  We study quantum information scrambling, specifically the growth of
Heisenberg operators, in large disordered spin chains using matrix product
operator dynamics to scan across the thermalization-localization quantum phase
transition. We observe ballistic operator growth for weak disorder, and a sharp
transition to a phase with sub-ballistic operator spreading. The critical
disorder strength for the ballistic to sub-ballistic transition is well below
the many body localization phase transition, as determined from finite size
scaling of energy eigenstate entanglement entropy in small chains. In contrast,
we find that the transition from sub-ballistic to logarithmic behavior at the
actual eigenstate localization transition is not resolved in our finite
numerics. These data are discussed in the context of a universal form for the
growing operator shape and substantiated with a simple phenomenological model
of rare regions.
