  Interacting many-body quantum systems show a rich array of physical phenomena
and dynamical properties, but are notoriously difficult to study: they are
challenging analytically and exponentially difficult to simulate on classical
computers. Small-scale quantum information processors hold the promise to
efficiently emulate these systems, but characterizing their dynamics is
experimentally challenging, requiring probes beyond simple correlation
functions and multi-body tomographic methods. Here, we demonstrate the
measurement of out-of-time-ordered correlators (OTOCs), one of the most
effective tools for studying quantum system evolution and processes like
quantum thermalization. We implement a 3x3 two-dimensional hard-core
Bose-Hubbard lattice with a superconducting circuit, study its
time-reversibility by performing a Loschmidt echo, and measure OTOCs that
enable us to observe the propagation of quantum information. A central
requirement for our experiments is the ability to coherently reverse time
evolution, which we achieve with a digital-analog simulation scheme. In the
presence of frequency disorder, we observe that localization can partially be
overcome with more particles present, a possible signature of many-body
localization in two dimensions.
