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

TL;DR

This paper advances Hamiltonian simulation by extending Trotter error bounds to multi-product formulas and introducing dynamic, error-robust formulas with optimized coefficients for near-term quantum computers.

Contribution

It extends Trotter error theory to multi-product formulas and introduces Minimax MPF, a dynamic, error-minimizing approach for improved quantum simulation accuracy.

Findings

Quadratic Trotter error reduction with no increased circuit depth

Dynamic MPF with time-dependent coefficients improves robustness

Rigorous error bounds for Minimax MPF

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.