It has long been known that in the usual black-box model, one cannot get super-polynomial quantum speedups without some promise on the inputs. In this thesis, we examine certain types of symmetric promises, and show that they also cannot give rise to super-polynomial quantum speedups. We conclude that exponential quantum speedups only occur given “structured” promises on the input.
Specifically, we show that there is a polynomial relationship of degree 12 between D(f) and Q(f) for any function f defined on permutations (elements of {0, 1, …, M-1}^M in which each alphabet element occurs exactly once). We generalize this result to all functions f defined on orbits of the symmetric group action S_n (which acts on an element of {0, 1, …, M-1}^n by permuting its entries). We also show that when M is constant, any function f defined on a “symmetric set” - one invariant under S_n - satisfies R(f) = O(Q(f)^{12(M-1)}).