  Photoelectron spectroscopy experiments in ionic solutions reveal important
electronic structure information, in which the interaction between hydrated
ions and water solvent can be inferred. Based on many-body perturbation theory
with GW approximation, we theoretically compute the quasiparticle electronic
structure of chloride anion solution, which is modeled by path-integral $ab$
$initio$ molecular dynamics simulation by taking account the nuclear quantum
effects (NQEs). The electronic levels of hydrated anion as well as water are
determined and compared to the recent experimental photoelectron spectra. It is
found that NQEs improve the agreement between theoretical prediction and
experiment because NQEs effectively weaken the hybridization of the between the
$\rm Cl^-$ anion and water. Our study indicates that NQEs plays a small but
non-negligible role in predicting the electronic structure of the aqueous
solvation of ions of the Hofmeister series.
