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.

中文翻译：

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Grothendieck 不等式表征与多项式方法相反


Aaronson 等人 （CCC'16） 令人惊讶的“与多项式方法相反”表明，任何有界二次多项式都可以通过 1 查询算法精确计算，直到与著名的 Grothendieck 常数相关的通用乘法因子。在这里，我们表明这样的结果并不能推广到四次多项式和 2 查询算法，即使我们允许加法近似。我们还表明，对于有界双线性形式，他们的结果隐含的加法近似是紧密的，这给出了 1 查询量子算法对 Grothendieck 常数的新表征。在此过程中，我们提供了对形式的完全有界规范及其双重规范的重新表述。