Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation

• URL: http://arxiv.org/abs/2306.12569v2
• Date: Fri, 9 Feb 2024 13:51:14 GMT
• Title: Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation
• Authors: Sergiy Zhuk, Niall Robertson and Sergey Bravyi
• Abstract summary: We extend the theory of Trotter error with commutator scaling to multi-product formulas. 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.
• Score: 3.2995359570845912
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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.

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