We actually have a few examples of an absolute speed up for quantum computers, though none of them are exponential and some of them are problems nobody cares to solve. For example, Grover's algorithm is superior (O(sqrt(n)) vs O(n)) to the best possible classical algorithm for the same problem.

If you are willing to accept comparisons of best known classical algorithms to best known quantum algorithms (without a guarantee that the existing algorithms are ideal) you can add others like factoring to the list, with exponential speed ups.