Endo (intern), Kurata (intern), and Nakagawa from QunaSys Inc. proposed a near-term quantum algorithm for computing the Green’s function, which is crucial for analyzing properties of molecules and materials. Now you can read the preprint of the paper on arXiv.

“Calculation of the Green's function on near-term quantum computers”

https://arxiv.org/abs/1909.12250

## Background

Near-term quantum computers with hundreds to thousands of qubits which are not fault-tolerant are approaching realization. Such computers are called noisy intermediate-scale quantum (NISQ) devices, and application of them to calculations in quantum chemistry and condensed matter physics is expected. In particular, the variational quantum eigensolver (VQE) algorithm which can find energy spectrum of molecules and materials is the most featured algorithm on NISQ devices.

## Motivation

Meanwhile, another important physical quantity other than energy spectrum has left a bit disregard in the recent development of NISQ algorithms: the Green’s function. It tells us crucial information on properties of molecules and materials, and allows us to calculate their responses to external field (e.g. electric resistivity). Although its ubiquitous character in the theory of computational quantum chemistry and condensed matter physics, no algorithm for evaluating the Green’ s function on NISQ devices has yet been proposed.

## Methods & Results

Endo (intern), Kurat a (intern), and Nakagawa from QunaSys Inc. proposed two methods to compute the Green’s function efficiently on NISQ devices. One of the methods is based on the variational quantum simulation (VQS) algorithm which calculates the time evolution of quantum systems on NISQ devices. We extend the original VQS algorithm and make a quantum circuit for evaluating the Green's function drastically simple. The other method takes advantage of the previous VQE algorithms that compute energy spectrum and transition amplitude, and calculates the Lehmann representation of the Green’s function. Both methods require shallow quantum circuits compatible with NISQ devices. We perform numerical simulation of our proposed methods by using fast quantum simulator Qulacs, and successfully reproduce the Green’s function of the Hubbard model, a prototypical model of electrons in solids.

## Outlook

It is expected that our new quantum algorithms will significantly broaden the possibility to utilize NISQ devices in chemistry and material research, because they allow the computation of the Green’s function for large systems which could not be tackled with classical computers.