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

##### Author(s)

Barak, Boaz; Brandao, Fernando G.S.L.; Harrow, Aram W.; Steurer, David; Zhou, Yuan; Kelner, Jonathan Adam; ... Show more Show less
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We study the computational complexity of approximating the 2-to-q norm of linear operators (defined as |A|[subscript 2->q] = max[subscript v≠ 0]|Av|[subscript q]/|v|[subscript 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: For any constant even integer q ≥ 4, a graph G is a small-set 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[superscript 2/q]) time, thus obtaining a different proof of the known subexponential algorithm for Small-Set-Expansion. 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[superscript 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. 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.

##### Description

Original manuscript October 30, 2012

##### Date issued

2012-05##### Department

Massachusetts Institute of Technology. Department of Mathematics; Massachusetts Institute of Technology. Department of Physics; McGovern Institute for Brain Research at MIT##### Journal

Proceedings of the 44th symposium on Theory of Computing (STOC '12)

##### Publisher

Association for Computing Machinery (ACM)

##### Citation

Boaz Barak, Fernando G.S.L. Brandao, Aram W. Harrow, Jonathan Kelner, David Steurer, and Yuan Zhou. 2012. Hypercontractivity, sum-of-squares proofs, and their applications. In Proceedings of the 44th symposium on Theory of Computing (STOC '12). ACM, New York, NY, USA, 307-326.

Version: Original manuscript

##### ISBN

9781450312455