Simultaneous estimation of multiple eigenvalues with short-depth quantum circuit on early fault-tolerant quantum computers

Zhiyan Ding; Lin Lin

doi:10.22331/q-2023-10-11-1136

Simultaneous estimation of multiple eigenvalues with short-depth quantum circuit on early fault-tolerant quantum computers

Zhiyan Ding¹ and Lin Lin^1,2,3

¹Department of Mathematics, University of California, Berkeley, CA 94720, USA
²Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
³Challenge Institute of Quantum Computation, University of California, Berkeley, CA 94720, USA

Published:	2023-10-11, volume 7, page 1136
Eprint:	arXiv:2303.05714v4
Doi:	https://doi.org/10.22331/q-2023-10-11-1136
Citation:	Quantum 7, 1136 (2023).

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

Abstract

We introduce a multi-modal, multi-level quantum complex exponential least squares (MM-QCELS) method to simultaneously estimate multiple eigenvalues of a quantum Hamiltonian on early fault-tolerant quantum computers. Our theoretical analysis demonstrates that the algorithm exhibits Heisenberg-limited scaling in terms of circuit depth and total cost. Notably, the proposed quantum circuit utilizes just one ancilla qubit, and with appropriate initial state conditions, it achieves significantly shorter circuit depths compared to circuits based on quantum phase estimation (QPE). Numerical results suggest that compared to QPE, the circuit depth can be reduced by around two orders of magnitude under several settings for estimating ground-state and excited-state energies of certain quantum systems.

Featured image: Multi-modal, Multi-level QCELS

Popular summary

Phase estimation is one of the most important quantum primitives. This paper focuses on designing phase estimation algorithms that can simultaneously estimate ground-and excited-state energies of a Hamiltonian, which are essential for understanding the optical and electronic properties of materials.

In our paper, we introduce the multi-modal, multi-level quantum complex exponential least squares (MM-QCELS) method for estimating multiple eigenvalues of a quantum Hamiltonian. Our approach employs a simple quantum circuit with only one ancilla qubit. We prove that the circuit depth and total cost of our method satisfies the Heisenberg-limited scaling. Furthermore, with suitable initial-state conditions, our circuit depth can be significantly shorter than that of quantum phase estimation (QPE) circuits. Therefore, this method is especially suitable for early fault-tolerant quantum computers.

► BibTeX data

@article{Ding2023simultaneous,
  doi = {10.22331/q-2023-10-11-1136},
  url = {https://doi.org/10.22331/q-2023-10-11-1136},
  title = {Simultaneous estimation of multiple eigenvalues with short-depth quantum circuit on early fault-tolerant quantum computers},
  author = {Ding, Zhiyan and Lin, Lin},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1136},
  month = oct,
  year = {2023}
}

► References

[1] D. W. Berry, B. L. Higgins, S. D. Bartlett, M. W. Mitchell, G. J. Pryde, and H. M. Wiseman. How to perform the most accurate possible phase measurements. Phys. Rev. A, 80(5):052114, 2009. doi:10.1103/PhysRevA.80.052114.
https://doi.org/10.1103/PhysRevA.80.052114

[2] P. Boufounos, V. Cevher, A. C. Gilbert, Y. Li, and M. J. Strauss. What's the frequency, kenneth?: Sublinear Fourier sampling off the grid. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 61–72, 2012. doi:10.1007/s00453-014-9918-0.
https://doi.org/10.1007/s00453-014-9918-0

[3] E. J. Candès and C. Fernandez-Granda. Towards a mathematical theory of super-resolution. Communications on Pure and Applied Mathematics, 67(6):906–956, 2014. doi:10.1002/cpa.21455.
https://doi.org/10.1002/cpa.21455

[4] S. Chen and A. Moitra. Algorithmic foundations for the diffraction limit. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, page 490–503, 2021. doi:10.1145/3406325.3451078.
https://doi.org/10.1145/3406325.3451078

[5] C. L. Cortes and S. K. Gray. Quantum Krylov subspace algorithms for ground- and excited-state energy estimation. Phys. Rev. A, 105:022417, 2022. doi:10.1103/PhysRevA.105.022417.
https://doi.org/10.1103/PhysRevA.105.022417

[6] Z. Ding and L. Lin. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum, 4:020331, May 2023. doi:10.1103/PRXQuantum.4.020331.
https://doi.org/10.1103/PRXQuantum.4.020331

[7] Y. Dong, L. Lin, and Y. Tong. Ground-state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices. PRX Quantum, 3:040305, 2022. doi:10.1103/PRXQuantum.3.040305.
https://doi.org/10.1103/PRXQuantum.3.040305

[8] M. F. Duarte and R. G. Baraniuk. Spectral compressive sensing. Applied and Computational Harmonic Analysis, 35(1):111–129, 2013. doi:10.1016/j.acha.2012.08.003.
https://doi.org/10.1016/j.acha.2012.08.003

[9] A. Dutkiewicz, B. M. Terhal, and T. E. O'Brien. Heisenberg-limited quantum phase estimation of multiple eigenvalues with few control qubits. Quantum, 6:830, 2022. doi:10.22331/q-2022-10-06-830.
https://doi.org/10.22331/q-2022-10-06-830

[10] E. N. Epperly, L. Lin, and Y. Nakatsukasa. A theory of quantum subspace diagonalization. SIAM Journal on Matrix Analysis and Applications, 43(3):1263–1290, 2022. doi:10.1137/21M145954X.
https://doi.org/10.1137/21M145954X

[11] V. Giovannetti, S. Lloyd, and L. Maccone. Advances in quantum metrology. Nature Photonics, 5(4):222–229, 2011. doi:10.1038/nphoton.2011.35.
https://doi.org/10.1038/nphoton.2011.35

[12] R. B. Griffiths and C.-S. Niu. Semiclassical Fourier transform for quantum computation. Phys. Rev. Lett., 76:3228–3231, 1996. doi:10.1103/PhysRevLett.76.3228.
https://doi.org/10.1103/PhysRevLett.76.3228

[13] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde. Entanglement-free Heisenberg-limited phase estimation. Nature, 450(7168):393–396, 2007. doi:10.1038/nature06257.
https://doi.org/10.1038/nature06257

[14] W. Huggins, J. Lee, U. Baek, B. O'Gorman, and K. Whaley. A non-orthogonal variational quantum eigensolver. New Journal of Physics, 22, 2020. doi:10.1088/1367-2630/ab867b.
https://doi.org/10.1088/1367-2630/ab867b

[15] Y. Jin, D. Liu, and Z. Song. Super-resolution and robust sparse continuous Fourier transform in any constant dimension: nearly linear time and sample complexity. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4667–4767, 2023. doi:10.1137/1.9781611977554.ch176.
https://doi.org/10.1137/1.9781611977554.ch176

[16] M. Kapralov. Sparse Fourier transform in any constant dimension with nearly-optimal sample complexity in sublinear time. In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, page 264–277, 2016. doi:10.1145/2897518.2897650.
https://doi.org/10.1145/2897518.2897650

[17] A. Y. Kitaev, A. Shen, and M. N. Vyalyi. Classical and quantum computation. American Mathematical Soc., 2002.

[18] K. Klymko, C. Mejuto-Zaera, S. J. Cotton, F. Wudarski, M. Urbanek, D. Hait, M. Head-Gordon, K. B. Whaley, J. Moussa, N. Wiebe, W. A. de Jong, and N. M. Tubman. Real-time evolution for ultracompact Hamiltonian eigenstates on quantum hardware. PRX Quantum, 3:020323, 2022. doi:10.1103/PRXQuantum.3.020323.
https://doi.org/10.1103/PRXQuantum.3.020323

[19] E. Knill, G. Ortiz, and R. D. Somma. Optimal quantum measurements of expectation values of observables. Phys. Rev. A, 75:012328, 2007. doi:10.1103/PhysRevA.75.012328.
https://doi.org/10.1103/PhysRevA.75.012328

[20] H. Li, H. Ni, and L. Ying. A note on spike localization for line spectrum estimation. preprint, 2023. doi:10.48550/arXiv.2303.00946.
https://doi.org/10.48550/arXiv.2303.00946

[21] H. Li, H. Ni, and L. Ying. On low-depth quantum algorithms for robust multiple-phase estimation. preprint, 2023. doi:10.48550/arXiv.2303.08099.
https://doi.org/10.48550/arXiv.2303.08099

[22] W. Li, W. Liao, and A. Fannjiang. Super-resolution limit of the esprit algorithm. IEEE Transactions on Information Theory, 66(7):4593–4608, 2020. doi:10.1109/TIT.2020.2974174.
https://doi.org/10.1109/TIT.2020.2974174

[23] L. Lin and Y. Tong. Heisenberg-limited ground state energy estimation for early fault-tolerant quantum computers. PRX Quantum, 3:010318, 2022. doi:10.1103/PRXQuantum.3.010318.
https://doi.org/10.1103/PRXQuantum.3.010318

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

[25] M. Motta, C. Sun, A. Tan, M. O’Rourke, E. Ye, A. Minnich, F. Brandão, and G. Chan. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics, 16:1–6, 2020. doi:10.1038/s41567-019-0704-4.
https://doi.org/10.1038/s41567-019-0704-4

[26] D. Nagaj, P. Wocjan, and Y. Zhang. Fast amplification of QMA. Quantum Inf. Comput., 9(11), 2009. doi:10.5555/2012098.2012106.
https://doi.org/10.5555/2012098.2012106

[27] H. Ni, H. Li, and L. Ying. On low-depth algorithms for quantum phase estimation. preprint, 2023. doi:10.48550/arXiv.2302.02454.
https://doi.org/10.48550/arXiv.2302.02454

[28] M. A. Nielsen and I. Chuang. Quantum computation and quantum information. Cambridge Univ. Pr., 2000. doi:10.5555/1972505.
https://doi.org/10.5555/1972505

[29] T. E. O'Brien, B. Tarasinski, and B. M. Terhal. Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments. New J. Phys., 21(2):023022, 2019. doi:10.1088/1367-2630/aafb8e.
https://doi.org/10.1088/1367-2630/aafb8e

[30] R. M. Parrish and P. L. McMahon. Quantum filter diagonalization: Quantum eigendecomposition without full quantum phase estimation. preprint, 2019. doi:10.48550/arXiv.1909.08925.
https://doi.org/10.48550/arXiv.1909.08925

[31] D. Poulin and P. Wocjan. Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer. Phys. Rev. Lett., 103:220502, Nov 2009. doi:10.1103/PhysRevLett.103.220502.
https://doi.org/10.1103/PhysRevLett.103.220502

[32] E. Price and Z. Song. A robust sparse Fourier transform in the continuous setting. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 583–600, 10 2015. doi:10.1109/FOCS.2015.42.
https://doi.org/10.1109/FOCS.2015.42

[33] K. Seki and S. Yunoki. Quantum power method by a superposition of time-evolved states. PRX Quantum, 2:010333, 2021. doi:10.1103/PRXQuantum.2.010333.
https://doi.org/10.1103/PRXQuantum.2.010333

[34] R. D. Somma. Quantum eigenvalue estimation via time series analysis. New J. Phys., 21(12):123025, 2019. doi:10.1088/1367-2630/ab5c60.
https://doi.org/10.1088/1367-2630/ab5c60

[35] N. H. Stair, R. Huang, and F. A. Evangelista. A multireference quantum Krylov algorithm for strongly correlated electrons. Journal of Chemical Theory and Computation, 16(4):2236–2245, 2020. doi:10.1021/acs.jctc.9b01125.
https://doi.org/10.1021/acs.jctc.9b01125

[36] M. E. Stroeks, J. Helsen, and B. M. Terhal. Spectral estimation for hamiltonians: a comparison between classical imaginary-time evolution and quantum real-time evolution. New Journal of Physics, 24(10):103024, 2022. doi:10.1088/1367-2630/ac919c.
https://doi.org/10.1088/1367-2630/ac919c

[37] Y. Subaşı, R. D. Somma, and D. Orsucci. Quantum algorithms for systems of linear equations inspired by adiabatic quantum computing. Phys. Rev. Lett., 122:060504, 2019. doi:10.1103/PhysRevLett.122.060504.
https://doi.org/10.1103/PhysRevLett.122.060504

[38] G. Tang, B. N. Bhaskar, P. Shah, and B. Recht. Compressed sensing off the grid. IEEE Transactions on Information Theory, 59(11):7465–7490, 2013. doi:10.1109/TIT.2013.2277451.
https://doi.org/10.1109/TIT.2013.2277451

[39] G. Wang, D. Stilck-Franca, R. Zhang, S. Zhu, and P. D. Johnson. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. preprint, 2022. doi:10.48550/arXiv.2209.06811.
https://doi.org/10.48550/arXiv.2209.06811

[40] Z. Yang and L. Xie. Achieving high resolution for super-resolution via reweighted atomic norm minimization. In 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 3646–3650, 2015. doi:10.1109/ICASSP.2015.7178651.
https://doi.org/10.1109/ICASSP.2015.7178651

[41] M. Zwierz, C. A. Pérez-Delgado, and P. Kok. General optimality of the Heisenberg limit for quantum metrology. Phys. Rev. Lett., 105:180402, 2010. doi:10.1103/PhysRevLett.105.180402.
https://doi.org/10.1103/PhysRevLett.105.180402

[42] M. Zwierz, C. A. Pérez-Delgado, and P. Kok. Ultimate limits to quantum metrology and the meaning of the Heisenberg limit. Phys. Rev. A, 85:042112, 2012. doi:10.1103/PhysRevA.85.042112.
https://doi.org/10.1103/PhysRevA.85.042112

Cited by

[1] Chiara Leadbeater, Nathan Fitzpatrick, David Muñoz Ramo, and Alex J. W. Thom, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 498 (2024) ISBN:979-8-3315-4137-8.

[2] Sven Danz, Mario Berta, Stefan Schröder, Pascal Kienast, Frank K. Wilhelm, and Alessandro Ciani, "Calculating response functions of coupled oscillators using quantum phase estimation", Physical Review Research 7 2, 023264 (2025).

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

[4] 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).

[5] Zhiyan Ding, Haoya Li, Lin Lin, HongKang Ni, Lexing Ying, and Ruizhe Zhang, "Quantum Multiple Eigenvalue Gaussian filtered Search: an efficient and versatile quantum phase estimation method", Quantum 8, 1487 (2024).

[6] Hardeep Bassi, Yizhi Shen, Harish S. Bhat, and Roel Van Beeumen, "From noisy observables to accurate ground-state energies: A quantum–classical signal subspace approach with denoising", APL Quantum 2 4, 046103 (2025).

[7] 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).

[8] Yizhi Shen, Daan Camps, Aaron Szasz, Siva Darbha, Katherine Klymko, David B. Williams--Young, Norm M. Tubman, and Roel Van Beeumen, "Estimating Eigenenergies from Quantum Dynamics: A Unified Noise-Resilient Measurement-Driven Approach", Quantum 9, 1836 (2025).

[9] Riki Toshio, Yutaro Akahoshi, Jun Fujisaki, Hirotaka Oshima, Shintaro Sato, and Keisuke Fujii, "Practical Quantum Advantage on Partially Fault-Tolerant Quantum Computer", Physical Review X 15 2, 021057 (2025).

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

[11] 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).

[12] Mario Motta, William Kirby, Ieva Liepuoniute, Kevin J Sung, Jeffrey Cohn, Antonio Mezzacapo, Katherine Klymko, Nam Nguyen, Nobuyuki Yoshioka, and Julia E Rice, "Subspace methods for electronic structure simulations on quantum computers", Electronic Structure 6 1, 013001 (2024).

[13] 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).

[14] Yuan-De Jin, Shi-Yu Zhang, Ulrik L. Andersen, and Wen-Long Ma, "General Approach to Error Detection of Bosonic Codes via Phase Estimation", Physical Review Letters 135 14, 140603 (2025).

[15] Avner Bensoussan, Elena Chachkarova, Karine Even-Mendoza, Sophie Fortz, and Connor Lenihan, "AccelerQ: Accelerating Quantum Eigensolvers with Machine Learning on Quantum Simulators", Proceedings of the ACM on Programming Languages 9 OOPSLA2, 2279 (2025).

[16] Hirofumi Nishi, Yuki Takei, Taichi Kosugi, Shunsuke Mieda, Yutaka Natsume, Takeshi Aoyagi, and Yu-ichiro Matsushita, "Encoded probabilistic imaginary-time evolution on a trapped-ion quantum computer for ground and excited states of spin qubits", Physical Review Applied 23 3, 034016 (2025).

[17] Gwonhak Lee, Seonghoon Choi, Joonsuk Huh, and Artur F. Izmaylov, "Efficient strategies for reducing sampling error in quantum Krylov subspace diagonalization", Digital Discovery 4 4, 954 (2025).

[18] Chiara Leadbeater, Nathan Fitzpatrick, David Muñoz Ramo, and Alex J W Thom, "Non-unitary Trotter circuits for imaginary time evolution", Quantum Science and Technology 9 4, 045007 (2024).

[19] Aaron Szasz, Ed Younis, and Wibe Albert de Jong, "Ground state energy and magnetization curve of a frustrated magnetic system from real-time evolution on a digital quantum processor", Quantum 9, 1704 (2025).

[20] Zhiyan Ding, Yulong Dong, Yu Tong, and Lin Lin, "Robust ground-state energy estimation under depolarizing noise", APL Computational Physics 2 2, 026111 (2026).

[21] Lexing Ying, "A perturbative analysis for noisy spectral estimation", Applied and Computational Harmonic Analysis 74, 101716 (2025).

[22] Zhiyan Ding, Ethan N. Epperly, Lin Lin, and Ruizhe Zhang, 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS) 2344 (2024) ISBN:979-8-3315-1674-1.

[23] Ruhee D'Cunha, Cristian L. Cortes, Laura Gagliardi, and Stephen K. Gray, "Fragment-based initialization for quantum subspace methods", Physical Review A 110 4, 042613 (2024).

[24] Zhiyan Ding, Lin Lin, Yilun Yang, and Ruizhe Zhang, "Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra", PRX Quantum 7 2, 020318 (2026).

[25] Changhao Yi, Cunlu Zhou, and Jun Takahashi, "Quantum Phase Estimation by Compressed Sensing", Quantum 8, 1579 (2024).

[26] Antonio Márquez Romero, Josh J. M. Kirsopp, Giuseppe Buonaiuto, and Michal Krompiec, "Study of nuclear magnetic resonance spectra with the multi-modal multi-level quantum complex exponential least squares algorithm", Physical Chemistry Chemical Physics 28 13, 7930 (2026).

[27] Woo-Ram Lee, Ryan Scott, and V. W. Scarola, "Hybrid quantum-gap-estimation algorithm using a filtered time series", Physical Review A 109 5, 052403 (2024).

[28] Erenay Karacan, "Phase estimation with compressed controlled time evolution", Physical Review A 113 4, 042420 (2026).

[29] Guoming Wang, Daniel Stilck França, Ruizhe Zhang, Shuchen Zhu, and Peter D. Johnson, "Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision", Quantum 7, 1167 (2023).

[30] 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).

[31] Shu Kanno, Kenji Sugisaki, Rei Sakuma, Jumpei Kato, Hajime Nakamura, and Naoki Yamamoto, "Tensor-based phase difference estimation on time-series analysis", Physical Review A 113 5, 052433 (2026).

[32] ACS In Focus (2025) ISBN:9780841295964.

[33] 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).

[34] Hongkang Ni, Haoya Li, and Lexing Ying, "On low-depth algorithms for quantum phase estimation", Quantum 7, 1165 (2023).

[35] Timothy Stroschein and Markus Reiher, "After 100 Years of Quantum Mechanics: Toward a Constructive Observation-Centered Perspective", Journal of Chemical Theory and Computation 22 13, 6268 (2026).

[36] Yuan-De Jin, Shi-Yu Zhang, and Wen-Long Ma, "Channel-based framework for phase estimation of multiple eigenvalues", Physical Review A 111 2, 022406 (2025).

[37] Haoya Li, Hongkang Ni, and Lexing Ying, "Adaptive low-depth quantum algorithms for robust multiple-phase estimation", Physical Review A 108 6, 062408 (2023).

[38] Maria Gabriela Jordão Oliveira and Nina Glaser, "Quantum block Krylov subspace projector algorithm for computing low-lying eigenenergies", Physical Review A 112 5, 052442 (2025).

[39] Yizhi Shen, Alex Buzali, Hong-Ye Hu, Katherine Klymko, Daan Camps, Susanne F. Yelin, and Roel Van Beeumen, "Efficient Measurement-Driven Eigenenergy Estimation with Classical Shadows", PRX Quantum 7 1, 010328 (2026).

[40] Woo-Ram Lee, Nathan M. Myers, and V. W. Scarola, "Noise-resilient and resource-efficient hybrid algorithm for robust quantum gap estimation", Physical Review A 111 2, 022421 (2025).

[41] Taehee Ko, Xiantao Li, and Chunhao Wang, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 669 (2024) ISBN:979-8-3315-4137-8.

[42] 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).

[43] Jacob S. Nelson and Andrew D. Baczewski, "Assessment of quantum phase estimation protocols for early fault-tolerant quantum computers", Physical Review A 110 4, 042420 (2024).

[44] Chusei Kiumi and Bálint Koczor, "TE-PAI: exact time evolution by sampling random circuits", Quantum Science and Technology 10 4, 045071 (2025).

[45] Kenji Sugisaki, "Projective Measurement-Based Quantum Phase Difference Estimation Algorithm for the Direct Computation of Eigenenergy Differences on a Quantum Computer", Journal of Chemical Theory and Computation 19 21, 7617 (2023).

[46] Amara Katabarwa, Katerina Gratsea, Athena Caesura, and Peter D. Johnson, "Early Fault-Tolerant Quantum Computing", PRX Quantum 5 2, 020101 (2024).

[47] Zhenning Liu, Xiantao Li, Chunhao Wang, and Jin-Peng Liu, "Toward end-to-end quantum simulation for protein dynamics", arXiv:2411.03972, (2024).

[48] Maria Dincă, Tim Chan, and Simon C. Benjamin, "Error mitigation for logical circuits using decoder confidence", arXiv:2512.15689, (2025).

[49] Pei Zeng, Guo Zheng, Qian Xu, and Liang Jiang, "Error-structure-tailored early fault-tolerant quantum computing", arXiv:2511.19983, (2025).

[50] Kenji Sugisaki, "Projective Quantum Phase Difference Estimation Algorithm for the Direct Computation of Eigenenergy Gaps on a Quantum Computer", arXiv:2307.09825, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-08 02:43:11) and SAO/NASA ADS (last updated successfully 2026-08-08 02:43:12). 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.