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.

翻译：我们给出了Arunachalam、Briët和Palazuelos（SICOMP'19）主要结果的新表述，并证明量子查询算法可由一类称为傅里叶完全有界多项式的新多项式刻画。我们猜想所有此类多项式均存在一个具有影响力的变量。该猜想弱于著名的Aaronson-Ambainis（AA）猜想（Theory of Computing'14），但与量子查询算法的经典模拟具有相同的推论。通过证明齐次傅里叶完全有界多项式满足AA猜想，我们验证了该猜想的一个新情形：若$d$次查询量子算法的输出为$2d$次齐次多项式$p$，则存在一个影响系数至少为$Var[p]^2$的变量。此外，我们给出了Bansal、Sinha和de Wolf（CCC'22与QIP'23）关于分块多重线性完全有界多项式存在影响变量结果的另一种证明。该证明更为简洁、获得了更优常数，且无需使用随机性。