The level of fitness a person needs depends on the physical demands of the trip: the duration of the trip as a whole, the number of hours active during the days, the amount of rest days, the amount and weight of the pack, the environment and the type of activity. It is important to have a clear picture of all this in order to be able to train specifically and increase the chances of successfully completing the climb.
It is not necessary to have the fitness of an Olympic athlete, but it does help in certain areas. People who are in good shape have less trouble getting up the mountain, experience a better state of mind because they are less tired and generally have a higher step speed. A good basic condition therefore helps the climber at altitude, but is not a necessity for getting up the mountain. It may just take a bit longer with a less good condition.