (2026).

[192] 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).

[193] William Kirby, Mario Motta, and Antonio Mezzacapo, "Exact and efficient Lanczos method on a quantum computer", Quantum 7, 1018 (2023).

[194] 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).

[195] 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).

[196] John M. Martyn, Zane M. Rossi, Andrew K. Tan, and Isaac L. Chuang, "Grand Unification of Quantum Algorithms", PRX Quantum 2 4, 040203 (2021).

[197] Lindsay Bassman, Miroslav Urbanek, Mekena Metcalf, Jonathan Carter, Alexander F. Kemper, and Wibe A. de Jong, "Simulating quantum materials with digital quantum computers", Quantum Science and Technology 6 4, 043002 (2021).

[198] A. Roggero, "Spectral-density estimation with the Gaussian integral transform", Physical Review A 102 2, 022409 (2020).

[199] Sam McArdle, András Gilyén, and Mario Berta, "A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits", arXiv:2209.12887, (2022).

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