= First-order unitary product-formula error bound
For <Hermitian matrices> $H_1,\ldots,H_m$ and $t\geq0$, let $S(t)=\prod_{j=1}^m e^{-itH_j}$. Then
$$
\boxed{\left\|S(t)-e^{-it\sum_jH_j}\right\|
\leq\frac{t^2}{2}\sum_{j<l}\|[H_j,H_l]\|.}
$$
Here every norm is the <spectral norm>. A short proof starts with two summands. Differentiate $F(s)=e^{-i(t-s)(A+B)}e^{-isA}e^{-isB}$; its derivative has norm at most $\|[e^{-isA},B]\|$. Differentiating $e^{-iuA}Be^{iuA}$ and integrating yields $\|[e^{-isA},B]\|\leq s\|[A,B]\|$, because <unitary operators> preserve the norm. Integrating $s$ from zero to $t$ gives $t^2\|[A,B]\|/2$. Inductively separate $H_1$ from the remaining sum, use the <triangle inequality> on their <commutator>, and apply the <telescoping bound for products of operators> to obtain the displayed many-term bound.
Repeating steps of size $t/k$ and telescoping across $k$ steps gives
$$
\left\|S(t/k)^k-e^{-it\sum_jH_j}\right\|
\leq\frac{t^2}{2k}\sum_{j<l}\|[H_j,H_l]\|.
$$
For a chain of $O(n)$ bounded nearest-neighbor terms, only $O(n)$ pairs fail to commute. Consequently first-order <product-formula Hamiltonian simulation> has error $O(nt^2/k)$ and uses $O(nk)$ constant-size gates. The general bound is also given in Proposition 9 of https://journals.aps.org/prx/pdf/10.1103/PhysRevX.11.011020[Childs and collaborators' analysis of Trotter error].
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