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Optimization of unitary transformations in Variational Quantum Algorithmsbenefits highly from efficient evaluation of cost function gradients withrespect to amplitudes of unitary generators. We propose several extensions ofthe parametric-shift-rule to formulating these gradients as linear combinationsof expectation values for generators with general eigen-spectrum (i.e. withmore than two eigenvalues). Our approaches are exact and do not use anyauxiliary qubits, instead they rely on a generator eigen-spectrum analysis. Twomain directions in the parametric-shift-rule extensions are 1) polynomialexpansion of the exponential unitary operator based on a limited number ofdifferent eigenvalues in the generator and 2) decomposition of the generator asa linear combination of low-eigenvalue operators (e.g. operators with only 2 or3 eigenvalues). These techniques have a range of scalings for the number ofneeded expectation values with the number of generator eigenvalues fromquadratic (for polynomial expansion) to linear and even $\log_2$ (for generatordecompositions). This allowed us to propose efficient differentiation schemessuperior to previous approaches for commonly used 2-qubit transformations (e.g.match-gates, transmon and fSim gates) and $\hat S^2$-conserving fermionicoperators for the variational quantum eigensolver.

Analytic gradients in variational quantum algorithms: Algebraic extensions of the parameter-shift rule to general unitary transformations