As soon as the user comes into battle, it transforms into its opponent. Broadly, the target's appearance (including form), types, Ability, stats, and moves are copied; the user retains its HP and level.
Source
If you're asking whether a Ditto will gain the ability to be be boosted by Protosynthesis/Quark Drive, than the answer is obviously yes. Imposter copies the ability of the target.
If you're asking whether a Ditto will copy the stat boost of the opposing Pokemon with Protosynthesis/Quark Drive, the answer is no. The increase in a stat here acts like a held item, similar to an Assault Vest or a Choice item. It increases the stat privately, (unless you're playing Showdown,) and an opponent can't copy that boost without stealing it with a move like Trick.
Hope this helped!
