  The discovery of the backpropagation algorithm ranks among one of the most
important moments in the history of machine learning, and has made possible the
training of large-scale neural networks through its ability to compute
gradients at roughly the same computational cost as model evaluation. Despite
its importance, a similar backpropagation-like scaling for gradient evaluation
of parameterised quantum circuits has remained elusive. Currently, the most
popular method requires sampling from a number of circuits that scales with the
number of circuit parameters, making training of large-scale quantum circuits
prohibitively expensive in practice. Here we address this problem by
introducing a class of structured circuits that are not known to be classically
simulable and admit gradient estimation with significantly fewer circuits. In
the simplest case -- for which the parameters feed into commuting quantum gates
-- these circuits allow for fast estimation of the gradient, higher order
partial derivatives and the Fisher information matrix. Moreover, specific
families of parameterised circuits exist for which the scaling of gradient
estimation is in line with classical backpropagation, and can thus be trained
at scale. In a toy classification problem on 16 qubits, such circuits show
competitive performance with other methods, while reducing the training cost by
about two orders of magnitude.
