  This article shows a proof of concept regarding the feasibility of ab initio
molecular simulation, wherein the wavefunctions and the positions of nuclei are
simultaneously determined by the quantum algorithm, as is realized by the
so-called Car-Parrinello method by classical computing. The approach used in
this article is of a hybrid style, which shall be realized by future
fault-tolerant quantum computer. First, the basic equations are approximated by
polynomials. Second, those polynomials are transformed to a specific form,
wherein all variables (representing the wavefunctions and the atomic
coordinates) are given by the transformations acting on a linear space of
monomials with finite dimension, and the unknown variables could be determined
as the eigenvalues of those transformation matrices. Third, the eigenvalues are
determined by quantum phase estimation. Following these three steps, namely,
symbolic, numeric, and quantum steps, we can determine the optimized electronic
and atomic structures of molecules.
