Accessing ground state and excited states energies in many-body system after symmetry restoration using quantum computers

We explore the possibility to perform symmetry restoration with the variation after projection technique on a quantum computer followed by additional post-processing. The final goal is to develop configuration interaction techniques based on many-body trial states pre-optimized on a quantum computer. We show how the projection method used for symmetry restoration can prepare optimized states that could then be employed as initial states for quantum or hybrid quantum-classical algorithms. We use the quantum phase estimation and quantum Krylov approaches for the post-processing. The latter method combined with the quantum variation after projection (Q-VAP) leads to very fast convergence towards the ground-state energy. The possibility to access excited states energies is also discussed. Illustrations of the different techniques are made using the pairing hamiltonian.