On converses to the polynomial method

TL;DR

This paper explores the limits of quantum algorithms approximating polynomials, showing that bounded quartic polynomials cannot be efficiently computed with few queries, and refines previous results with explicit examples and error bounds.

Contribution

It corrects a previous construction, provides explicit examples using additive combinatorics, and extends the non-approximability result to include small additive errors.

Findings

Bounded quadratic polynomials can be computed exactly with one quantum query.

Bounded quartic polynomials cannot be approximated by 2-query quantum algorithms.

The non-approximability result holds even with small additive errors.

Abstract

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.