  Accurate electronic structure calculations might be one of the most
anticipated applications of quantum computing.The recent landscape of quantum
simulations within the Hartree-Fock approximation raises the prospect of
substantial theory and hardware developments in this context.Here we propose a
general quantum circuit for M{\o}ller-Plesset perturbation theory (MPPT)
calculations, which is a popular and powerful post-Hartree-Fock method widly
harnessed in solving electronic structure problems. MPPT improves on the
Hartree-Fock method by including electron correlation effects wherewith
Rayleigh-Schrodinger perturbation theory. Given the Hartree-Fock results, the
proposed circuit is designed to estimate the second order energy corrections
with MPPT methods. In addition to demonstration of the theoretical scheme, the
proposed circuit is further employed to calculate the second order energy
correction for the ground state of Helium atom, and the total error rate is
around 2.3%. Experiments on IBM 27-qubit quantum computers express the
feasibility on near term quantum devices, and the capability to estimate the
second order energy correction accurately. In imitation of the classical MPPT,
our approach is non-heuristic, guaranteeing that all parameters in the circuit
are directly determined by the given Hartree-Fock results. Moreover, the
proposed circuit shows a potential quantum speedup comparing to the traditional
MPPT calculations. Our work paves the way forward the implementation of more
intricate post-Hartree-Fock methods on quantum hardware, enriching the toolkit
solving electronic structure problems on quantum computing platforms.
