3 MOVES
If you play: 1. Qc3+ Kb1 2. Bc2 ...
Then 2. ... Ka1
is not forced. With 2. ... Kc1, white's Bishop is blocking the Queen's attack. The Queen can no longer move to an attacking position without risking the Queen. But then you could still have 3. Bc3 Kb1 4. Qc2 Ka1 5. Qa2#
Please correct me if I made an error or if my notation is wrong: I've not commented here before.