The Structure of Promises in Quantum Speedups

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 $\{0,1,\dots, M-1\}^n$ in which each alphabet element occurs exactly once). We generalize this result to all functions $f$ defined on orbits of the symmetric group action $S_n$ (which acts on an element of $\{0,1,\dots, M-1\}^n$ 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 $R(f)=O(Q(f)^{12(M-1)})$.