Quantum speedups need structure

Nathan Keller; Ohad Klein

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arXiv · 1911.03748

Quantum speedups need structure

Nathan Keller·Ohad Klein

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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.

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Nathan Keller, Ohad Klein. 2019-11-09. Quantum speedups need structure. https://arxiv.org/abs/1911.03748

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