Quantum computing methods for electronic states of the water molecule

We compare recently proposed methods to compute the electronic state energies of the water molecule on a quantum computer. The methods include the phase estimation algorithm based on Trotter decomposition, the phase estimation algorithm based on the direct implementation of the Hamiltonian, direct measurement based on the implementation of the Hamiltonian and a specific variational quantum eigensolver, Pairwise VQE. We explain how each method works and compare the simulation results in terms of gate complexity and the number of measurements. In conclusion, among methods based on the phase estimation algorithm, the second order direct method provides the most efficient circuit implementations in terms of the gate complexity. With large scale quantum computation, the second order direct method seems to be better for large molecule systems. Moreover, Pairwise VQE serves the most practical method for near-term applications on the current available quantum computers. Finally, the possibility of extending the calculation to excited states and resonances is discussed. This work is posted on arXiv: Quantum computing methods for electronic states of the water molecule.