Lagrange Interpolation Approach for General Parameter-Shift Rule

A parameter-shift rule is a circuit-based approach to evaluate the analytic differential in various variational quantum algorithms. Meanwhile, general parameter-shift rules were discussed under the stochastic approach, the finite Fourier series, and the polynomial expansion approach. Here, we introduce the Lagrange interpolation approach for practically deriving the general parameter-shift rule. Our method is exact and applicable for any nondegenerate spectrum of the generator. We illustrate the approach for the well-known two-term, four-term, and later extend to a general multiple-term parameter-shift rule with any-order derivative. We further demonstrate the multiple-term parameter-shift rule in the variational quantum eigensolver for finding the ground state many-body systems.