  In a fast scrambling many-body quantum system, information is spread and
entanglement is built up on a timescale that grows logarithmically with the
system size. This is of fundamental interest in understanding the dynamics of
many-body systems, as well as in efficiently producing entangled resource
states and error-correcting codes. In this work, we identify a dynamical
transition marking the onset of scrambling in quantum circuits with different
levels of long-range connectivity. In particular, we show that as a function of
the interaction range for circuits of different structures, the tripartite
mutual information exhibits a scaling collapse around a critical point between
two clearly defined regimes of different dynamical behaviour. We study this
transition analytically in a related long-range Brownian circuit model and show
how the transition can be mapped onto the statistical mechanics of a long-range
Ising model in a particular region of parameter space. This mapping predicts
mean-field critical exponents $\nu = -1/(1+s_c)$, which are consistent with the
critical exponents extracted from Clifford circuit numerics. In addition to
systems with conventional power-law interactions, we identify the same
phenomenon in deterministic, sparse circuits that can be realised in
experiments with neutral atom arrays.
