No, you can’t use the existing foundations (but there are exceptions).
Our building codes are stringently protective against seismic collapse, and most one-story houses have foundations that are just not capable of supporting a second story’s weight and sideways movement in a quake.
Many lots (and therefore houses) in Los Angeles are narrow, and allow new supporting posts and walls to be located on (or rather, in) the exterior walls of the house. The new second-story floor structure can then extend from one exterior wall to the other, completely bypassing the first floor’s interior walls, almost like a bridge floating above.
But if there are interior walls on the first floor, the structural engineer will often use those for intermediate supports, thus reducing the size (and cost) of the new floor beams above. In this case new interior foundations will be needed to support the new interior posts (or walls), and this may call for removal (and replacement) of some of the existing first story flooring.
Exceptions? Some one-story houses were designed and built to support a second story at a later time. In these cases–as long as the structural engineering was designed to work with the current building codes–a second story can be added without additional foundations. In other cases the existing structure may be strong enough that simple reinforcement of some walls, and a relatively modest amount of foundation work may be all that is needed.