  Electronic structure methods typically benefit from symmetry breaking and
restoration, specially in the strong correlation regime. The same goes for
Ans\"atze on a quantum computer. We develop a unitary coupled cluster method on
the antisymmetrized geminal power (AGP) -- a state formally equivalent to the
number-projected Bardeen--Cooper--Schrieffer wavefunction. We demonstrate our
method for the single-band Fermi--Hubbard Hamiltonian in one and two
dimensions. We also explore post-selection as a state preparation step to
obtain correlated AGP and prove that it scales no worse than
$\mathcal{O}(\sqrt{M})$ in the number of measurements, thereby making it a less
expensive alternative to gauge integration to restore particle number symmetry.
