  In recent years, we have witnessed an explosion of experimental tools by
which quantum systems can be manipulated in a controlled and coherent way. One
of the most important goals now is to build quantum simulators, which would
open up the possibility of exciting experiments probing various theories in
regimes that are not achievable under normal lab circumstances. Here we present
a novel approach to gain detailed control on the quantum simulation of strongly
correlated quantum many-body systems by constructing the explicit quantum
circuits that diagonalize their dynamics. We show that the exact quantum
circuits underlying some of the most relevant many-body Hamiltonians only need
a finite amount of local gates. As a particularly simple instance, the full
dynamics of a one-dimensional Quantum Ising model in a transverse field with
four spins is shown to be reproduced using a quantum circuit of only six local
gates. This opens up the possibility of experimentally producing strongly
correlated states, their time evolution at zero time and even thermal
superpositions at zero temperature. Our method also allows to uncover the exact
circuits corresponding to models that exhibit topological order and to
stabilizer states.
