Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation

Dong An; Di Fang; Lin Lin

doi:10.22331/q-2022-04-15-690

Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation

Dong An¹, Di Fang^2,3,5, and Lin Lin^2,4,5

¹Joint Center for Quantum Information and Computer Science (QuICS), University of Maryland, College Park, MD 20742, USA
²Department of Mathematics, University of California, Berkeley, CA 94720, USA
³Simons Institute for the Theory of Computing, University of California, Berkeley, CA 94720, USA
⁴Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
⁵Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA

Published:	2022-04-15, volume 6, page 690
Eprint:	arXiv:2111.03103v2
Doi:	https://doi.org/10.22331/q-2022-04-15-690
Citation:	Quantum 6, 690 (2022).

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

Abstract

We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.

► BibTeX data

@article{An2022timedependent,
  doi = {10.22331/q-2022-04-15-690},
  url = {https://doi.org/10.22331/q-2022-04-15-690},
  title = {Time-dependent {H}amiltonian {S}imulation of {H}ighly {O}scillatory {D}ynamics and {S}uperconvergence for {S}chr{\"{o}}dinger {E}quation},
  author = {An, Dong and Fang, Di and Lin, Lin},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {690},
  month = apr,
  year = {2022}
}

► References

[1] T. Albash and D. A. Lidar. Adiabatic quantum computation. Rev. Mod. Phys., 90:015002, 2018. doi:10.1103/RevModPhys.90.015002.
https://doi.org/10.1103/RevModPhys.90.015002

[2] D. An, D. Fang, and L. Lin. Time-dependent unbounded Hamiltonian simulation with vector norm scaling. Quantum, 5:459, may 2021. doi:10.22331/q-2021-05-26-459.
https://doi.org/10.22331/q-2021-05-26-459

[3] D. W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders. Efficient quantum algorithms for simulating sparse Hamiltonians. Commun. Math. Phys., 270(2):359–371, 2007. doi:10.1007/s00220-006-0150-x.
https://doi.org/10.1007/s00220-006-0150-x

[4] D. W. Berry and A. M. Childs. Black-box Hamiltonian simulation and unitary implementation. Quantum Information & Computation, 12(1-2):29–62, 2012. doi:10.26421/QIC12.1-2.
https://doi.org/10.26421/QIC12.1-2

[5] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma. Exponential improvement in precision for simulating sparse Hamiltonians. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 283–292, 2014. doi:10.1145/2591796.2591854.
https://doi.org/10.1145/2591796.2591854

[6] D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma. Simulating Hamiltonian dynamics with a truncated Taylor series. Phys. Rev. Lett., 114:090502, 2015. doi:10.1103/PhysRevLett.114.090502.
https://doi.org/10.1103/PhysRevLett.114.090502

[7] D. W. Berry, A. M. Childs, and R. Kothari. Hamiltonian simulation with nearly optimal dependence on all parameters. Proceedings of the 56th IEEE Symposium on Foundations of Computer Science, pages 792–809, 2015. doi:10.1109/FOCS.2015.54.
https://doi.org/10.1109/FOCS.2015.54

[8] D. W. Berry, A. M. Childs, Y. Su, X. Wang, and N. Wiebe. Time-dependent Hamiltonian simulation with $l^{1}$-norm scaling. Quantum, 4:254, 2020. doi:10.22331/q-2020-04-20-254.
https://doi.org/10.22331/q-2020-04-20-254

[9] D. W. Berry, R. Cleve, and S. Gharibian. Gate-efficient discrete simulations of continuous-time quantum query algorithms. Quantum Information and Computation, 14(1-2):1–30, 2014. doi:10.26421/QIC14.1-2-1.
https://doi.org/10.26421/QIC14.1-2-1

[10] S. Blanes, F. Casas, and M. Thalhammer. High-order commutator-free quasi-magnus exponential integrators for non-autonomous linear evolution equations. Computer Physics Communications, 220:243–262, 2017. URL: https://www.sciencedirect.com/science/article/pii/S0010465517302357, doi:https://doi.org/10.1016/j.cpc.2017.07.016.
https://doi.org/10.1016/j.cpc.2017.07.016
https://www.sciencedirect.com/science/article/pii/S0010465517302357

[11] A. Boulkhemair. L2 Estimates for Weyl Quantization. Journal of Functional Analysis, 165(1):173–204, 1999. doi:10.1006/jfan.1999.3423.
https://doi.org/10.1006/jfan.1999.3423

[12] R. L. Burden, J. D. Faires, and A. C. Reynolds. Numerical analysis. Brooks Cole, 2000.

[13] E. Campbell. Random compiler for fast Hamiltonian simulation. Phys. Rev. Lett., 123(7):070503, 2019. doi:10.1103/PhysRevLett.123.070503.
https://doi.org/10.1103/PhysRevLett.123.070503

[14] S. Chakraborty, A. Gilyén, and S. Jeffery. The power of block-encoded matrix powers: improved regression techniques via faster hamiltonian simulation. arXiv:1804.01973, 2018. 10.4230/LIPIcs.ICALP.2019.33.
https://doi.org/10.4230/LIPIcs.ICALP.2019.33
arXiv:1804.01973

[15] C.-F. Chen, H.-Y. Huang, R. Kueng, and J. A. Tropp. Quantum simulation via randomized product formulas: Low gate complexity with accuracy guarantees. 2020. arXiv:2008.11751.
https://doi.org/10.1103/PRXQuantum.2.040305
arXiv:2008.11751

[16] Y.-H. Chen, A. Kalev, and I. Hen. Quantum algorithm for time-dependent hamiltonian simulation by permutation expansion. PRX Quantum, 2:030342, Sep 2021. URL: https://link.aps.org/doi/10.1103/PRXQuantum.2.030342, doi:10.1103/PRXQuantum.2.030342.
https://doi.org/10.1103/PRXQuantum.2.030342

[17] A. M. Childs, D. Maslov, Y. Nam, N. J. Ross, and Y. Su. Toward the first quantum simulation with quantum speedup. Proc. Nat. Acad. Sci., 115:9456–9461, 2018. doi:10.1073/pnas.1801723115.
https://doi.org/10.1073/pnas.1801723115

[18] A. M. Childs, A. Ostrander, and Y. Su. Faster quantum simulation by randomization. Quantum, 3:182, 2019. doi:10.22331/q-2019-09-02-182.
https://doi.org/10.22331/q-2019-09-02-182

[19] A. M. Childs and Y. Su. Nearly optimal lattice simulation by product formulas. Phys. Rev. Lett., 123(5):050503, 2019. doi:10.1103/PhysRevLett.123.050503.
https://doi.org/10.1103/PhysRevLett.123.050503

[20] A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu. Theory of trotter error with commutator scaling. Phys. Rev. X, 11:011020, 2021. doi:10.1103/PhysRevX.11.011020.
https://doi.org/10.1103/PhysRevX.11.011020

[21] A. M. Childs and N. Wiebe. Hamiltonian simulation using linear combinations of unitary operations. Quantum Information and Computation, 12, Nov 2012. URL: http://dx.doi.org/10.26421/QIC12.11-12, doi:10.26421/qic12.11-12.
https://doi.org/10.26421/QIC12.11-12

[22] H. O. Cordes. On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. Journal of Functional Analysis, 18(2):115–131, 1975. doi:10.1016/0022-1236(75)90020-8.
https://doi.org/10.1016/0022-1236(75)90020-8

[23] D. Dong and I. R. Petersen. Quantum control theory and applications: a survey. IET Control Theory & Applications, 4(12):2651–2671, 2010. doi:10.1049/iet-cta.2009.0508.
https://doi.org/10.1049/iet-cta.2009.0508

[24] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser. Quantum computation by adiabatic evolution. 2000. arXiv:quant-ph/0001106.
arXiv:quant-ph/0001106

[25] A. Gilyén, S. Arunachalam, and N. Wiebe. Optimizing quantum optimization algorithms via faster quantum gradient computation. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1425–1444, 2019. doi:10.1137/1.9781611975482.87.
https://doi.org/10.1137/1.9781611975482.87

[26] A. Gilyén, Y. Su, G. H. Low, and N. Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 193–204, 2019. doi:10.1145/3313276.3316366.
https://doi.org/10.1145/3313276.3316366

[27] E. Hairer, S. P. Nørsett, and G. Wanner. Solving ordinary differential equation I: nonstiff problems, volume 8. Springer, 1987. doi:10.1007/978-3-540-78862-1.
https://doi.org/10.1007/978-3-540-78862-1

[28] M. Hochbruck and C. Lubich. On Magnus integrators for time-dependent Schrödinger equations. SIAM J. Numer. Anal., 41(3):945–963, 2003. doi:10.1137/S0036142902403875.
https://doi.org/10.1137/S0036142902403875

[29] J. Huyghebaert and H. De Raedt. Product formula methods for time-dependent Schrödinger problems. J. Phys. A, 23(24):5777–5793, 1990. doi:10.1088/0305-4470/23/24/019.
https://doi.org/10.1088/0305-4470/23/24/019

[30] A. Iserles and S. P. Nørsett. On the solution of linear differential equations in lie groups. Phil. Trans. R. Soc. A., 357:983–1019, 1999. doi:10.1098/rsta.1999.0362.
https://doi.org/10.1098/rsta.1999.0362

[31] M. Kieferová, A. Scherer, and D. W. Berry. Simulating the dynamics of time-dependent hamiltonians with a truncated dyson series. Physical Review A, 99(4), Apr 2019. URL: http://dx.doi.org/10.1103/PhysRevA.99.042314, doi:10.1103/physreva.99.042314.
https://doi.org/10.1103/PhysRevA.99.042314

[32] I. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, G.-L. Chan, and R. Babbush. Quantum Simulation of Electronic Structure with Linear Depth and Connectivity. Phys. Rev. Lett., 120(11):110501, 2018. doi:10.1103/PhysRevLett.120.110501.
https://doi.org/10.1103/PhysRevLett.120.110501

[33] I. D. Kivlichan, N. Wiebe, R. Babbush, and A. Aspuru-Guzik. Bounding the costs of quantum simulation of many-body physics in real space. J. Phys. A Math. Theor., 50:305301, 2017. doi:10.1088/1751-8121/aa77b8.
https://doi.org/10.1088/1751-8121/aa77b8

[34] A. W. Knapp. Basic Real Analysis. Springer Science & Business Media, 2005. doi:10.1007/0-8176-4441-5.
https://doi.org/10.1007/0-8176-4441-5

[35] G. H. Low. Hamiltonian simulation with nearly optimal dependence on spectral norm. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 491–502, 2019. doi:10.1145/3313276.3316386.
https://doi.org/10.1145/3313276.3316386

[36] G. H. Low and I. L. Chuang. Optimal Hamiltonian simulation by quantum signal processing. Phys. Rev. Lett., 118:010501, 2017. doi:10.1103/PhysRevLett.118.010501.
https://doi.org/10.1103/PhysRevLett.118.010501

[37] G. H. Low and I. L. Chuang. Hamiltonian simulation by qubitization. Quantum, 3:163, Jul 2019. URL: http://dx.doi.org/10.22331/q-2019-07-12-163, doi:10.22331/q-2019-07-12-163.
https://doi.org/10.22331/q-2019-07-12-163

[38] G. H. Low and N. Wiebe. Hamiltonian simulation in the interaction picture. 2019. arXiv:1805.00675.
arXiv:1805.00675

[39] A. Ma, A. B. Magann, T.-S. Ho, and H. Rabitz. Optimal control of coupled quantum systems based on the first-order magnus expansion: Application to multiple dipole-dipole-coupled molecular rotors. Physical Review A, 102(1), Jul 2020. URL: http://dx.doi.org/10.1103/PhysRevA.102.013115, doi:10.1103/physreva.102.013115.
https://doi.org/10.1103/PhysRevA.102.013115

[40] W. Magnus. On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math., 7:649–673, 1954. doi:10.1002/cpa.3160070404.
https://doi.org/10.1002/cpa.3160070404

[41] A. Martinez. An introduction to semiclassical and microlocal analysis, volume 994. Springer, 2002. doi:10.1007/978-1-4757-4495-8.
https://doi.org/10.1007/978-1-4757-4495-8

[42] J. Mizrahi, B. Neyenhuis, K. G. Johnson, W. C. Campbell, C. Senko, D. Hayes, and C. Monroe. Quantum control of qubits and atomic motion using ultrafast laser pulses. Applied Physics B, 114(1-2):45–61, Nov 2013. URL: http://dx.doi.org/10.1007/s00340-013-5717-6, doi:10.1007/s00340-013-5717-6.
https://doi.org/10.1007/s00340-013-5717-6

[43] M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty. Optimal control, geometry, and quantum computing. Phys. Rev. A, 73(6):062323, 2006. doi:10.1103/PhysRevA.73.062323.
https://doi.org/10.1103/PhysRevA.73.062323

[44] D. Poulin, A. Qarry, R. Somma, and F. Verstraete. Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space. Phys. Rev. Lett., 106(17):170501, 2011. doi:10.1103/PhysRevLett.106.170501.
https://doi.org/10.1103/PhysRevLett.106.170501

[45] A. Rajput, A. Roggero, and N. Wiebe. Hybridized methods for quantum simulation in the interaction picture, 2021. arXiv:2109.03308.
arXiv:2109.03308

[46] B. Şahinoğlu and R. D. Somma. Hamiltonian simulation in the low energy subspace. 2020. arXiv:2006.02660.
https://doi.org/10.1038/s41534-021-00451-w
arXiv:2006.02660

[47] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 3. Princeton Univ. Pr., 1993.

[48] E. M. Stein and R. Shakarchi. Functional analysis. Princeton University Press, 2011. doi:10.1515/9781400840557.
https://doi.org/10.1515/9781400840557

[49] Y. Su, D. W. Berry, N. Wiebe, N. Rubin, and R. Babbush. Fault-tolerant quantum simulations of chemistry in first quantization, 2021. arXiv:2105.12767, doi:10.1103/PRXQuantum.2.040332.
https://doi.org/10.1103/PRXQuantum.2.040332
arXiv:2105.12767

[50] M. Thalhammer. A fourth-order commutator-free exponential integrator for nonautonomous differential equations. SIAM Journal on Numerical Analysis, 44(2):851–864, 2006. URL: http://www.jstor.org/stable/40232777, doi:10.1137/05063042.
https://doi.org/10.1137/05063042
http://www.jstor.org/stable/40232777

[51] M. C. Tran, S.-K. Chu, Y. Su, A. M. Childs, and A. V. Gorshkov. Destructive error interference in product-formula lattice simulation. Phys. Rev. Lett., 124(22):220502, 2020. doi:10.1103/PhysRevLett.124.220502.
https://doi.org/10.1103/PhysRevLett.124.220502

[52] L. Wahlbin. Superconvergence in Galerkin finite element methods. Springer, 2006. doi:10.1007/BFb0096835.
https://doi.org/10.1007/BFb0096835

[53] D. Wecker, M. B. Hastings, N. Wiebe, B. K. Clark, C. Nayak, and M. Troyer. Solving strongly correlated electron models on a quantum computer. Phys. Rev. A, 92:062318, 2015. doi:10.1103/PhysRevA.92.062318.
https://doi.org/10.1103/PhysRevA.92.062318

[54] N. Wiebe, D. Berry, P. Høyer, and B. C. Sanders. Higher order decompositions of ordered operator exponentials. J. Phys. A, 43(6):065203, 2010. doi:10.1088/1751-8113/43/6/065203.
https://doi.org/10.1088/1751-8113/43/6/065203

[55] M. Zworski. Semiclassical Analysis. American Mathematical Society, 2012. doi:10.1090/gsm/138.
https://doi.org/10.1090/gsm/138

Cited by

[1] Yu Cao, Shi Jin, and Nana Liu, "Unifying framework for quantum simulation algorithms for time-dependent Hamiltonian dynamics", Physical Review Research 7 4, 043186 (2025).

[2] Michelle Wynne Sze, Yao Tang, Silas Dilkes, David Muñoz Ramo, Ross Duncan, and Nathan Fitzpatrick, "Hamiltonian dynamics simulation using linear combination of unitaries on an ion trap quantum computer", Quantum Science and Technology 11 1, 015023 (2026).

[3] Dong An, Jin-Peng Liu, Daochen Wang, and Qi Zhao, "Quantum Differential Equation Solvers: Limitations and Fast-Forwarding", Communications in Mathematical Physics 406 8, 189 (2025).

[4] Nhat A. Nghiem, "Efficient quantum algorithm for simulating a time-dependent Hamiltonian that commutes at different times", Physical Review A 112 4, 042603 (2025).

[5] Andrew M. Childs, Jiaqi Leng, Tongyang Li, Jin-Peng Liu, and Chenyi Zhang, "Quantum simulation of real-space dynamics", Quantum 6, 860 (2022).

[6] Osama Muhammad Raisuddin and Suvranu De, "A Review of Quantum Scientific Computing Algorithms Relevant to Computational Mechanics", Archives of Computational Methods in Engineering 33 1, 745 (2026).

[7] Alan Bidart, Prateek Vaish, Tilas Kabengele, Yaoqi Pang, Yuan Liu, and Brenda M. Rubenstein, "Quantum Computing Beyond Ground-State Electronic Structure: A Review of Progress Toward Quantum Chemistry Out of the Ground State", Annual Review of Physical Chemistry 77 1, 417 (2026).

[8] Shi Jin, Nana Liu, and Yue Yu, "Quantum circuits for the heat equation with physical boundary conditions via Schrödingerization", Journal of Computational Physics 538, 114138 (2025).

[9] Di Fang and Albert Tres Vilanova, "Observable Error Bounds of the Time-Splitting Scheme for Quantum-Classical Molecular Dynamics", SIAM Journal on Numerical Analysis 61 1, 26 (2023).

[10] Julien Zylberman, Thibault Fredon, Nuno F Loureiro, and Fabrice Debbasch, "Trotter-based quantum algorithm for solving transport equations with exponentially fewer time-steps", Quantum Science and Technology 11 2, 025015 (2026).

[11] Shi Jin, Nana Liu, and Chuwen Ma, "Quantum simulation of Maxwell’s equations via Schrödingerisation", ESAIM: Mathematical Modelling and Numerical Analysis 58 5, 1853 (2024).

[12] Tatsuhiko N. Ikeda, Hideki Kono, and Keisuke Fujii, "Measuring Trotter error and its application to precision-guaranteed Hamiltonian simulations", Physical Review Research 6 3, 033285 (2024).

[13] 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).

[14] Dong An, Andrew M. Childs, and Lin Lin, "Quantum Algorithm for Linear Non-unitary Dynamics with Near-Optimal Dependence on All Parameters", Communications in Mathematical Physics 407 1, 19 (2026).

[15] Chuwen Ma, Shi Jin, Nana Liu, Kezhen Wang, and Lei Zhang, "Schrödingerization-Based Quantum Circuits For Maxwell’s Equations With Time-Dependent Source Terms", Journal of Scientific Computing 108 2, 43 (2026).

[16] Jin-Peng Liu and Lin Lin, "Dense outputs from quantum simulations", Journal of Computational Physics 514, 113213 (2024).

[17] Shi Jin, Nana Liu, and Wei Wei, "Quantum Algorithms for Stochastic Differential Equations: A Schrödingerisation Approach", Journal of Scientific Computing 104 2, 56 (2025).

[18] Jacob Watkins, Nathan Wiebe, Alessandro Roggero, and Dean Lee, "Time-Dependent Hamiltonian Simulation Using Discrete-Clock Constructions", PRX Quantum 5 4, 040316 (2024).

[19] Shi Jin, Nana Liu, and Yue Yu, "Quantum simulation of partial differential equations: Applications and detailed analysis", Physical Review A 108 3, 032603 (2023).

[20] Osama Muhammad Raisuddin and Suvranu De, "qRLS: quantum relaxation for linear systems in finite element analysis", Engineering with Computers 40 5, 3273 (2024).

[21] Zachary Morrell, Marc Vuffray, Sidhant Misra, and Carleton Coffrin, 2024 IEEE International Conference on Quantum Computing and Engineering (QCE) 764 (2024) ISBN:979-8-3315-4137-8.

[22] Dong An, Jin-Peng Liu, and Lin Lin, "Linear Combination of Hamiltonian Simulation for Nonunitary Dynamics with Optimal State Preparation Cost", Physical Review Letters 131 15, 150603 (2023).

[23] Di Fang and Conrad Qu, "Uniform Semiclassical Observable Error Bound of Trotter–Suzuki Splitting: A Simple Algebraic Proof", SIAM Journal on Numerical Analysis 64 3, 876 (2026).

[24] Di Fang, Lin Lin, and Yu Tong, "Time-marching based quantum solvers for time-dependent linear differential equations", Quantum 7, 955 (2023).

[25] Di Fang, Diyi Liu, and Rahul Sarkar, "Time-Dependent Hamiltonian Simulation via Magnus Expansion: Algorithm and Superconvergence", Communications in Mathematical Physics 406 6, 128 (2025).

[26] Shi Jin, Xiantao Li, Nana Liu, and Yue Yu, "Quantum Simulation for Quantum Dynamics with Artificial Boundary Conditions", SIAM Journal on Scientific Computing 46 4, B403 (2024).

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

[28] J. E. Elenewski, C. M. Camara, and A. Kalev, "Prospects for NMR spectral prediction on fault-tolerant quantum computers", Physical Review Research 8 1, 013315 (2026).

[29] Pablo Antonio Moreno Casares, Modjtaba Shokrian Zini, and Juan Miguel Arrazola, "Quantum simulation of time-dependent Hamiltonians via commutator-free quasi-Magnus operators", Quantum 8, 1567 (2024).

[30] Yu Cao, Shi Jin, and Nana Liu, "Quantum simulation for time-dependent Hamiltonians—with applications to non-autonomous ordinary and partial differential equations", Journal of Physics A: Mathematical and Theoretical 58 15, 155304 (2025).

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

[32] Jiaqi Leng, Joseph Li, Yuxiang Peng, and Xiaodi Wu, "Expanding Hardware-Efficiently Manipulable Hilbert Space via Hamiltonian Embedding", Quantum 9, 1857 (2025).

[33] Di Fang, Xiaoxu Wu, and Avy Soffer, "On the Trotter Error in Many-body Quantum Dynamics with Coulomb Potentials", Communications in Mathematical Physics 407 4, 83 (2026).

[34] Yonah Borns-Weil and Di Fang, "Uniform Observable Error Bounds of Trotter Formulae for the Semiclassical Schrödinger Equation", Multiscale Modeling & Simulation 23 1, 255 (2025).

[35] 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).

[36] Shi Jin, Nana Liu, and Chuwen Ma, "On Schrödingerization-Based Quantum Algorithms for Linear Dynamical Systems with Inhomogeneous Terms", SIAM Journal on Numerical Analysis 63 4, 1861 (2025).

[37] Yonah Borns-Weil, Di Fang, and Jiaqi Zhang, "Discrete Superconvergence Analysis for Quantum Magnus Algorithms of Unbounded Hamiltonian Simulation", Communications in Mathematical Physics 407 2, 29 (2026).

[38] Hongzheng Zhao, Marin Bukov, Markus Heyl, and Roderich Moessner, "Adaptive Trotterization for Time-Dependent Hamiltonian Quantum Dynamics Using Piecewise Conservation Laws", Physical Review Letters 133 1, 010603 (2024).

[39] Rundi Lu, Hao-En Li, Zhengwei Liu, and Jin-Peng Liu, "Infinite-dimensional Extension of the Linear Combination of Hamiltonian Simulation: Theorems and Applications", arXiv:2502.19688, (2025).

[40] Kunal Sharma and Minh C. Tran, "Hamiltonian Simulation in the Interaction Picture Using the Magnus Expansion", arXiv:2404.02966, (2024).

[41] Xiantao Li, "Some error analysis for the quantum phase estimation algorithms", Journal of Physics A Mathematical General 55 32, 325303 (2022).

[42] Di Fang and Conrad Qu, "Uniform semiclassical observable error bound of Trotter-Suzuki splitting: a simple algebraic proof", arXiv:2507.02783, (2025).

[43] Osama Muhammad Raisuddin and Suvranu De, "Quantum Relaxation for Linear Systems in Finite Element Analysis", arXiv:2308.01377, (2023).

[44] Pablo Antonio Moreno Casares, Modjtaba Shokrian Zini, and Juan Miguel Arrazola, "Quantum simulation of time-dependent Hamiltonians via commutator-free quasi-Magnus operators", arXiv:2403.13889, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-07-17 07:12:00) and SAO/NASA ADS (last updated successfully 2026-07-16 17:42:04). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-07-17 07:12:00: Cannot retrieve data from ADS due to rate limitations.

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.