First-order two-local Hamiltonian simulation ($O(M+M^3{t}^2/ϵ)$)

For a sum of $M$ Hermitian two-local terms of norm below one, repeat a first-order Lie-Trotter product formula step $r$ times. The first-order unitary product-formula error bound gives total error at most $t^2{\sum }_{j<k}\Vert [H_j,H_k]\Vert /(2r)<M(M-1)t^2/(2r)$ for $M>1$. Taking $r=\max (1,\lceil M(M-1)t^2/(2ϵ)\rceil )$ uses $Mr$ arbitrary two-qubit gates. At unit time and $M=O(n^2)$ this is a sufficient $O(n^6/ϵ)$ bound; additional commutation structure can improve it.