  Variational quantum algorithms are ubiquitous in applications of noisy
intermediate-scale quantum computers. Due to the structure of conventional
parametrized quantum gates, the evaluated functions typically are finite
Fourier series of the input parameters. In this work, we use this fact to
derive new, general parameter-shift rules for single-parameter gates, and
provide closed-form expressions to apply them. These rules are then extended to
multi-parameter quantum gates by combining them with the stochastic
parameter-shift rule. We perform a systematic analysis of quantum resource
requirements for each rule, and show that a reduction in resources is possible
for higher-order derivatives. Using the example of the quantum approximate
optimization algorithm, we show that the generalized parameter-shift rule can
reduce the number of circuit evaluations significantly when computing
derivatives with respect to parameters that feed into many gates. Our approach
additionally reproduces reconstructions of the evaluated function up to a
chosen order, leading to known generalizations of the Rotosolve optimizer and
new extensions of the quantum analytic descent optimization algorithm.
