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arXiv · 2212.08559

Grothendieck inequalities characterize converses to the polynomial method

Jop Briët·Francisco Escudero Gutiérrez·Sander Gribling
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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.

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Jop Briët, Francisco Escudero Gutiérrez, Sander Gribling. 2022-12-16. Grothendieck inequalities characterize converses to the polynomial method. https://doi.org/10.22331/q-2024-11-18-1526

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