We show that several quantum circuit families can be simulated efficiently classically if it is promised that their output distribution is approximately sparse i.e. the distribution is close to one where only a polynomially small, a priori unknown subset of the measurement probabilities are nonzero. Classical simulations are thereby obtained for quantum circuits which---without the additional sparsity promise---are considered hard to simulate. Our results apply in particular to a family of Fourier sampling circuits (which have structural similarities to Shor's factoring algorithm) but also to several other circuit families, such as IQP circuits. Our results provide examples of quantum circuits that cannot achieve exponential speed-ups due to the presence of too much destructive interference i.e. too many cancelations of amplitudes. The crux of our classical simulation is an efficient algorithm for approximating the significant Fourier coefficients of a class of states called computationally tractable states. The latter result may have applications beyond the scope of this work. In the proof we employ and extend sparse approximation techniques, in particular the Kushilevitz-Mansour algorithm, in combination with probabilistic simulation methods for quantum circuits.