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arXiv · 2203.00212

Influence in Completely Bounded Block-multilinear Forms and Classical Simulation of Quantum Algorithms

Nikhil Bansal·Makrand Sinha·Ronald de Wolf
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Abstract

The Aaronson-Ambainis conjecture (Theory of Computing '14) says that every low-degree bounded polynomial on the Boolean hypercube has an influential variable. This conjecture, if true, would imply that the acceptance probability of every $d$-query quantum algorithm can be well-approximated almost everywhere (i.e., on almost all inputs) by a $\mathrm{poly}(d)$-query classical algorithm. We prove a special case of the conjecture: in every completely bounded degree-$d$ block-multilinear form with constant variance, there always exists a variable with influence at least $1/\mathrm{poly}(d)$. In a certain sense, such polynomials characterize the acceptance probability of quantum query algorithms, as shown by Arunachalam, Bri\"et and Palazuelos (SICOMP '19). As a corollary we obtain efficient classical almost-everywhere simulation for a particular class of quantum algorithms that includes for instance $k$-fold Forrelation. Our main technical result relies on connections to free probability theory.

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Nikhil Bansal, Makrand Sinha, Ronald de Wolf. 2022-03-01. Influence in Completely Bounded Block-multilinear Forms and Classical Simulation of Quantum Algorithms. https://arxiv.org/abs/2203.00212

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