  Over the last century, a large number of physical and mathematical
developments paired with rapidly advancing technology have allowed the field of
quantum chemistry to advance dramatically. However, the lack of computationally
efficient methods for the exact simulation of quantum systems on classical
computers presents a limitation of current computational approaches. We report,
in detail, how a set of pre-computed molecular integrals can be used to
explicitly create a quantum circuit, i.e. a sequence of elementary quantum
operations, that, when run on a quantum computer, to obtain the energy of a
molecular system with fixed nuclear geometry using the quantum phase estimation
algorithm. We extend several known results related to this idea and discuss the
adiabatic state preparation procedure for preparing the input states used in
the algorithm. With current and near future quantum devices in mind, we provide
a complete example using the hydrogen molecule, of how a chemical Hamiltonian
can be simulated using a quantum computer.
