Influences of Fourier Completely Bounded Polynomials and Classical Simulation of Quantum Algorithms

TL;DR

This paper characterizes quantum query algorithms using Fourier completely bounded polynomials, explores their influential variables, and proves new cases of the Aaronson-Ambainis conjecture with implications for classical simulation.

Contribution

It introduces Fourier completely bounded polynomials as a new characterization of quantum algorithms and proves a new case of the Aaronson-Ambainis conjecture for homogeneous polynomials.

Findings

Quantum query algorithms are characterized by Fourier completely bounded polynomials.

Homogeneous Fourier completely bounded polynomials have influential variables.

New simpler proof for influential variables in block-multilinear completely bounded polynomials.

Abstract

We give a new presentation of the main result of Arunachalam, Bri\"et and Palazuelos (SICOMP'19) and show that quantum query algorithms are characterized by a new class of polynomials which we call Fourier completely bounded polynomials. We conjecture that all such polynomials have an influential variable. This conjecture is weaker than the famous Aaronson-Ambainis (AA) conjecture (Theory of Computing'14), but has the same implications for classical simulation of quantum query algorithms. We prove a new case of the AA conjecture by showing that it holds for homogeneous Fourier completely bounded polynomials. This implies that if the output of $d$-query quantum algorithm is a homogeneous polynomial $p$ of degree $2d$, then it has a variable with influence at least $Var[p]^2$. In addition, we give an alternative proof of the results of Bansal, Sinha and de Wolf (CCC'22 and QIP'23) showing that block-multilinear completely bounded polynomials have influential variables. Our proof is simpler, obtains better constants and does not use randomness.