Quantum speedups need structure

TL;DR

This paper proves a conjecture linking polynomial influence to variable importance, which implies that quantum algorithms can be efficiently simulated classically on most inputs, highlighting limitations of quantum speedups.

Contribution

It establishes a lower bound on variable influence for multilinear polynomials, confirming a conjecture that supports classical simulation of most quantum algorithms.

Findings

Proves the influence lower bound for multilinear polynomials.

Shows quantum algorithms can be approximated classically on most inputs.

Supports limitations on quantum speedups for query-based algorithms.

Abstract

We prove the following conjecture, raised by Aaronson and Ambainis in 2008: Let $f:\{-1,1\}^n \rightarrow [-1,1]$ be a multilinear polynomial of degree $d$. Then there exists a variable $x_i$ whose influence on $f$ is at least $\mathrm{poly}(\mathrm{Var}(f)/d)$. As was shown by Aaronson and Ambainis, this result implies the following well-known conjecture on the power of quantum computing, dating back to 1999: Let $Q$ be a quantum algorithm that makes $T$ queries to a Boolean input and let $\epsilon,\delta > 0$. Then there exists a deterministic classical algorithm that makes $\mathrm{poly}(T,1/\epsilon,1/\delta)$ queries to the input and that approximates $Q$'s acceptance probability to within an additive error $\epsilon$ on a $1-\delta$ fraction of inputs. In other words, any quantum algorithm can be simulated on most inputs by a classical algorithm which is only polynomially slower, in terms of query complexity.