Green's function approach to quantum chemistry on a quantum computer via dynamical self-energy mapping

We propose a low-depth circuit for quantum chemistry based on the dynamical self-energy mapping (DSEM) paradigm, where a sparse Hamiltonian is created that shares the same dynamical part of the self energy as that of the molecule (evaluated with a simple approximate method). DSEM then assumes the exact self-energy of the sparse model will approximate the dynamical part of the self-energy of the molecule accurately. As a first step in this approach, we illustrate how this procedure can work on a quantum computer, by simulating the following process classically: (i) generate the lesser and greater Green's functions in the time domain; (ii) process the temporal data with compressive sensing to determine the retarded Green's function in the frequency domain; (iii) extract the imaginary part of the self-energy by determining the weights and frequencies of the corresponding delta functions, and (iv) verify the accuracy by computing the ground-state energy. We illustrate part of this algorithm with the Hubbard model on a ring.