  Local Hamiltonians, $H_k$, describe non-trivial $k$-body interactions in
quantum many-body systems. Here, we address the dynamical simulatability of a
$k$-local Hamiltonian by a simpler one, $H_{k'}$, with $k'<k$, under the
realistic constraint that both Hamiltonians act on the same Hilbert space. When
it comes to exact simulation, we build upon known methods to derive examples of
$H_k$ and $H_{k'}$ that simulate the same physics. We also address the most
realistic case of approximate simulation. There, we upper-bound the error up to
which a Hamiltonian can simulate another one, regardless of their internal
structure, and prove, by means of an example, that the accuracy of a
$(k'=2)$-local Hamiltonian to simulate $H_{k}$ with $k>2$ increases with $k$.
Finally, we propose a method to search for the $k'$-local Hamiltonian that
simulates, with the highest possible precision, the short time dynamics of a
given $H_k$ Hamiltonian.
