Product formula approximations of the time-evolution operator on quantum computers are of great interest due to their simplicity, and good scaling with system size by exploiting commutativity between Hamiltonian terms. However, product formulas exhibit poor scaling with the time $t$ and error $epsilon$ of simulation as the gate cost of a single step scales exponentially with the order $m$ of accuracy. We introduce well-conditioned multiproduct formulas, which are a linear combination of product formulas, where a single step has polynomial cost $mathcal{O}(m^2log{(m)})$ and succeeds with probability $Omega(1/operatorname{log}^2{(m)})$. Our multiproduct formulas imply a simple and generic simulation algorithm that simultaneously exploits commutativity in arbitrary systems and has a worst-case cost $mathcal{O}(tlog^{2}{(t/epsilon)})$ which is optimal up to poly-logarithmic factors. In contrast, prior Trotter and post-Trotter Hamiltonian simulation algorithms realize only one of these two desirable features. A key technical result of independent interest is our solution to a conditioning problem in previous multiproduct formulas that amplified numerical errors by $e^{Omega(m)}$ in the classical setting, and led to a vanishing success probability $e^{-Omega(m)}$ in the quantum setting.
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