When is this flight? I ask because there are a number of moving parts going on with airlines and terminals at LAX.

Firstly, AA uses T4 but it also uses four gates in T6. The T6 gates are mainly used for pmUS flights so there's a good chance your PHX flight may arrive T6 instead of T4. Note, however, that those AA T6 gates are moving to a T5 later this month as part of a much larger series of moves (that will take place over many months) to allow Delta to relocate all operations from T5 to T2/T3.

I'm not clear if Qatar will remain in T2 following these moves, or if they will relocate to TBIT or perhaps even T4.

That said, if your flight is in the next few days, you will arrive at either T4 or T6. Although these terminals are connected airside (and are connected to TBIT airside), they are not connected to T2 airside so you will have to exit and reclear at T2.

I will need others to weight on on how easiest to make this connection, but wanted to make sure you are aware of some pending terminal changes in case your flight is still in the future. (Exact dates of all the airline moves have not been publicly announced AFAIK)