The many-body Green’s function approach is widely used in Condensed Matter Physics (and, in particular, in ab initio approaches) to describe the properties and evolution of a many-body interacting system. Typical applications range from one- to two-body evolution in a system of interacting electrons in atomic (molecules, two-dimensional and bulk) systems mediated by photons and/or phonons[1,2]. Some of the main advantages of these many-body approaches is the possibility of directly connecting the theoretical simulations to the experimental findings such as, for instance, the Photo Emission Spectroscopy (PES), Absorption spectrum and non-linear harmonic spectra, photoexcitations, quenches in Mott and excitonic insulators and high-temperature superconductors. Moreover, the description of the phonon dynamics such as dephasing and thermalization can be formally included in the theoretical Green function framework. 
In recent years, quantum computing has emerged as a potentially transformative technology to tackle many-body problems.  Hybrid quantum-classical schemes such as the Suzuki-Trotter decomposition, Lanczos algorithms offer strategies to overcome the exponential scaling that plagues classical approaches [3].  

However, developing powerful and reliable quantum computers remains a significant technical challenge. This is why much of today's research focuses on noisy intermediate-scale quantum (NISQ) devices — smaller, less sophisticated quantum computers that are already available, albeit with limited performance. In order to maximise the potential of NISQ devices, scientists have explored various methods based on Green’s functions, a mathematical tool used to study the behaviour of particles in complex systems.These approaches include advanced techniques like the use of the Lehmann representation and an excited state searching method,  the quantum Equation Of Motion technique, a mixed Variational Quantum Eigensolver (VQE) and Variational Linear Equation Solver (VLES), Coupled Cluster Green’s function method, Krilov Variational Quantum Algorithm, Cartan decomposition, and local Variational Quantum Compilation (LVQC) for real-time Green function [4-6].
Although these proof-of-principle studies have demonstrated simulations of small molecules or simplified models in NISQ, the application of quantum algorithms to compute many-body Green’s functions and integrate them into materials science workflows is still in its infancy.
Critical challenges remain in algorithm development, error mitigation, resource estimation, and linking quantum outputs to experimentally measurable quantities.