The Need for Structure in Quantum Speedups

TL;DR

This paper investigates the conditions under which quantum algorithms can achieve exponential speedups, establishing lower bounds for symmetric problems and proposing a conjecture linking polynomial influence to classical simulation of quantum algorithms.

Contribution

It proves a lower bound on quantum query complexity for symmetric problems and introduces a conjecture on polynomial influence that suggests classical simulation of quantum algorithms is often possible.

Findings

Quantum query complexity for symmetric problems is at least the 7th root of classical complexity.

A conjecture that low-degree polynomials have highly influential variables.

Implication that P might equal BQP relative to a random oracle.

Abstract

Is there a general theorem that tells us when we can hope for exponential speedups from quantum algorithms, and when we cannot? In this paper, we make two advances toward such a theorem, in the black-box model where most quantum algorithms operate. First, we show that for any problem that is invariant under permuting inputs and outputs (like the collision or the element distinctness problems), the quantum query complexity is at least the 7th root of the classical randomized query complexity. (An earlier version of this paper gave the 9th root.) This resolves a conjecture of Watrous from 2002. Second, inspired by recent work of O'Donnell et al. (2005) and Dinur et al. (2006), we conjecture that every bounded low-degree polynomial has a "highly influential" variable. Assuming this conjecture, we show that every T-query quantum algorithm can be simulated on most inputs by a poly(T)-query classical algorithm, and that one essentially cannot hope to prove P!=BQP relative to a random oracle.