  For noisy intermediate-scale quantum (NISQ) devices only a moderate number of
qubits with a limited coherence is available thus enabling only shallow
circuits and a few time evolution steps in the currently performed quantum
computations. Here, we present how to bypass this challenge in practical
molecular chemistry simulations on NISQ devices by employing a
classical-quantum hybrid algorithm allowing us to produce a sparse Hamiltonian
which contains only $\mathcal{O}(n^2)$ terms in a Gaussian orbital basis when
compared to the $\mathcal{O}(n^4)$ terms of a standard Hamiltonian, where $n$
is the number of orbitals in the system. Classical part of this hybrid entails
parameterization of the sparse, fictitious Hamiltonian in such a way that it
recovers the self-energy of the original molecular system. Quantum machine then
uses this fictitious Hamiltonian to calculate the self-energy of the system. We
show that the developed hybrid algorithm yields very good total energies for
small molecular test cases while reducing the depth of the quantum circuit by
at least an order of magnitude when compared with simulations involving a full
Hamiltonian.
