  Optimization of unitary transformations in Variational Quantum Algorithms
benefits highly from efficient evaluation of cost function gradients with
respect to amplitudes of unitary generators. We propose several extensions of
the parametric-shift-rule to formulating these gradients as linear combinations
of expectation values for generators with general eigen-spectrum (i.e. with
more than two eigenvalues). Our approaches are exact and do not use any
auxiliary qubits, instead they rely on a generator eigen-spectrum analysis. Two
main directions in the parametric-shift-rule extensions are 1) polynomial
expansion of the exponential unitary operator based on a limited number of
different eigenvalues in the generator and 2) decomposition of the generator as
a linear combination of low-eigenvalue operators (e.g. operators with only 2 or
3 eigenvalues). These techniques have a range of scalings for the number of
needed expectation values with the number of generator eigenvalues from
quadratic (for polynomial expansion) to linear and even $\log_2$ (for generator
decompositions). This allowed us to propose efficient differentiation schemes
superior to previous approaches for commonly used 2-qubit transformations (e.g.
match-gates, transmon and fSim gates) and $\hat S^2$-conserving fermionic
operators for the variational quantum eigensolver.
