  The dynamical generation of complex correlations in quantum many-body systems
is of renewed interest in the context of quantum chaos, where the
out-of-time-ordered (OTO) correlation function appears as a convenient measure
of scrambling. To detect the the transition from scrambling to many-body
localization, the latter of which has limited dynamical complexity and is often
classically simulatable, we develop both exact and approximate methods to
compute OTO correlators for arbitrary universal quantum circuits. We take
advantage of the mapping of quantum circuits to the dynamics of interacting
fermions in one dimension, as Gaussian time evolution supplemented by quartic
interaction gates. In this framework, the OTO correlator can be calculated
exactly as a superposition of exponentially many Gaussian-fermionic
trajectories in the number of interaction gates. We develop a
variationally-optimized, Gaussian approximation to the spatial propagation of
an initially-local operator by restriction to the fastest-traveling fermionic
modes, in a similar spirit as light-front computational methods in quantum
field theory. We demonstrate that our method can detect the many-body
localization transitions of generally time-dependent dynamics without the need
for perturbatively weak interactions.
