  Quantum scrambling describes the spreading of local information into many
degrees of freedom in quantum systems. This provides the conceptual connection
among diverse phenomena ranging from thermalizing quantum dynamics to models of
black holes. Here we experimentally probe the exponential scrambling of a
multi-particle system near a bistable point in phase space and utilize it for
entanglement-enhanced metrology. We use a time-reversal protocol to observe a
simultaneous exponential growth of both the metrological gain and the
out-of-time-order correlator, thereby experimentally verifying the relation
between quantum metrology and quantum information scrambling. Our experiments
demonstrate that fast-scrambling dynamics capable of exponentially fast
entanglement generation are useful for practical metrology, resulting in 6.8(4)
dB gain beyond the Standard Quantum Limit.
