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与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族刻画。我们猜想所有此类多项式均具有一个影响性变量。该猜想弱于著名的Aaronson-Ambainis（AA）猜想（Theory of Computing'14），但对其在量子查询算法经典模拟中的推论保持一致。通过证明齐次傅里叶完全有界多项式满足该猜想，我们证实了AA猜想的一个新情形。这表明若d次查询量子算法的输出为度数为2d的齐次多项式p，则存在一个影响力至少为Var[p]²的变量。此外，我们给出了Bansal、Sinha與de Wolf（CCC'22和QIP'23）研究成果的替代性证明，证明了分块多重线性完全有界多项式具有影响性变量。该证明方法更简洁、获得了更好的常数且无需使用随机性。