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

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 / ε by combining Trotterization and the linear combination of unitaries (LCU). It achieves poly-logarithmic cost in 1 / ε 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 p o l y l o g ( 1 / ε ) -scaling like the LCU. Our results will provide improved or accurate error and cost also for various algorithms using interpolation or extrapolation of Trotterization.