  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.
