  We propose to use a quantum adiabatic and simulated-annealing framework to
compute theground state of small molecules. The initial Hamiltonian of our
algorithms is taken to be themaximum commuting Hamiltonian that consists of a
maximal set of commuting terms in the fullHamiltonian of molecules in the Pauli
basis. We consider two variants. In the first method, weperform the adiabatic
evolution on the obtained time- or path-dependent Hamiltonian with theinitial
state as the ground state of the maximum commuting Hamiltonian. However, this
methoddoes suffer from the usual problems of adiabatic quantum computation due
to degeneracy andenergy-level crossings along the Hamiltonian path. This
problem is mitigated by a Zeno method,i.e., via a series of eigenstate
projections used in the quantum simulated annealing, with the path-dependent
Hamiltonian augmented by a sum of Pauli X terms, whose contribution vanishes at
thebeginning and the end of the path. In addition to the ground state, the low
lying excited states canbe obtained using this quantum Zeno approach with equal
accuracy to that of the ground state.
