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.