  Systems of interacting quantum spins show a rich spectrum of quantum phases
and display interesting many-body dynamics. Computing characteristics of even
small systems on conventional computers poses significant challenges. A quantum
simulator has the potential to outperform standard computers in calculating the
evolution of complex quantum systems. Here, we perform a digital quantum
simulation of the paradigmatic Heisenberg and Ising interacting spin models
using a two transmon-qubit circuit quantum electrodynamics setup. We make use
of the exchange interaction naturally present in the simulator to construct a
digital decomposition of the model-specific evolution and extract its full
dynamics. This approach is universal and efficient, employing only resources
which are polynomial in the number of spins and indicates a path towards the
controlled simulation of general spin dynamics in superconducting qubit
platforms.
