On the sum-of-squares degree of symmetric quadratic functions

TL;DR

This paper investigates the sum-of-squares degree of symmetric quadratic functions over the boolean hypercube, providing bounds and applications in complexity theory, optimization, and quantum algorithms.

Contribution

It establishes bounds on the sum-of-squares degree for specific symmetric functions and applies these results to improve understanding of extension complexity, Positivstellensatz refutations, and quantum query complexity.

Findings

Lower bounds on sum-of-squares degree for approximation in various norms.

Optimality of Grigoriev's Positivstellensatz degree lower bound.

Bounds on quantum query complexity for approximating these functions.

Abstract

We study how well functions over the boolean hypercube of the form $f_k(x)=(|x|-k)(|x|-k-1)$ can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in $\ell_{\infty}$-norm as well as in $\ell_1$-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on $\ell_1$-approximation of $f_k$; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from his work; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.