  The idea of the out-of-time-order correlator (OTOC) has recently emerged in
the study of both condensed matter systems and gravitational systems. It not
only plays a key role in investigating the holographic duality between a
strongly interacting quantum system and a gravitational system, but also
diagnoses the chaotic behavior of many-body quantum systems and characterizes
the information scrambling. Based on the OTOCs, three different concepts --
quantum chaos, holographic duality, and information scrambling -- are found to
be intimately related to each other. Despite of its theoretical importance, the
experimental measurement of the OTOC is quite challenging and so far there is
no experimental measurement of the OTOC for local operators. Here we report the
measurement of OTOCs of local operators for an Ising spin chain on a nuclear
magnetic resonance quantum simulator. We observe that the OTOC behaves
differently in the integrable and non-integrable cases. Based on the recent
discovered relationship between OTOCs and the growth of entanglement entropy in
the many-body system, we extract the entanglement entropy from the measured
OTOCs, which clearly shows that the information entropy oscillates in time for
integrable models and scrambles for non-intgrable models. With the measured
OTOCs, we also obtain the experimental result of the butterfly velocity, which
measures the speed of correlation propagation. Our experiment paves a way for
experimentally studying quantum chaos, holographic duality, and information
scrambling in many-body quantum systems with quantum simulators.
