  We discuss here some aspects related to the symmetries of a quantum many-body
problem when trying to treat it on a quantum computer. Several features related
to symmetry conservation, symmetry breaking, and possible symmetry restoration
are reviewed. After briefly discussing some of the standard symmetries relevant
for many-particle systems, we discuss the advantage of encoding some symmetries
directly in quantum ans\"atze, especially to reduce the quantum register size.
It is, however, well-known that the use of symmetry-breaking states can also be
a unique way to incorporate specific internal correlations when a spontaneous
symmetry breaking occurs. These aspects are discussed in the quantum computing
context. Ultimately, an accurate description of quantum systems can be achieved
only when the initially broken symmetries are properly restored. We review
several methods explored previously to perform symmetry restoration on a
quantum computer, for instance, the ones based on symmetry filtering by quantum
phase estimation and by an iterative independent set of Hadamard tests. We
propose novel methods that pave the new directions to perform symmetry
restoration, like those based on the purification of the state employing the
linear combination of unitaries (LCU) approach.
