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
paper, 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}n in which each alphabet element occurs
exactly once). We generalize this result to all functions f defined on orbits
of the symmetric group action Sn (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 Sn -
satisfies R(f)=O(Q(f)12(M−1)).Comment: 15 page