  Perturbation theory (PT) might be one of the most powerful and fruitful tools
for both physicists and chemists, which has led to a wide variety of
applications. Over the past decades, advances in quantum computing provide
opportunities for alternatives to classical methods. Recently, a general
quantum circuit estimating the low order PT corrections has been proposed. In
this article, we revisit the quantum circuits for PT calculations, and develop
the methods for higher order PT corrections of eigenenergy, especially the 3rd
and 4th order corrections. We present the feasible quantum circuit to estimate
each term in these PT corrections. There are two the fundamental operations in
the proposed circuit. One approximates the perturbation terms, the other
approximates the inverse of unperturbed energy difference. The proposed method
can be generalized to higher order PT corrections.
