  Scrambling, a process in which quantum information spreads over a complex
quantum system becoming inaccessible to simple probes, happens in generic
chaotic quantum many-body systems, ranging from spin chains, to metals, even to
black holes. Scrambling can be measured using out-of-time-ordered correlators
(OTOCs), which are closely tied to the growth of Heisenberg operators. In this
work, we present a general method to calculate OTOCs of local operators in
local one-dimensional systems based on approximating Heisenberg operators as
matrix-product operators (MPOs). Contrary to the common belief that such tensor
network methods work only at early times, we show that the entire early growth
region of the OTOC can be captured using an MPO approximation with modest bond
dimension. We analytically establish the goodness of the approximation by
showing that if an appropriate OTOC is close to its initial value, then the
associated Heisenberg operator has low entanglement across a given cut. We use
the method to study scrambling in a chaotic spin chain with $201$ sites. Based
on this data and OTOC results for black holes, local random circuit models, and
non-interacting systems, we conjecture a universal form for the dynamics of the
OTOC near the wavefront. We show that this form collapses the chaotic spin
chain data over more than fifteen orders of magnitude.
