  The set equality problem is to decide whether two sets $A$ and $B$ 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 quality problem. It was an open problem to find any $w(1)$ query lower
bound when sets $A$ and $B$ are given by quantum oracles with functions $a$ and
$b$.
  We will prove $\Omega(\frac{n^{1/3}}{\log^{1/3} n})$ lower bound for the set
equality problem when the set of the preimages are very small for every element
in $A$ and $B$.

Quantum Physics