Calculating the Single-Particle Many-body Green's Functions via the Quantum Singular Value Transform Algorithm

TL;DR

This paper explores using the Quantum Singular Value Transform algorithm to compute single-particle Green's functions, focusing on matrix inversion approximation and efficient circuit design within quantum simulations.

Contribution

It demonstrates a noise-free simulation of QSVT for Green's function calculation and introduces a new circuit construction to reduce gate complexity.

Findings

Successful simulation of Green's function for the Anderson model

Polynomial approximation impacts at zero in matrix inversion

Reduced gate count in the proposed circuit design

Abstract

The Quantum Singular Value Transformation (QSVT) is a technique that provides a unified framework for describing many of the quantum algorithms discovered to date. We implement a noise-free simulation of the technique to investigate how it can be used to perform matrix inversion, which is an important step in calculating the single-particle Green's function in the Lehmann representation. Due to the inverse function not being defined at zero, we explore the effect of approximating f(x)=1/x with a polynomial. This is carried out by calculating the single-particle Green's function of the two-site single-impurity Anderson model. We also propose a new circuit construction for the linear combination of unitaries block encoding technique, that reduces the number of single and two-qubit gates required.