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

Kaoru Mizuta

doi:10.22331/q-2026-01-19-1974

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

Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo 113-8656, Japan
Photon Science Center, Graduate School of Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo 113-8656, Japan
RIKEN Center for Quantum Computing (RQC), Hirosawa 2-1, Wako, Saitama 351-0198, Japan

Published:	2026-01-19, volume 10, page 1974
Editor:	Tongyang Li
Eprint:	arXiv:2507.06557v4
Doi:	https://doi.org/10.22331/q-2026-01-19-1974
Citation:	Quantum 10, 1974 (2026).

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Abstract

A 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}
}

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Cited by

[1] Tim Möbus, "Multi-product Zeno effect achieving higher order convergence rates", Quantum 10, 2148 (2026).

[2] 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).

[3] Kaoru Mizuta, Tatsuhiko N. Ikeda, and Keisuke Fujii, "Explicit error bounds with commutator scaling for time-dependent product and multi-product formulas", arXiv:2410.14243, (2024).

[4] Di Fang, Diyi Liu, and Shuchen Zhu, "High-order Magnus Expansion for Hamiltonian Simulation", arXiv:2509.06054, (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 03:06:37) and SAO/NASA ADS (last updated successfully 2026-08-12 03:06:38). The list may be incomplete as not all publishers provide suitable and complete citation data.

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