Quantum computing methods for electronic states of the water molecule

TL;DR

This paper compares quantum computing methods for calculating water molecule electronic states, analyzing their efficiency, accuracy, and practicality for near-term and large-scale quantum computers.

Contribution

It provides a detailed comparison of multiple quantum algorithms for molecular energy calculations, highlighting the most efficient and practical methods for different quantum computing scenarios.

Findings

Second order direct phase estimation is most gate-efficient for large molecules.

Pairwise VQE is most practical for current quantum computers.

Analysis of error and gate complexity for each method.

Abstract

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. After deriving the Hamiltonian using STO-3G basis, we first explain how each method works and then compare the simulation results in terms of gate complexity and the number of measurements for the ground state of the water molecule with different O-H bond lengths. Moreover, we present the analytical analyses of the error and the gate-complexity for each method. While the required number of qubits for each method is almost the same, the number of gates and the error vary a lot. 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.