Grothendieck inequalities characterize converses to the polynomial method

TL;DR

This paper explores the limitations of the polynomial method in quantum computing, showing that certain bounds do not extend to higher-degree polynomials and establishing new characterizations of the Grothendieck constant.

Contribution

It demonstrates the non-generalizability of the converse to the polynomial method to quartic polynomials and 2-query algorithms, and provides new insights into the Grothendieck constant and related norms.

Findings

The converse to the polynomial method does not extend to quartic polynomials.

Additive approximation bounds are tight for bounded bilinear forms.

New reformulations of the completely bounded norm and its dual are provided.

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. 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.