Methods, systems and apparatus for estimating the fidelity of a quantum computing system. In one aspect, a method includes defining one or more random quantum circuits, wherein a noisy experimental implementation of each random quantum circuit is approximated by a depolarizing channel with respective polarization parameter; generating, for each defined random quantum circuit and by the quantum computing system, a set of experimental data, wherein data items in the set of experimental data comprise measured bit strings corresponding to experimental implementations of the random quantum circuit; determining, for each of the one or more random quantum circuits, an estimate of the respective polarization parameter, comprising maximizing a log-likelihood of the polarization parameter conditioned on the respective set of experimental data using series inversion; and determining an estimate of the fidelity of the quantum computing system based on the determined estimates of respective polarization parameters.