We develop an efficient and robust approach to Hamiltonian identification for multipartite quantum systems based on the method of compressed sensing. This work demonstrates that with only
O(s log(d)) experimental configurations, consisting of random local preparations and measurements, one can estimate the Hamiltonian of a d-dimensional system, provided that the Hamiltonian is nearly
s-sparse in a known basis. We numerically simulate the performance of this algorithm for three- and four-body interactions in spin-coupled quantum dots and atoms in optical lattices. Furthermore, we apply the algorithm to characterize Hamiltonian fine structure and unknown system-bath interactions.