Quantum lower bounds of set equality problems

The set equality problem is to decide whether two sets given by functions a and b such that A = {a(1), a(2), ..., a(n)} and B = {b(1), b(2), ..., b(n)} are equal or disjoint, under the promise that one of these is the case. Some other problems, like the Graph Isomorphism problem, is solvable by reduction to the set equality problem. This motivates to find a polynomial quantum lower bound for the set equality problem, showing that this is not the way to solve the Graph Isomorphism problem by a quantum computer. We will prove Ω( n 1/3 log1/3 n ) lower bound for set equality problem when the set of the preimages are very small for every element in A and B. To find any w(1) lower bound of set equality problem when a and b are one-to-one functions still remains open.