  The out-of-time-ordered correlation (OTOC) and entanglement are two
physically motivated and widely used probes of the "scrambling" of quantum
information, a phenomenon that has drawn great interest recently in quantum
gravity and many-body physics. We argue that the corresponding notions of
scrambling can be fundamentally different, by proving an asymptotic separation
between the time scales of the saturation of OTOC and that of entanglement
entropy in a random quantum circuit model defined on graphs with a tight
bottleneck, such as tree graphs. Our result counters the intuition that a
random quantum circuit mixes in time proportional to the diameter of the
underlying graph of interactions. It also provides a more rigorous
justification for an argument in our previous work arXiv:1807.04363, that black
holes may be slow information scramblers, which in turn relates to the black
hole information problem. The bounds we obtained for OTOC are interesting in
their own right in that they generalize previous studies of OTOC on lattices to
the geometries on graphs in a rigorous and general fashion.
