  Quantum chemistry calculations on a quantum computer frequently suffer from
symmetry breaking: the situation when a state of assumed spin and number of
electrons is contaminated with contributions of undesired symmetry. The
situation may even culminate in convergence to a state of completely unexpected
symmetry, e.g. that of for a neutral species while a cation was expected.
Previously, the constrained variational quantum eigensolver (CVQE) approach was
proposed to alleviate this problem [Ryabinkin et al. (2018), J. Chem. Theory
Comput. DOI:10.1021/acs.jctc.8b00943] here we analyze alternative, more robust
solutions. In particular, we investigate how symmetry information can be
incorporated directly into qubit Hamiltonians. We identify three essentially
different techniques, the symmetry projection, spectral shift, and spectral
reflection methods, which are all capable of solving the problem albeit at
different computational cost, measured as the length of the resulting qubit
operators. On the examples of LiH and H$_2$O molecules we show that the
spectral shift method, which is equivalent to penalizing states of wrong
symmetry, is the most efficient, followed by spectral reflection, and symmetry
projection.
