A fundamental problem in characterization of complex quantum systems is the exponential growth in the required physical resources with the size of system. We develop an efficient method for complete estimation of an unknown quantum process/Hamiltonian with a polynomial number of experimental configurations via employing techniques known
as compressed sensing. We demonstrate that by O(s \log d) random local preparations and measurements settings one can fully identify a quantum process/Hamiltonian for a d-dimensional system, if it is known to be nearly s-sparse in a basis. We present the first experimental implementation of this method for two- and four-photon quantum optical systems with a significant reduction in physical resources compare to known tomography techniques. Moreover, we demonstrate robustness of this technique by performing efficient high-fidelity estimation of two-qubit photonic phase gates under various decoherence strengths.