Hypercontractivity, sum-of-squares proofs, and their applications

Abstract

We study the computational complexity of approximating the 2-toq norm of linear operators (defined as A 2→q = max v 0 Av| q / v 2 ) for q > 2, as well as connections between this question and issues arising in quantum information theory and the study of Khot's Unique Games Conjecture (UGC). We show the following:1. For any constant even integer q 4, a graph G is a smallset expander if and only if the projector into the span of the top eigenvectors of G's adjacency matrix has bounded 2 → q norm. As a corollary, a good approximation to the 2 → q norm will refute the Small-Set Expansion Conjecture -a close variant of the UGC. We also show that such a good approximation can be obtained in exp(n 2/q ) time, thus obtaining a different proof of the known subexponential algorithm for Small-Set Expansion.2. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy certify an upper bound on the 2 → 4 norm of the projector to low degree polynomials over the Boolean cube, as well certify the unsatisfiability of the "noisy cube" and "short code" based instances of Unique Games considered by prior works. This improves on the previous upper bound of exp(log O(1) n) rounds (for the "short code"), as well as separates the "Sum of Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to require ω(1) rounds.3. We show reductions between computing the 2 → 4 norm and computing the injective tensor norm of a tensor, a problem with connections to quantum information theory. Three corollaries are: (i) the 2 → 4 norm is NP-hard to approximate to precision inverse-polynomial in the dimension, (ii) the 2 → 4 norm does not have a good approximation (in the sense above) unless 3-SAT can be solved in time exp( √ n poly log(n)), and (iii) known algorithms for the quantum separability problem imply a non-trivial additive approximation for the 2 → 4 norm.

An Improved Semidefinite Programming Hierarchy for Testing Entanglement

“…A more sophisticated example of this scaling based on an M arising from a Bell test related to the unique games problem is in [30]. One of the major open questions in this area is whether low levels of SDP hierarchies such as DPS can resolve hard optimizations problems of intermediate complexity such as the unique games problem [15]. A point x ∈ V is a singular point if the matrix J has less than full rank at x. V is smooth if it has no singular points.…”

“…Second, h Sep is closely related to the problems of estimating the 2 → 4 norm of a matrix, finding the least-expanding small set in a graph and estimating the optimum value of a unique game [15]. These problems in turn relate to the approximation complexity of constraint satisfaction problems, which are an extremely general class of discrete optimization problems.…”

On the Hardest Problem Formulations for the $$0/1$$ Lasserre Hierarchy

“…3 In particular for 0/1 problems L vanishes on the truncated ideal generated by (2) is a relaxation since one can take L to be the evaluation map f → f (x * ) for any optimal solution x * . Relaxation (2) can be equivalently formulated in terms of moment matrices [24].…”

The Proof-Search Problem between Bounded-Width Resolution and Bounded-Degree Semi-algebraic Proofs

“…For these reasons, SA + k and LS + k are called the semi-definite versions of SA k and LS k . The catchy acronym SoS (for sum-of-squares) is also used for certain versions of SA + k (see [35], [5]). …”