Generalized quantum circuit differentiation rules

Variational quantum algorithms that are used for machine learning rely on the ability to automatically differentiate parametrized circuits with respect underlying parameters. Here we propose rules differentiating (unitaries) arbitrary generators. Unlike standard parameter-shift rule valid unitaries generated by operators spectra limited at most two unique eigenvalues (represented involutory and idempotent operators), our approach also works generators a generic nondegenerate spectrum. Based spectral decomposition, derive simple recipe allows explicit derivative evaluation. The corresponds weighted sum of measured expectations shifted number function evaluations is equal positive nonzero gaps (eigenvalue differences) generator. We apply relevant examples two-qubit gates, among others showing so-called fermionic simulation (fSim) gate can be differentiated using four measurements. Additionally, present generalized differentiation case Pauli-string generators, based distinct shifts (here referred as triangulation approach), analyze variance measurements in different scenarios. Our work offers toolbox efficient hardware-oriented needed circuit optimization operator-based representation.