 
Summary: The Need for Structure in Quantum Speedups
Scott Aaronson #
MIT
Andris Ambainis +
University of Latvia
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 blackbox 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 9 th root of the classical randomized query complexity. This resolves a conjecture of
Watrous from 2002.
Second, inspired by recent work of O'Donnell et al. and Dinur et al., we conjecture that every
bounded lowdegree 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.
1 Introduction
Perhaps the central lesson gleaned from fifteen years of quantum algorithms research is this:
