Computing Ground State Properties with Early Fault-Tolerant Quantum Computers

Significant effort in applied quantum computing has been devoted to the problem of ground state energy estimation for molecules and materials. Yet, for many applications of practical value, additional properties of the ground state must be estimated. These include Green's functions used to compute electron transport in materials and the one-particle reduced density matrices used to compute electric dipoles of molecules. In this paper, we propose a quantum-classical hybrid algorithm to efficiently estimate such ground state properties with high accuracy. The quantum portion of the algorithm employs a simple, low-depth quantum circuit and the classical portion contains a novel post-processing procedure to extract the information of the ground state property from quantum sample data. We provide an analysis of various costs (circuit repetitions, maximal evolution time, and expected total runtime) as a function of target accuracy, spectral gap, and initial ground state overlap. This algorithm suggests a concrete approach to using early fault-tolerant quantum computers for carrying out industry-relevant molecular and materials calculations.