A Near-Term Quantum Algorithm for Computing Molecular and Materials Properties based on Recursive Variational Series Methods

Determining properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is: how might we use imperfect near-term quantum computers to solve problems of practical value? We propose a quantum algorithm to estimate properties of molecules using near-term quantum devices. The method is a recursive variational series estimation method where we expand an operator of interest in terms of Chebyshev polynomials, and evaluate each term in the expansion using a variational quantum algorithm. We test our method on filter operators: the spectral density operator (SDO) $\delta(E-\hat{H})$ and Green's filter operator (GFO) $(E-\hat{H})^{-1}$, where the poles correspond to eigenvalues of the system Hamiltonian. A useful property of the SDO and GFO is to ``filter'' out eigenenergies and eigenstates in the spectral regions of interest, for example to analyze optical spectra. Furthermore, the GFO is useful to study for example quantum many-body systems and quantum transport by calculating the Green's functions in the frequency domain. We discuss two implementations based on our recursive variational series estimation method and a recent study of quantum algorithms to compute Green's functions using the Lanczos recursions [F. Jamet \emph{et al.}, arXiv:2105.13298 (2021)]. Numerically, we show that in the present of sampling and device noise, our method is significantly more noise resilient. In conclusion, we find there is a potentially useful application on near-term quantum computers for evaluating expectation values.