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

TL;DR

This paper analyzes the efficiency of multi-product formulas (MPF) for Hamiltonian simulation, addressing the limitations of previous error bounds and demonstrating that MPF can achieve system-size scaling comparable to Trotterization while maintaining polylogarithmic error dependence.

Contribution

The paper introduces an alternative error bound for MPF that fully exploits locality and commutator scaling, establishing its cost as comparable to Trotterization in system size with polylogarithmic error scaling.

Findings

Proves MPF achieves Trotterization-like system-size scaling.

Derives an error bound that exploits locality and commutator structure.

Shows MPF maintains polylogarithmic error scaling with improved accuracy.

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.