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

• URL: http://arxiv.org/abs/2507.06557v2
• Date: Thu, 31 Jul 2025 08:13:36 GMT
• Title: On the commutator scaling in Hamiltonian simulation with multi-product formulas
• Authors: Kaoru Mizuta,
• Abstract summary: We show an alternative commutator-scaling error of MPF and derive its size-efficient cost properly inheriting the advantage in Trotterization.<n>We prove that Hamiltonian simulation by MPF certainly achieves the cost whose system-size dependence is as large as Trotterization.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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, arXiv:1907.11679 (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, arXiv:2403.08922 (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.

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