Multi-product formulas 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 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. Secondly, we introduce dynamic multi-product formulas with time-dependent coefficients chosen to minimize a certain efficiently computable proxy for the Trotter error. Numerical simulations suggest that the error achieved by the dynamic multi-product formulas is close to the optimal one.

翻译：多乘积公式是Trotter电路的线性组合，能以更少的Trotter步骤实现高保真度的哈密顿时间演化模拟。本文提出两项贡献以推动多乘积公式更适用于近期量子模拟。首先，我们将Childs、Su、Tran等人发展的具有对易子标度的Trotter误差理论推广至多乘积公式。该结果表明，与常规乘积公式相比，多乘积公式能在任意时间区间上将Trotter误差实现二次降低，且无需增加所需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，我们引入具有时变系数的动态多乘积公式，通过最小化某个可有效计算的Trotter误差代理量来选择系数。数值模拟表明，动态多乘积公式达到的误差接近最优值。