  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.
