= First-order two-local Hamiltonian simulation
{title2=$O(M+M^3t^2/\epsilon)$}
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 $t^2\sum_{j<k}\|[H_j,H_k]\|/(2r)<M(M-1)t^2/(2r)$ for $M>1$. Taking $r=\max(1,\lceil M(M-1)t^2/(2\epsilon)\rceil)$ uses $Mr$ arbitrary two-qubit gates. At unit time and $M=O(n^2)$ this is a sufficient $O(n^6/\epsilon)$ bound; additional commutation structure can improve it.
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