Hardness Results for Weaver's Discrepancy Problem

Abstract: Marcus, Spielman and Srivastava (Annals of Mathematics 2014) solved the Kadison--Singer Problem by proving a strong form of Weaver's conjecture: they showed that for all $\alpha > 0$ and all lists of vectors of norm at most $\sqrt{\alpha}$ whose outer products sum to the identity, there exists a signed sum of those outer products with operator norm at most $\sqrt{8 \alpha} + 2 \alpha.$ We prove that it is NP-hard to distinguish such a list of vectors for which there is a signed sum that equals the zero matrix from those in which every signed sum has operator norm at least $\kappa \sqrt{\alpha}$, for some absolute constant $\kappa > 0.$ Thus, it is NP-hard to construct a signing that is a constant factor better than that guaranteed to exist. For $\alpha = 1/4$, we prove that it is NP-hard to distinguish whether there is a signed sum that equals the zero matrix from the case in which every signed sum has operator norm at least $1/4$.