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“多项式方法逆命题”表明，任何有界二次多项式都可以在期望中被一个 1-查询算法精确计算，其误差仅涉及著名的 Grothendieck 常数相关的全局乘法因子。本文证明，即使允许加法近似，该结果也无法推广到四次多项式与 2-查询算法。我们还证明了他们结果所隐含的加法近似对于有界双线性形式是紧的，这为 Grothendieck 常数提供了一个基于 1-查询量子算法的新刻画。在此过程中，我们给出了形式的完全有界范数及其对偶范数的重新表述。