A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the famous Grothendieck constant. Here we show that such a result does not generalize to quartic polynomials and 2-query algorithms, even when we allow for additive approximations. We also show that the additive approximation implied by their result is tight for bounded bilinear forms, which gives a new characterization of the Grothendieck constant in terms of 1-query quantum algorithms. Along the way we provide reformulations of the completely bounded norm of a form, and its dual norm.

翻译：Aaronson等人（CCC'16）提出的一个令人惊讶的"多项式方法逆问题"表明，任何有界二次多项式都可以通过一个1次查询算法精确计算其期望值，其误差上限与著名的Grothendieck常数相关的一个普适乘法因子有关。本文证明，即使允许加性逼近，该结论也无法推广至四次多项式与2次查询算法。我们还证明了该结论对有界双线性形式所隐含的加性逼近是紧的，这给出了Grothendieck常数在1次查询量子算法意义下的新刻画。在此过程中，我们提供了形式的有界完全范数及其对偶范数的重新表述。