Quantum Zeno approach for molecular energies with maximum commuting initial Hamiltonians

We propose to use a quantum adiabatic and simulated-annealing framework to compute the ground state of small molecules. The initial Hamiltonian of our algorithms is taken to be the maximum commuting Hamiltonian that consists of a maximal set of commuting terms in the full Hamiltonian of molecules in the Pauli basis. We consider two variants. In the first method, we perform the adiabatic evolution on the obtained time- or path-dependent Hamiltonian with the initial state as the ground state of the maximum commuting Hamiltonian. However, this method does suffer from the usual problems of adiabatic quantum computation due to degeneracy and energy-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 the beginning and the end of the path. In addition to the ground state, the low lying excited states can be obtained using this quantum Zeno approach with equal accuracy to that of the ground state.