  Operator spreading under unitary time evolution has attracted a lot of
attention recently, as a way to probe many-body quantum chaos. While quantities
such as out-of-time-ordered correlators (OTOC) do distinguish interacting from
non-interacting systems, it has remained unclear to what extent they can truly
diagnose chaotic {\it vs} integrable dynamics in many-body quantum systems.
Here, we analyze operator spreading in generic 1D many-body quantum systems
using a combination of matrix product operator (MPO) and analytical techniques,
focusing on the operator {\em right-weight}. First, we show that while small
bond dimension MPOs allow one to capture the exponentially-decaying tail of the
operator front, in agreement with earlier results, they lead to significant
quantitative and qualitative errors for the actual front -- defined by the
maximum of the right-weight. We find that while the operator front broadens
diffusively in both integrable and chaotic interacting spin chains, the precise
shape and scaling of the height of the front in integrable systems is anomalous
for all accessible times. We interpret these results using a quasiparticle
picture. This provides a sharp, though rather subtle signature of many-body
quantum chaos in the operator front.
