Simplifying the simulation of local Hamiltonian dynamics

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.