  We use out-of-time-order commutator (OTOC) to diagnose the propagation of
chaos in one dimensional long-range power law interaction system. We map the
evolution of OTOC to a classical stochastic dynamics problem and use a Brownian
quantum circuit to exactly derive the master equation. We vary two parameters:
the number of qubits $N$ on each site (the onsite Hilbert space dimension) and
the power law exponent $\alpha$. Three light cone structures of OTOC appear at
$N = 1$: (1) logarithmic when $0.5<\alpha\lesssim 0.8$, (2) sublinear power law
when $0.8 \lesssim \alpha \lesssim 1.5$ and (3) linear when $\alpha \gtrsim
1.5$. The OTOC scales as $\exp(\lambda t)/x^{2\alpha} $ and $t^{2 \alpha /
\zeta} / x^{ 2 \alpha} $ respectively beyond the light cones in the first two
cases. When $\alpha \geq 2$, the OTOC has essentially the same diffusive
broadening as systems with short-range interactions, suggesting a complete
recovery of locality. In the large $N$ limit, it is always a logarithmic light
cone asymptotically, although a linear light cone can appear before the
transition time for $ \alpha \gtrsim 1.5$. This implies the locality is never
fully recovered for finite $\alpha$. Our result provides a unified physical
picture for the chaos dynamics in long-range power law interaction system.
