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

• Main Authors: de Wolf, Ronald, Yuen, Henry, Lee, Troy, Prakash, Anupam
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2018
• Subjects: Sum-of-squares Degree; Approximation Theory; DRNTU::Science::Physics
• Online Access: https://hdl.handle.net/10356/90218; http://hdl.handle.net/10220/47238
• Institution: Nanyang Technological University

We study how well functions over the boolean hypercube of the form f_k(x)=(lxl-k)(lxl-k-1) can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in l_{infinity}-norm as well as in l_1-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer [Lee/Raghavendra/Steurer, STOC 2015] 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 l_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 [Grigoriev, Comp. Compl. 2001]; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.