  Quantum simulation of quantum chemistry is one of the most compelling
applications of quantum computing. It is of particular importance in areas
ranging from materials science, biochemistry and condensed matter physics.
Here, we propose a full quantum eigensolver (FQE) algorithm to calculate the
molecular ground energies and electronic structures using quantum gradient
descent. Compared to existing classical-quantum hybrid methods such as
variational quantum eigensolver (VQE), our method removes the classical
optimizer and performs all the calculations on a quantum computer with faster
convergence. The gradient descent iteration depth has a favorable complexity
that is logarithmically dependent on the system size and inverse of the
precision. Moreover, the FQE can be further simplified by exploiting
perturbation theory for the calculations of intermediate matrix elements, and
obtain results with a precision that satisfies the requirement of chemistry
application. The full quantum eigensolver can be implemented on a near-term
quantum computer. With the rapid development of quantum computing hardware, FQE
provides an efficient and powerful tool to solve quantum chemistry problems.
