Complexity limitations on quantum computation

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP.

1. BQP is low for PP, i.e., PP^BQP=PP.
2. There exists a relativized world where P=BQP and the polynomial-time hierarchy is infinite.
3. There exists a relativized world where BQP does not have complete sets.
4. There exists a relativized world where P=BQP but P is not equal to UP intersect coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.