This list was put on QL a while back. Not sure how accurate it is, but it seems to include the usual suspects;

American Staffordshire Terrier
Boerboel
Boston Terrier
Boxer
Bull Terrier
Bulldog
Chinese Shar-Pei
Doberman
Great Dane
Japanese Akita
Neapolitan Mastiff
Pug
Rottweiler
Staffordshire Bull Terrier

Vets may be able to confirm. Try www.dohavets.com

You also need to make sure the apartment/villa/compound where you live is happy for you to have a dog. Not all of them are.

Also, I notice you're from the UK. Be aware that, as it currently stands, to take your dog back from Qatar to the UK will require it to go into 6 months quarantine. This is because Qatar is not part of the PETS scheme, though hopefully this will change.