  The vast majority of dynamical systems in classical physics are chaotic and
exhibit the butterfly effect: a minute change in initial conditions can soon
have exponentially large effects elsewhere. But this phenomenon is difficult to
reconcile with quantum mechanics. One of the main goals in the field of quantum
chaos is to establish a correspondence between the dynamics of classical
chaotic systems and their quantum counterparts. In isolated systems in the
absence of decoherence, there is such a correspondence in dynamics, but it
usually persists only over a short time window, after which quantum
interference washes out classical chaos. We demonstrate that quantum mechanics
can also play the opposite role and generate exponential instabilities in
classically non-chaotic systems within this early-time window. Our calculations
employ the out-of-time-ordered correlator (OTOC) -- a diagnostic that reduces
to the Lyapunov exponent in the classical limit, but is well defined for
general quantum systems. Specifically, we show that a variety of classically
non-chaotic models, such as polygonal billiards, whose classical Lyapunov
exponents are always zero, demonstrate a Lyapunov-like exponential growth of
the OTOC at early times with Planck's-constant-dependent rates. This behavior
is sharply contrasted with the slow early-time growth of the analog of the OTOC
in the systems' classical counterparts. These results suggest that
classical-to-quantum correspondence in dynamics is violated in the OTOC even
before quantum interference develops.
