Multi-product formulas (MPF) are linear combinations of Trotter circuits offering high-quality simulation of Hamiltonian time evolution with fewer Trotter steps. Here we report two contributions aimed at making multi-product formulas more viable for near-term quantum simulations. First, we extend the theory of Trotter error with commutator scaling developed by Childs, Su, Tran et al. to multi-product formulas. Our result implies that multi-product formulas can achieve a quadratic reduction of Trotter error in 1-norm (nuclear norm) on arbitrary time intervals compared with the regular product formulas without increasing the required circuit depth or qubit connectivity. The number of circuit repetitions grows only by a constant factor. Second, we introduce dynamic multi-product formulas with time-dependent coefficients chosen to minimize a certain efficiently computable proxy for the Trotter error. We use a minimax estimation method to make dynamic multi-product formulas robust to uncertainty from algorithmic errors, sampling and hardware noise. We call this method Minimax MPF and we provide a rigorous bound on its error.

翻译：多乘积公式（MPF）是Trotter电路的线性组合，能以较少Trotter步数实现哈密顿时间演化的高质量模拟。本文提出两项贡献，旨在使多乘积公式更适用于近期量子模拟。首先，我们将Childs、Su、Tran等人发展的基于交换子标度的Trotter误差理论推广至多乘积公式。结果表明，相较于常规乘积公式，多乘积公式可在任意时间区间内将Trotter误差的1-范数（核范数）降低二次量级，且无需增加电路深度或量子比特连通性。电路重复次数仅增加常数因子。其次，我们引入动态多乘积公式，其系数随时间变化，通过最小化某个高效可计算的Trotter误差代理量实现优化。我们采用极小极大估计方法，使动态多乘积公式对算法误差、采样噪声及硬件噪声具有鲁棒性，并将该方法命名为Minimax MPF，同时给出其误差的严格界。