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

TL;DR

This paper proves a special case of the Aaronson-Ambainis conjecture, showing that certain quantum algorithms can be approximated classically on most inputs, using techniques from free probability theory.

Contribution

It establishes influence bounds for completely bounded block-multilinear forms, linking them to quantum query algorithms and enabling classical simulation of some quantum processes.

Findings

Existence of influential variables in bounded block-multilinear forms

Classical almost-everywhere simulation for specific quantum algorithms

Application to quantum algorithms like k-fold Forrelation

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.