  The purpose of this experiment was to use the known analytical techniques to
study the creation, simulation, and measurements of molecular Hamiltonians. The
techniques used consisted of the Linear Combination of Atomic Orbitals (LCAO),
the Linear Combination of Unitaries (LCU), and the Phase Estimation Algorithm
(PEA). The molecules studied were $H_2$ with and without spin, as well as
$He_2$ without spin. Hamiltonians were created under the LCAO basis, and
reconstructed using the Jordan-Winger transform in order to create a linear
combination of Pauli spin operators. The lengths of each molecular Hamiltonian
greatly increased from the $H_2$ without spin, to $He_2$. This resulted in a
reduced ability to simulate the Hamiltonians under ideal conditions. Thus, only
low orders of l = 1 and l = 2 were used when expanding the Hamiltonian in
accordance to the LCU method of simulation. The resulting Hamiltonians were
measured using PEA, and plotted against function of $\frac{2\pi(K)}{N}$ and the
probability distribution of each register. The resolution of the graph was
dependent on the amount of registers, N, being used. However, the reduction of
order hardly changed the image of the $H_2$ graphs. Qualitative comparisons
between the three molecules were drawn.
