You have the swimming pools in Aspire and also Hamad aquatic center (close to each other). You have an olympic size swimming pool opposite Al-Sadd Sport Club.

Almost every 5 stars hotels have swimming pools (I would recommend Ritz for its inside one, and Grand Hyatt for its outside one) but also big companies (Qatar Foundation has a olympic swimming pool but only available to its workers. If you have a friend working at QF, u can go with him).
If u have a friend in Zig-Zag towers or in some compounds (beverly Hills for example), they also have nice swimming pools)