Complexity class for quantum computer with commutative gates

BQP is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. In quantum computer allowed operations can be represented by unitary matrices. In general, unitary matrices do not commute with each other.

My question is what would be BQP analog for a quantum computer where only unitary matrices that commute with each other are allowed?

• Commuting Pauli circuits are pretty limited, doesn't IQP have to do with circuits of gates commuting with $X$? – Niel de Beaudrap Aug 30 '14 at 20:12