quantum-computing

On the commutator scaling in Hamiltonian simulation with multi-product formulas

Kaoru Mizuta

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⚡ Quantum Brief

Kaoru Mizuta’s January 2026 study resolves a critical gap in multi-product formula (MPF) Hamiltonian simulation, proving its efficiency in system size N while maintaining polylogarithmic error scaling in 1/ε. Previous MPF error bounds relied on unbounded nested commutators, eliminating locality benefits and prohibiting size-efficient scaling—a flaw in prior analyses like Aftab et al. (2024). The paper introduces a truncated commutator framework via Floquet-Magnus expansion, restoring locality advantages and matching Trotterization’s N-scaling without sacrificing LCU’s polylog(1/ε) error suppression. This breakthrough ensures MPF achieves optimal cost in both N and 1/ε, validating its promise for near-term quantum simulators and algorithms using Trotter interpolation/extrapolation. The work directly impacts quantum chemistry, lattice models, and error-mitigated simulations by providing tighter, practical error bounds for hardware implementations.

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AbstractA multi-product formula (MPF) is a promising approach for Hamiltonian simulation efficiently both in the system size $N$ and the inverse allowable error $1/\varepsilon$ by combining Trotterization and the linear combination of unitaries (LCU). It achieves poly-logarithmic cost in $1/\varepsilon$ like LCU [G. H. Low, V. Kliuchnikov, N. Wiebe, (2019)]. The efficiency in $N$ is expected to come from the commutator scaling in Trotterization, and this appears to be confirmed by the error bound of MPF expressed by nested commutators [J. Aftab, D. An, K. Trivisa, (2024)]. However, we point out that the efficiency of MPF in the system size $N$ is not exactly resolved yet in that the present error bound expressed by nested commutators is incompatible with the size-efficient complexity reflecting the commutator scaling. The problem is that $q$-fold nested commutators with arbitrarily large $q$ are involved in their requirement and error bound. The benefit of commutator scaling by locality is absent, and the cost efficient in $N$ becomes prohibited in general. In this paper, we show an alternative commutator-scaling error of MPF and derive its size-efficient cost properly inheriting the advantage in Trotterization. The requirement and the error bound in our analysis, derived by techniques from the Floquet-Magnus expansion, have a certain truncation order in the nested commutators and can fully exploit the locality. We prove that Hamiltonian simulation by MPF certainly achieves the cost whose system-size dependence is as large as Trotterization while keeping the $\mathrm{polylog}(1/\varepsilon)$-scaling like the LCU. Our results will provide improved or accurate error and cost also for various algorithms using interpolation or extrapolation of Trotterization.► BibTeX data@article{Mizuta2026commutatorscalingin, doi = {10.22331/q-2026-01-19-1974}, url = {https://doi.org/10.22331/q-2026-01-19-1974}, title = {On the commutator scaling in {H}amiltonian simulation with multi-product formulas}, author = {Mizuta, Kaoru}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1974}, month = jan, year = {2026} }► References [1] Masuo Suzuki. ``General theory of fractal path integrals with applications to many‐body theories and statistical physics''. J. Math. Phys. 32, 400–407 (1991). https://doi.org/10.1063/1.529425 [2] Seth Lloyd. ``Universal Quantum Simulators''. Science 273, 1073–1078 (1996). https://doi.org/10.1126/science.273.5278.1073 [3] R Somma. ``A trotter-suzuki approximation for lie groups with applications to hamiltonian simulation''. Journal of Mathematical Physics 57, 062202 (2015). https://doi.org/10.1063/1.4952761 [4] Andrew M. Childs and Yuan Su. ``Nearly optimal lattice simulation by product formulas''. Phys. Rev. Lett. 123, 050503 (2019). https://doi.org/10.1103/PhysRevLett.123.050503 [5] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. ``Theory of trotter error with commutator scaling''. Phys. Rev. X 11, 011020 (2021). https://doi.org/10.1103/PhysRevX.11.011020 [6] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. ``Simulating Hamiltonian Dynamics with a Truncated Taylor Series''. Phys. Rev. Lett. 114, 090502 (2015). https://doi.org/10.1103/PhysRevLett.114.090502 [7] Guang Hao Low and Isaac L Chuang. ``Optimal Hamiltonian Simulation by Quantum Signal Processing''. Phys. Rev. Lett. 118, 010501 (2017). https://doi.org/10.1103/PhysRevLett.118.010501 [8] Guang Hao Low and Isaac L Chuang. ``Hamiltonian simulation by qubitization''. Quantum 3, 163 (2019). https://doi.org/10.22331/q-2019-07-12-163 [9] András Gilyén, Yuan Su, Guang Hao Low, and Nathan 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. STOC 2019 (2019). https://doi.org/10.1145/3313276.3316366 [10] Andrew M. Childs and Nathan Wiebe. ``Hamiltonian simulation using linear combinations of unitary operations''. Quantum Info. Comput. 12, 901–924 (2012). https://doi.org/10.26421/qic12.11-12 [11] Guang Hao Low, Vadym Kliuchnikov, and Nathan Wiebe. ``Well-conditioned multiproduct Hamiltonian simulation'' (2019). arXiv:1907.11679. arXiv:1907.11679 [12] Paul K. Faehrmann, Mark Steudtner, Richard Kueng, Maria Kieferova, and Jens Eisert. ``Randomizing multi-product formulas for Hamiltonian simulation''. Quantum 6, 806 (2022). https://doi.org/10.22331/q-2022-09-19-806 [13] Almudena Carrera Vazquez, Ralf Hiptmair, and Stefan Woerner. ``Enhancing the quantum linear systems algorithm using Richardson extrapolation''. ACM Transactions on Quantum Computing 3, 1–37 (2022). https://doi.org/10.1145/3490631 [14] 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). https://doi.org/10.22331/q-2023-07-25-1067 [15] Sergiy Zhuk, Niall F. Robertson, and Sergey Bravyi. ``Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation''. Phys. Rev. Res. 6, 033309 (2024). https://doi.org/10.1103/PhysRevResearch.6.033309 [16] Junaid Aftab, Dong An, and Konstantina Trivisa. ``Multi-product Hamiltonian simulation with explicit commutator scaling'' (2024). arXiv:2403.08922. arXiv:2403.08922 [17] Dmitry A Abanin, Wojciech De Roeck, and François Huveneers. ``Exponentially Slow Heating in Periodically Driven Many-Body Systems''. Phys. Rev. Lett. 115, 256803 (2015). https://doi.org/10.1103/PhysRevLett.115.256803 [18] Tomotaka Kuwahara, Takashi Mori, and Keiji Saito. ``Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems''. Ann. Phys. 367, 96–124 (2016). https://doi.org/10.1016/j.aop.2016.01.012 [19] Takashi Mori, Tomotaka Kuwahara, and Keiji Saito. ``Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems''. Phys. Rev. Lett. 116, 120401 (2016). https://doi.org/10.1103/PhysRevLett.116.120401 [20] Dmitry A Abanin, Wojciech De Roeck, Wen Wei Ho, and François Huveneers. ``Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems''. Phys. Rev. B 95, 014112 (2017). https://doi.org/10.1103/PhysRevB.95.014112 [21] Dmitry Abanin, Wojciech De Roeck, Wen Wei Ho, and François Huveneers. ``A rigorous theory of Many-Body prethermalization for periodically driven and closed quantum systems''. Commun. Math. Phys. 354, 809–827 (2017). https://doi.org/10.1007/s00220-017-2930-x [22] Suguru Endo, Qi Zhao, Ying Li, Simon Benjamin, and Xiao Yuan. ``Mitigating algorithmic errors in a Hamiltonian simulation''. Phys. Rev. A 99, 012334 (2019). https://doi.org/10.1103/PhysRevA.99.012334 [23] Gumaro Rendon. ``All you need is Trotter'' (2023). arXiv:2311.01533. arXiv:2311.01533 [24] Gumaro Rendon, Jacob Watkins, and Nathan Wiebe. ``Improved accuracy for Trotter simulations using Chebyshev interpolation''. Quantum 8, 1266 (2024). https://doi.org/10.22331/q-2024-02-26-1266 [25] James D. Watson and Jacob Watkins. ``Exponentially Reduced Circuit Depths Using Trotter Error Mitigation''. PRX Quantum 6, 030325 (2025). https://doi.org/10.1103/kw39-yxq5 [26] James D. Watson. ``Randomly Compiled Quantum Simulation with Exponentially Reduced Circuit Depths'' (2024). arXiv:2411.04240. arXiv:2411.04240 [27] Shantanav Chakraborty, Soumyabrata Hazra, Tongyang Li, Changpeng Shao, Xinzhao Wang, and Yuxin Zhang. ``Quantum singular value transformation without block encodings: Near-optimal complexity with minimal ancilla'' (2025). arXiv:2504.02385. arXiv:2504.02385 [28] Taozhi Guo, Gumaro Rendon, and Rutuja Kshirsagar. ``Canonical Partition Function on a Quantum Computer through Trotter Interpolation'' (2025). arXiv:2506.09318. arXiv:2506.09318 [29] Kaoru Mizuta and Tomotaka Kuwahara. ``Trotterization is Substantially Efficient for Low-Energy States''. Phys. Rev. Lett. 135, 130602 (2025). https://doi.org/10.1103/q87n-5xhz [30] Siu A Chin. ``Multi-product splitting and Runge-Kutta-Nyström integrators''. Celest. Mech. Dyn. Astron. 106, 391–406 (2010). https://doi.org/10.1007/s10569-010-9255-9 [31] Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. ``Quantum amplitude amplification and estimation''. Quantum Computation and InformationPage 53–74 (2002). https://doi.org/10.1090/conm/305/05215 [32] S Blanes, F Casas, J A Oteo, and J Ros. ``The Magnus expansion and some of its applications''. Phys. Rep. 470, 151–238 (2009). https://doi.org/10.1016/j.physrep.2008.11.001 [33] Achilleas Lazarides, Arnab Das, and Roderich Moessner. ``Equilibrium states of generic quantum systems subject to periodic driving''. Phys. Rev. E 90, 012110 (2014). https://doi.org/10.1103/PhysRevE.90.012110 [34] Luca D’Alessio and Marcos Rigol. ``Long-time behavior of isolated periodically driven interacting lattice systems''. Phys. Rev. X 4, 041048 (2014). https://doi.org/10.1103/PhysRevX.4.041048 [35] Kunal Sharma and Minh C Tran. ``Hamiltonian Simulation in the Interaction Picture Using the Magnus Expansion'' (2024). arXiv:2404.02966. arXiv:2404.02966 [36] Ana Arnal, Fernando Casas, and Cristina Chiralt. ``A Note on the Baker–Campbell–Hausdorff Series in Terms of Right-Nested Commutators''. Mediterranean Journal of Mathematics 18, 53 (2021). https://doi.org/10.1007/s00009-020-01681-6 [37] Jeongwan Haah, Matthew B Hastings, Robin Kothari, and Guang Hao Low. ``Quantum algorithm for simulating real time evolution of lattice Hamiltonians''. SIAM J. Comput.Pages FOCS18–250–FOCS18–284 (2021). https://doi.org/10.1137/18M1231511 [38] Elliott H Lieb and Derek W Robinson. ``The finite group velocity of quantum spin systems''. Commun. Math. Phys. 28, 251–257 (1972). https://doi.org/10.1007/BF01645779 [39] Minh C. Tran, Andrew Y. Guo, Yuan Su, James R. Garrison, Zachary Eldredge, Michael Foss-Feig, Andrew M. Childs, and Alexey V. Gorshkov. ``Locality and digital quantum simulation of power-law interactions''. Phys. Rev. X 9, 031006 (2019). https://doi.org/10.1103/PhysRevX.9.031006 [40] Jacob Watkins, Nathan Wiebe, Alessandro Roggero, and Dean Lee. ``Time-Dependent Hamiltonian Simulation Using Discrete-Clock Constructions''. PRX Quantum 5, 040316 (2024). https://doi.org/10.1103/PRXQuantum.5.040316 [41] Kaoru Mizuta, Tatsuhiko N. Ikeda, and Keisuke Fujii. ``Explicit error bounds with commutator scaling for time-dependent product and multi-product formulas'' (2024). arXiv:2410.14243. arXiv:2410.14243 [42] Yu Cao, Shi Jin, and Nana Liu. ``Unifying framework for quantum simulation algorithms for time-dependent hamiltonian dynamics''. Phys. Rev. Res. 7, 043186 (2025). https://doi.org/10.1103/fkh5-b669Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-19 17:00:23: Could not fetch cited-by data for 10.22331/q-2026-01-19-1974 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-19 17:00:23: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractA multi-product formula (MPF) is a promising approach for Hamiltonian simulation efficiently both in the system size $N$ and the inverse allowable error $1/\varepsilon$ by combining Trotterization and the linear combination of unitaries (LCU). It achieves poly-logarithmic cost in $1/\varepsilon$ like LCU [G. H. Low, V. Kliuchnikov, N. Wiebe, (2019)]. The efficiency in $N$ is expected to come from the commutator scaling in Trotterization, and this appears to be confirmed by the error bound of MPF expressed by nested commutators [J. Aftab, D. An, K. Trivisa, (2024)]. However, we point out that the efficiency of MPF in the system size $N$ is not exactly resolved yet in that the present error bound expressed by nested commutators is incompatible with the size-efficient complexity reflecting the commutator scaling. The problem is that $q$-fold nested commutators with arbitrarily large $q$ are involved in their requirement and error bound. The benefit of commutator scaling by locality is absent, and the cost efficient in $N$ becomes prohibited in general. In this paper, we show an alternative commutator-scaling error of MPF and derive its size-efficient cost properly inheriting the advantage in Trotterization. The requirement and the error bound in our analysis, derived by techniques from the Floquet-Magnus expansion, have a certain truncation order in the nested commutators and can fully exploit the locality. We prove that Hamiltonian simulation by MPF certainly achieves the cost whose system-size dependence is as large as Trotterization while keeping the $\mathrm{polylog}(1/\varepsilon)$-scaling like the LCU. Our results will provide improved or accurate error and cost also for various algorithms using interpolation or extrapolation of Trotterization.► BibTeX data@article{Mizuta2026commutatorscalingin, doi = {10.22331/q-2026-01-19-1974}, url = {https://doi.org/10.22331/q-2026-01-19-1974}, title = {On the commutator scaling in {H}amiltonian simulation with multi-product formulas}, author = {Mizuta, Kaoru}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1974}, month = jan, year = {2026} }► References [1] Masuo Suzuki. ``General theory of fractal path integrals with applications to many‐body theories and statistical physics''. J. Math. Phys. 32, 400–407 (1991). https://doi.org/10.1063/1.529425 [2] Seth Lloyd. ``Universal Quantum Simulators''. Science 273, 1073–1078 (1996). https://doi.org/10.1126/science.273.5278.1073 [3] R Somma. ``A trotter-suzuki approximation for lie groups with applications to hamiltonian simulation''. Journal of Mathematical Physics 57, 062202 (2015). https://doi.org/10.1063/1.4952761 [4] Andrew M. Childs and Yuan Su. ``Nearly optimal lattice simulation by product formulas''. Phys. Rev. Lett. 123, 050503 (2019). https://doi.org/10.1103/PhysRevLett.123.050503 [5] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu. ``Theory of trotter error with commutator scaling''. Phys. Rev. X 11, 011020 (2021). https://doi.org/10.1103/PhysRevX.11.011020 [6] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. ``Simulating Hamiltonian Dynamics with a Truncated Taylor Series''. Phys. Rev. Lett. 114, 090502 (2015). https://doi.org/10.1103/PhysRevLett.114.090502 [7] Guang Hao Low and Isaac L Chuang. ``Optimal Hamiltonian Simulation by Quantum Signal Processing''. Phys. Rev. Lett. 118, 010501 (2017). https://doi.org/10.1103/PhysRevLett.118.010501 [8] Guang Hao Low and Isaac L Chuang. ``Hamiltonian simulation by qubitization''. Quantum 3, 163 (2019). https://doi.org/10.22331/q-2019-07-12-163 [9] András Gilyén, Yuan Su, Guang Hao Low, and Nathan 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. STOC 2019 (2019). https://doi.org/10.1145/3313276.3316366 [10] Andrew M. Childs and Nathan Wiebe. ``Hamiltonian simulation using linear combinations of unitary operations''. Quantum Info. Comput. 12, 901–924 (2012). https://doi.org/10.26421/qic12.11-12 [11] Guang Hao Low, Vadym Kliuchnikov, and Nathan Wiebe. ``Well-conditioned multiproduct Hamiltonian simulation'' (2019). arXiv:1907.11679. arXiv:1907.11679 [12] Paul K. Faehrmann, Mark Steudtner, Richard Kueng, Maria Kieferova, and Jens Eisert. ``Randomizing multi-product formulas for Hamiltonian simulation''. Quantum 6, 806 (2022). https://doi.org/10.22331/q-2022-09-19-806 [13] Almudena Carrera Vazquez, Ralf Hiptmair, and Stefan Woerner. ``Enhancing the quantum linear systems algorithm using Richardson extrapolation''. ACM Transactions on Quantum Computing 3, 1–37 (2022). https://doi.org/10.1145/3490631 [14] 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). https://doi.org/10.22331/q-2023-07-25-1067 [15] Sergiy Zhuk, Niall F. Robertson, and Sergey Bravyi. ``Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation''. Phys. Rev. Res. 6, 033309 (2024). https://doi.org/10.1103/PhysRevResearch.6.033309 [16] Junaid Aftab, Dong An, and Konstantina Trivisa. ``Multi-product Hamiltonian simulation with explicit commutator scaling'' (2024). arXiv:2403.08922. arXiv:2403.08922 [17] Dmitry A Abanin, Wojciech De Roeck, and François Huveneers. ``Exponentially Slow Heating in Periodically Driven Many-Body Systems''. Phys. Rev. Lett. 115, 256803 (2015). https://doi.org/10.1103/PhysRevLett.115.256803 [18] Tomotaka Kuwahara, Takashi Mori, and Keiji Saito. ``Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems''. Ann. Phys. 367, 96–124 (2016). https://doi.org/10.1016/j.aop.2016.01.012 [19] Takashi Mori, Tomotaka Kuwahara, and Keiji Saito. ``Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems''. Phys. Rev. Lett. 116, 120401 (2016). https://doi.org/10.1103/PhysRevLett.116.120401 [20] Dmitry A Abanin, Wojciech De Roeck, Wen Wei Ho, and François Huveneers. ``Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems''. Phys. Rev. B 95, 014112 (2017). https://doi.org/10.1103/PhysRevB.95.014112 [21] Dmitry Abanin, Wojciech De Roeck, Wen Wei Ho, and François Huveneers. ``A rigorous theory of Many-Body prethermalization for periodically driven and closed quantum systems''. Commun. Math. Phys. 354, 809–827 (2017). https://doi.org/10.1007/s00220-017-2930-x [22] Suguru Endo, Qi Zhao, Ying Li, Simon Benjamin, and Xiao Yuan. ``Mitigating algorithmic errors in a Hamiltonian simulation''. Phys. Rev. A 99, 012334 (2019). https://doi.org/10.1103/PhysRevA.99.012334 [23] Gumaro Rendon. ``All you need is Trotter'' (2023). arXiv:2311.01533. arXiv:2311.01533 [24] Gumaro Rendon, Jacob Watkins, and Nathan Wiebe. ``Improved accuracy for Trotter simulations using Chebyshev interpolation''. Quantum 8, 1266 (2024). https://doi.org/10.22331/q-2024-02-26-1266 [25] James D. Watson and Jacob Watkins. ``Exponentially Reduced Circuit Depths Using Trotter Error Mitigation''. PRX Quantum 6, 030325 (2025). https://doi.org/10.1103/kw39-yxq5 [26] James D. Watson. ``Randomly Compiled Quantum Simulation with Exponentially Reduced Circuit Depths'' (2024). arXiv:2411.04240. arXiv:2411.04240 [27] Shantanav Chakraborty, Soumyabrata Hazra, Tongyang Li, Changpeng Shao, Xinzhao Wang, and Yuxin Zhang. ``Quantum singular value transformation without block encodings: Near-optimal complexity with minimal ancilla'' (2025). arXiv:2504.02385. arXiv:2504.02385 [28] Taozhi Guo, Gumaro Rendon, and Rutuja Kshirsagar. ``Canonical Partition Function on a Quantum Computer through Trotter Interpolation'' (2025). arXiv:2506.09318. arXiv:2506.09318 [29] Kaoru Mizuta and Tomotaka Kuwahara. ``Trotterization is Substantially Efficient for Low-Energy States''. Phys. Rev. Lett. 135, 130602 (2025). https://doi.org/10.1103/q87n-5xhz [30] Siu A Chin. ``Multi-product splitting and Runge-Kutta-Nyström integrators''. Celest. Mech. Dyn. Astron. 106, 391–406 (2010). https://doi.org/10.1007/s10569-010-9255-9 [31] Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. ``Quantum amplitude amplification and estimation''. Quantum Computation and InformationPage 53–74 (2002). https://doi.org/10.1090/conm/305/05215 [32] S Blanes, F Casas, J A Oteo, and J Ros. ``The Magnus expansion and some of its applications''. Phys. Rep. 470, 151–238 (2009). https://doi.org/10.1016/j.physrep.2008.11.001 [33] Achilleas Lazarides, Arnab Das, and Roderich Moessner. ``Equilibrium states of generic quantum systems subject to periodic driving''. Phys. Rev. E 90, 012110 (2014). https://doi.org/10.1103/PhysRevE.90.012110 [34] Luca D’Alessio and Marcos Rigol. ``Long-time behavior of isolated periodically driven interacting lattice systems''. Phys. Rev. X 4, 041048 (2014). https://doi.org/10.1103/PhysRevX.4.041048 [35] Kunal Sharma and Minh C Tran. ``Hamiltonian Simulation in the Interaction Picture Using the Magnus Expansion'' (2024). arXiv:2404.02966. arXiv:2404.02966 [36] Ana Arnal, Fernando Casas, and Cristina Chiralt. ``A Note on the Baker–Campbell–Hausdorff Series in Terms of Right-Nested Commutators''. Mediterranean Journal of Mathematics 18, 53 (2021). https://doi.org/10.1007/s00009-020-01681-6 [37] Jeongwan Haah, Matthew B Hastings, Robin Kothari, and Guang Hao Low. ``Quantum algorithm for simulating real time evolution of lattice Hamiltonians''. SIAM J. Comput.Pages FOCS18–250–FOCS18–284 (2021). https://doi.org/10.1137/18M1231511 [38] Elliott H Lieb and Derek W Robinson. ``The finite group velocity of quantum spin systems''. Commun. Math. Phys. 28, 251–257 (1972). https://doi.org/10.1007/BF01645779 [39] Minh C. Tran, Andrew Y. Guo, Yuan Su, James R. Garrison, Zachary Eldredge, Michael Foss-Feig, Andrew M. Childs, and Alexey V. Gorshkov. ``Locality and digital quantum simulation of power-law interactions''. Phys. Rev. X 9, 031006 (2019). https://doi.org/10.1103/PhysRevX.9.031006 [40] Jacob Watkins, Nathan Wiebe, Alessandro Roggero, and Dean Lee. ``Time-Dependent Hamiltonian Simulation Using Discrete-Clock Constructions''. PRX Quantum 5, 040316 (2024). https://doi.org/10.1103/PRXQuantum.5.040316 [41] Kaoru Mizuta, Tatsuhiko N. Ikeda, and Keisuke Fujii. ``Explicit error bounds with commutator scaling for time-dependent product and multi-product formulas'' (2024). arXiv:2410.14243. arXiv:2410.14243 [42] Yu Cao, Shi Jin, and Nana Liu. ``Unifying framework for quantum simulation algorithms for time-dependent hamiltonian dynamics''. Phys. Rev. Res. 7, 043186 (2025). https://doi.org/10.1103/fkh5-b669Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-19 17:00:23: Could not fetch cited-by data for 10.22331/q-2026-01-19-1974 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-19 17:00:23: No response from ADS or unable to decode the received json data when getting the list of citing works.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.

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