  Simulating quantum dynamics beyond the reach of classical computers is one of
the main envisioned applications of quantum computers. The most promising
quantum algorithms to this end in the near-term are the simplest, which use the
Trotter formula and its higher-order variants to approximate the dynamics of
interest. The approximation error of these algorithms is often poorly
understood, even in the most basic cases, which are particularly relevant for
experiments. Recent studies have reported anomalously low approximation error
with unexpected scaling in such cases, which they attribute to quantum
interference between the errors from different steps of the algorithm. Here we
provide a simpler picture of these effects by relating the Trotter formula to
its second-order variant. Our method generalizes state-of-the-art error bounds
without the technical caveats of prior studies, and elucidates how each part of
the total error arises from the underlying quantum circuit. We compare our
bound to the true error numerically, and find a close match over many orders of
magnitude in the simulation parameters. Our findings reduce the required
circuit depth for the most basic quantum simulation algorithms, and illustrate
a useful method for bounding simulation error more broadly.
