Mapping functional connectivity between neurons is an essential step towards probing the neural computations mediating behavior. The ability to consistently and robustly determine synaptic connectivity maps in large populations of interconnected neurons is a significant challenge in terms of yield, accuracy and experimental time. Here we developed a compressive sensing approach to reconstruct synaptic connectivity maps based on random two photon (2p) cell-targeted optogenetic stimulation and membrane voltage readout of many putative postsynaptic neurons. Using a biophysical network model of interconnected populations of excitatory and inhibitory neurons, we found that the mapping can be achieved with far fewer measurements than the standard pairwise sequential approach. We characterized the recall and precision probabilities as a function of network observability, sparsity, number of neurons stimulated per trial, off-target stimulation, synaptic reliability, propagation latency and network topology. We found that that network sparsity and synaptic reliability were primary determinants of the performance. In particular, in a network with 10% probability of neuronal connectivity, functional connections were recovered with >85% recall and >80% precision in half the trials that would be required for single cell stimulation. Our results suggest a rapid and efficient method to reconstruct functional connectivity of brain networks where sparsity is predominantly present.