If we think we're being visited by aliens, then this suggests that this is the kind of thing we should be on the lookout for here.
The big challenge faced by our approach is that the laser array is limited to earth orbit. As it pushes the spacecraft away from earth and the laser rapidly loses efficiency. However if you had laser arrays over the planned route of the spacecraft you can keep adding velocity. However getting that infrastructure into place decades to millennia. You start with the array in home world orbit and then start building further and further out arrays. You also want the boost stations at the destination to slow spacecraft or change their directions.