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

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