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Grothendieck inequalities characterize converses to the polynomial method

Jop Briët, Francisco Escudero Gutiérrez, Sander Gribling

TL;DR

It is shown 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, but this result does not generalize to quartic polynomials and 2-query algorithms, even when the authors allow for additive approximations.

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.

Grothendieck inequalities characterize converses to the polynomial method

TL;DR

It is shown 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, but this result does not generalize to quartic polynomials and 2-query algorithms, even when the authors allow for additive approximations.

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.
Paper Structure (24 sections, 31 theorems, 74 equations)

This paper contains 24 sections, 31 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Our contributions.
  3. Our main technical result
  4. Organization
  5. Preliminaries
  6. Notation
  7. Polynomials
  8. Tensors, norms and quantum query complexity
  9. Block-multilinear forms
  10. $\mathcal{E}(p,t)$ for block-multilinear forms
  11. Reformulation of the completely bounded norm of a form
  12. Contractivity of the projector $\Pi_Q$.
  13. Putting everything together
  14. Separations between infinity and completely bounded norms
  15. Lower bounding the completely bounded norm
  16. ...and 9 more sections

Key Result

Theorem 1.1

There exists an absolute constant $C\in(0,1]$ such that $\mathcal{E}(Cp,1)=0$ for every bounded polynomial $p$ of degree at most $2$.

Theorems & Definitions (41)

  • Theorem 1.1: Quadratic multiplicative converse Aaronson2015PolynomialsQQ
  • Corollary 1.2: Quadratic additive converse
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: Grothendieck's theorem
  • Theorem 1.7: Informal version of \ref{['theo:lowerSDP']}
  • Theorem 2.1: Quantum query algorithms are completely bounded forms
  • Definition 2.2
  • Definition 2.3
  • ...and 31 more