Similar to this list over at cstheory, I'm looking for a list of computational problems in the query complexity model for which it is known that no super-polynomial quantum speedups exist. What are some natural problems for which quantum computers are known to have only 'limited' advantage?
Disclaimer: I am not sure if this is your are looking for, so I do not pretend this to be an answer.
As far as I know, one example is calculation of a binary function parity. Binary function has odd parity in case it returns one for odd number of inputs. In other case it has even parity. In this task, a quantum computer is as fast as classical computer.