Abstract
The accurate implementation of quantum gates is essential for the realization of quantum algorithms and digital quantum simulations. This accuracy may be increased on noisy hardware through the variational optimization of gates, however the experimental realization of such a protocol is impeded by the large effort required to estimate the fidelity of an implemented gate. With a hierarchy of approximations we find a faithful approximation to the quantum process fidelity that can be estimated experimentally with reduced effort. Its practical use is demonstrated with the optimization of a three-qubit quantum gate on a commercially available quantum processor.