We posted a preprint that extensively analyzes penalty methods in variational quantum eigensolver to obtain a specific symmetric quantum state.

Penalty methods for variational quantum eigensolver,
https://arxiv.org/abs/2010.13951.

We explain technical details in our tech blog post.

-> Published as Phys. Rev. Research 3, 013197 (2021)
Phys. Rev. Research 3, 013197 (2021) - Penalty methods for a variational quantum eigensolver (aps.org)