Minimum Trotterization Formulas for a Time-Dependent Hamiltonian

Tatsuhiko N. Ikeda; Asir Abrar; Isaac L. Chuang; Sho Sugiura

doi:10.22331/q-2023-11-06-1168

Minimum Trotterization Formulas for a Time-Dependent Hamiltonian

Tatsuhiko N. Ikeda^1,2,3, Asir Abrar⁴, Isaac L. Chuang⁵, and Sho Sugiura^4,6

¹RIKEN Center for Quantum Computing, Wako, Saitama 351-0198, Japan
²Department of Physics, Boston University, Boston, Massachusetts 02215, USA
³Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
⁴Physics and Informatics Laboratory, NTT Research, Inc.,940 Stewart Dr., Sunnyvale, California, 94085, USA
⁵Department of Physics, Department of Electrical Engineering and Computer Science, and Co-Design Center for Quantum Advantage, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
⁶Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, 02139, MA, USA

Published:	2023-11-06, volume 7, page 1168
Eprint:	arXiv:2212.06788v4
Doi:	https://doi.org/10.22331/q-2023-11-06-1168
Citation:	Quantum 7, 1168 (2023).

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

Abstract

When a time propagator $e^{\delta t A}$ for duration $\delta t$ consists of two noncommuting parts $A=X+Y$, Trotterization approximately decomposes the propagator into a product of exponentials of $X$ and $Y$. Various Trotterization formulas have been utilized in quantum and classical computers, but much less is known for the Trotterization with the time-dependent generator $A(t)$. Here, for $A(t)$ given by the sum of two operators $X$ and $Y$ with time-dependent coefficients $A(t) = x(t) X + y(t) Y$, we develop a systematic approach to derive high-order Trotterization formulas with minimum possible exponentials. In particular, we obtain fourth-order and sixth-order Trotterization formulas involving seven and fifteen exponentials, respectively, which are no more than those for time-independent generators. We also construct another fourth-order formula consisting of nine exponentials having a smaller error coefficient. Finally, we numerically benchmark the fourth-order formulas in a Hamiltonian simulation for a quantum Ising chain, showing that the 9-exponential formula accompanies smaller errors per local quantum gate than the well-known Suzuki formula.

► BibTeX data

@article{Ikeda2023minimum,
  doi = {10.22331/q-2023-11-06-1168},
  url = {https://doi.org/10.22331/q-2023-11-06-1168},
  title = {Minimum {T}rotterization {F}ormulas for a {T}ime-{D}ependent {H}amiltonian},
  author = {Ikeda, Tatsuhiko N. and Abrar, Asir and Chuang, Isaac L. and Sugiura, Sho},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1168},
  month = nov,
  year = {2023}
}

► References

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

[2] S. Blanes and P.C. Moan. Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. Journal of Computational and Applied Mathematics, 142 (2): 313–330, 2002. https://doi.org/10.1016/S0377-0427(01)00492-7.
https://doi.org/10.1016/S0377-0427(01)00492-7

[3] S. Blanes, F. Casas, J.A. Oteo, and J. Ros. The Magnus expansion and some of its applications. Physics Reports, 470 (5): 151–238, 2009. https://doi.org/10.1016/j.physrep.2008.11.001.
https://doi.org/10.1016/j.physrep.2008.11.001

[4] Sergey Bravyi, David P. DiVincenzo, and Daniel Loss. Schrieffer–Wolff transformation for quantum many-body systems. Annals of Physics, 326 (10): 2793–2826, 2011. https://doi.org/10.1016/j.aop.2011.06.004.
https://doi.org/10.1016/j.aop.2011.06.004

[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.
https://doi.org/10.1103/PhysRevX.11.011020

[6] Etienne Forest and Ronald D. Ruth. Fourth-order symplectic integration. Physica D: Nonlinear Phenomena, 43 (1): 105–117, 1990. https://doi.org/10.1016/0167-2789(90)90019-L.
https://doi.org/10.1016/0167-2789(90)90019-L

[7] Naomichi Hatano and Masuo Suzuki. Finding Exponential Product Formulas of Higher Orders, pages 37–68. Springer Berlin Heidelberg, Berlin, Heidelberg, 2005. ISBN 978-3-540-31515-5. https://doi.org/10.1007/11526216_2.
https://doi.org/10.1007/11526216_2

[8] J Huyghebaert and H De Raedt. Product formula methods for time-dependent Schrodinger problems. Journal of Physics A: Mathematical and General, 23 (24): 5777, 1990. https://doi.org/10.1088/0305-4470/23/24/019.
https://doi.org/10.1088/0305-4470/23/24/019

[9] Tatsuhiko N. Ikeda and Keisuke Fujii. Trotter24: A precision-guaranteed adaptive stepsize trotterization for hamiltonian simulations. arXiv:2307.05406, 2023. https://doi.org/10.48550/arXiv.2307.05406.
https://doi.org/10.48550/arXiv.2307.05406
arXiv:2307.05406

[10] A Iserles, A Marthinsen, and S P Nørsett. On the Implementation of the Method of Magnus Series for Linear Differential Equations. BIT Numerical Mathematics, 39 (2): 281–304, 1999. https://doi.org/10.1023/A:1022393913721.
https://doi.org/10.1023/A:1022393913721

[11] Tobias Jahnke and Christian Lubich. Error Bounds for Exponential Operator Splittings. BIT Numerical Mathematics, 40 (4): 735–744, 2000. https://doi.org/10.1023/A:1022396519656.
https://doi.org/10.1023/A:1022396519656

[12] Tosio Kato. On the Trotter-Lie Product Formula. Proceedings of the Japan Academy, 50 (9): 694–698, 1974. https://doi.org/10.3792/pja/1195518790.
https://doi.org/10.3792/pja/1195518790

[13] 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.
https://doi.org/10.1103/PhysRevLett.118.010501

[14] Guang Hao Low and Nathan Wiebe. Hamiltonian simulation in the interaction picture. arXiv:1805.00675, 2018. https://doi.org/10.48550/arXiv.1805.00675.
https://doi.org/10.48550/arXiv.1805.00675
arXiv:1805.00675

[15] John M. Martyn, Zane M. Rossi, Andrew K. Tan, and Isaac L. Chuang. Grand Unification of Quantum Algorithms. PRX Quantum, 2: 040203, 2021. https://doi.org/10.1103/PRXQuantum.2.040203.
https://doi.org/10.1103/PRXQuantum.2.040203

[16] Kaoru Mizuta and Keisuke Fujii. Optimal Hamiltonian simulation for time-periodic systems. Quantum, 7: 962, 2023. https://doi.org/10.22331/q-2023-03-28-962.
https://doi.org/10.22331/q-2023-03-28-962

[17] I P Omelyan, I M Mryglod, and R Folk. Optimized Forest–Ruth- and Suzuki-like algorithms for integration of motion in many-body systems. Computer Physics Communications, 146 (2): 188–202, 2002. https://doi.org/10.1016/S0010-4655(02)00451-4.
https://doi.org/10.1016/S0010-4655(02)00451-4

[18] Johann Ostmeyer. Optimised trotter decompositions for classical and quantum computing. Journal of Physics A: Mathematical and Theoretical, 56 (28): 285303, 2023. https://doi.org/10.1088/1751-8121/acde7a.
https://doi.org/10.1088/1751-8121/acde7a

[19] David Poulin, Angie Qarry, Rolando Somma, and Frank Verstraete. Quantum Simulation of Time-Dependent Hamiltonians and the Convenient Illusion of Hilbert Space. Phys. Rev. Lett., 106: 170501, 2011. https://doi.org/10.1103/PhysRevLett.106.170501.
https://doi.org/10.1103/PhysRevLett.106.170501

[20] J. R. Schrieffer and P. A. Wolff. Relation between the Anderson and Kondo Hamiltonians. Phys. Rev., 149: 491–492, 1966. https://doi.org/10.1103/PhysRev.149.491.
https://doi.org/10.1103/PhysRev.149.491

[21] Andrew T Sornborger, Phillip Stancil, and Michael R Geller. Toward prethreshold gate-based quantum simulation of chemical dynamics: using potential energy surfaces to simulate few-channel molecular collisions. Quantum Information Processing, 17 (5): 106, 2018. https://doi.org/10.1007/s11128-018-1878-x.
https://doi.org/10.1007/s11128-018-1878-x

[22] Masuo Suzuki. Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Physics Letters A, 146 (6): 319–323, 1990. https://doi.org/10.1016/0375-9601(90)90962-N.
https://doi.org/10.1016/0375-9601(90)90962-N

[23] Masuo Suzuki. General Decomposition Theory of Ordered Exponentials. Proceedings of the Japan Academy, Series B, 69 (7): 161–166, 1993. https://doi.org/10.2183/pjab.69.161.
https://doi.org/10.2183/pjab.69.161

[24] H. F. Trotter. On the Product of Semi-Groups of Operators. Proceedings of the American Mathematical Society, 10 (4): 545–551, 1959. https://doi.org/10.2307/2033649.
https://doi.org/10.2307/2033649

[25] Jacob Watkins, Nathan Wiebe, Alessandro Roggero, and Dean Lee. Time-dependent Hamiltonian Simulation Using Discrete Clock Constructions. arXiv:2203.11353, 2022. https://doi.org/10.48550/arXiv.2203.11353.
https://doi.org/10.48550/arXiv.2203.11353
arXiv:2203.11353

[26] Nathan Wiebe, Dominic Berry, Peter Høyer, and Barry C Sanders. Higher order decompositions of ordered operator exponentials. Journal of Physics A: Mathematical and Theoretical, 43 (6): 065203, jan 2010. https://doi.org/10.1088/1751-8113/43/6/065203.
https://doi.org/10.1088/1751-8113/43/6/065203

[27] Haruo Yoshida. Construction of higher order symplectic integrators. Physics Letters A, 150 (5): 262–268, 1990. https://doi.org/10.1016/0375-9601(90)90092-3.
https://doi.org/10.1016/0375-9601(90)90092-3

[28] Hongzheng Zhao, Marin Bukov, Markus Heyl, and Roderich Moessner. Making trotterization adaptive and energy-self-correcting for nisq devices and beyond. PRX Quantum, 4: 030319, 2023a. https://doi.org/10.1103/PRXQuantum.4.030319.
https://doi.org/10.1103/PRXQuantum.4.030319

[29] Hongzheng Zhao, Marin Bukov, Markus Heyl, and Roderich Moessner. Adaptive trotterization for time-dependent hamiltonian quantum dynamics using instantaneous conservation laws. arXiv:2307.10327, 2023b. https://doi.org/10.48550/arXiv.2307.10327.
https://doi.org/10.48550/arXiv.2307.10327
arXiv:2307.10327

Cited by

[1] Tim Möbus, "On Strong Bounds for Trotter and Zeno Product Formulas with Bosonic Applications", Quantum 8, 1424 (2024).

[2] Andrew D. King, Alberto Nocera, Marek M. Rams, Jacek Dziarmaga, Roeland Wiersema, William Bernoudy, Jack Raymond, Nitin Kaushal, Niclas Heinsdorf, Richard Harris, Kelly Boothby, Fabio Altomare, Mohsen Asad, Andrew J. Berkley, Martin Boschnak, Kevin Chern, Holly Christiani, Samantha Cibere, Jake Connor, Martin H. Dehn, Rahul Deshpande, Sara Ejtemaee, Pau Farre, Kelsey Hamer, Emile Hoskinson, Shuiyuan Huang, Mark W. Johnson, Samuel Kortas, Eric Ladizinsky, Trevor Lanting, Tony Lai, Ryan Li, Allison J. R. MacDonald, Gaelen Marsden, Catherine C. McGeoch, Reza Molavi, Travis Oh, Richard Neufeld, Mana Norouzpour, Joel Pasvolsky, Patrick Poitras, Gabriel Poulin-Lamarre, Thomas Prescott, Mauricio Reis, Chris Rich, Mohammad Samani, Benjamin Sheldan, Anatoly Smirnov, Edward Sterpka, Berta Trullas Clavera, Nicholas Tsai, Mark Volkmann, Alexander M. Whiticar, Jed D. Whittaker, Warren Wilkinson, Jason Yao, T. J. Yi, Anders W. Sandvik, Gonzalo Alvarez, Roger G. Melko, Juan Carrasquilla, Marcel Franz, and Mohammad H. Amin, "Beyond-classical computation in quantum simulation", Science 388 6743, 199 (2025).

[3] Christopher F. Kane, Siddharth Hariprakash, Neel S. Modi, Michael Kreshchuk, and Christian W Bauer, "Block encoding bosons by signal processing", Quantum 9, 1747 (2025).

[4] 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).

[5] 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).

[6] Nacer Eddine Belaloui, Abdellah Tounsi, Abdelmouheymen Rabah Khamadja, Mohamed Messaoud Louamri, Achour Benslama, David E. Bernal Neira, and Mohamed Taha Rouabah, "Ground-State Energy Estimation on Current Quantum Hardware through the Variational Quantum Eigensolver: A Practical Study", Journal of Chemical Theory and Computation 21 14, 6777 (2025).

[7] Patrick Draper, Luis Hidalgo, and Anton Ilderton, "Hamiltonian truncation and quantum simulation of strong-field QED beyond tree level", Physical Review D 113 5, 056010 (2026).

[8] Luis Hidalgo and Patrick Draper, "Quantum simulations for strong-field QED", Physical Review D 109 7, 076004 (2024).

[9] 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).

[10] Vu Tuan Hai, Nguyen Tan Viet, Jesus Urbaneja, Nguyen Vu Linh, Lan Nguyen Tran, and Le Bin Ho, "Multi-target quantum compilation algorithm", Machine Learning: Science and Technology 5 4, 045057 (2024).

[11] 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).

[12] 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).

[13] Pooja Siwach, Kaytlin Harrison, and A. Baha Balantekin, "Collective neutrino oscillations on a quantum computer with hybrid quantum-classical algorithm", Physical Review D 108 8, 083039 (2023).

[14] Nacer Eddine Belaloui, Abdellah Tounsi, Rabah Abdelmouheymen Khamadja, Mohamed Messaoud Louamri, Achour Benslama, David E. Bernal Neira, and Mohamed Taha Rouabah, "Ground State Energy Estimation on Current Quantum Hardware Through The Variational Quantum Eigensolver: A Comprehensive Study", arXiv:2412.02606, (2024).

[15] Shuo Zhou, Zhaokai Pan, Weiyuan Gong, and Tongyang Li, "Time-Dependent Hamiltonian Simulation in the Low-Energy Subspace", arXiv:2601.01550, (2026).

[16] 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 2026-08-09 03:30:53) and SAO/NASA ADS (last updated successfully 2026-08-09 03:30:54). 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.