The well-conditioned multi-product formula (MPF), proposed by [Low, Kliuchnikov, and Wiebe, 2019], is a simple high-order time-independent Hamiltonian simulation algorithm that implements a linear combination of standard product formulas of low order. While the MPF aims to simultaneously exploit commutator scaling among Hamiltonians and achieve near-optimal time and precision dependence, its lack of a rigorous error bound on the nested commutators renders its practical advantage ambiguous. In this work, we conduct a rigorous complexity analysis of the well-conditioned MPF, demonstrating explicit commutator scaling and near-optimal time and precision dependence at the same time. Using our improved complexity analysis, we present several applications of practical interest where the MPF based on a second-order product formula can achieve a polynomial speedup in both system size and evolution time, as well as an exponential speedup in precision, compared to second-order and even higher-order product formulas. Compared to post-Trotter methods, the MPF based on a second-order product formula can achieve polynomially better scaling in system size, with only poly-logarithmic overhead in evolution time and precision.

翻译：由[Low, Kliuchnikov和Wiebe, 2019]提出的良态多乘积公式(MPF)是一种简单的高阶时间无关哈密顿量模拟算法，它实现了低阶标准乘积公式的线性组合。尽管MPF旨在同时利用哈密顿量之间的对易子标度并实现近最优的时间与精度依赖性，但其嵌套对易子误差界的缺失使其实际优势尚不明确。在本工作中，我们对良态MPF进行了严格的复杂度分析，同时展示了显式的对易子标度以及近最优的时间与精度依赖性。利用改进的复杂度分析，我们提出了若干实际应用场景：基于二阶乘积公式的MPF可在系统规模和演化时间上实现多项式加速，并在精度上实现指数加速，其性能超越二阶乃至更高阶乘积公式。与后Trotter方法相比，基于二阶乘积公式的MPF可在系统规模上实现多项式量级的更优标度，且仅产生关于演化时间和精度的多对数开销。