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. A natural question posed there asks if bounded quartic polynomials can be approximated by $2$-query quantum algorithms. Arunachalam, Palazuelos and the first author showed that there is no direct analogue of the result of Aaronson et al. in this case. We improve on this result in the following ways: First, we point out and fix a small error in the construction that has to do with a translation from cubic to quartic polynomials. Second, we give a completely explicit example based on techniques from additive combinatorics. Third, we show that the result still holds when we allow for a small additive error. For this, we apply an SDP characterization of Gribling and Laurent (QIP'19) for the completely-bounded approximate degree.

翻译：Aaronson等人（CCC'16）提出的一个令人惊讶的“多项式方法逆定理”表明：任何有界二次多项式都可以通过一个1-查询算法精确计算其期望值，仅涉及一个与著名Grothendieck常数相关的普适乘法因子。该工作自然提出的一个问题是：有界四次多项式是否可以被2-查询量子算法近似。Arunachalam、Palazuelos与首位作者证明在此情形下不存在Aaronson等人结果的直接类比。我们从以下方面改进了该结果：首先，指出并修正了构造中涉及从三次多项式到四次多项式转换的一个小错误；其次，基于加性组合学技术给出了一个完全显式的例子；第三，证明了当允许小加性误差时该结果仍然成立。为此，我们应用了Gribling与Laurent（QIP'19）关于完全有界近似度的SDP刻画。