  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.
