The Structure of Promises in Quantum Speedups

TL;DR

This paper investigates the limitations of quantum speedups under symmetric promises, demonstrating that exponential quantum advantages require specific structured input promises, with polynomial bounds relating classical and quantum complexities.

Contribution

It establishes polynomial bounds between classical and quantum complexities for functions with symmetric promises, highlighting the necessity of structured promises for exponential quantum speedups.

Findings

Polynomial relationship of degree 12 between $D(f)$ and $Q(f)$ for permutation-based functions.

Generalization of bounds to functions on orbits of the symmetric group.

Bound on randomized complexity $R(f)$ in terms of quantum complexity $Q(f)$ for symmetric sets.

Abstract

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 $R(f)=O(Q(f)^{12(M-1)})$.