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

Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, David Steurer, Yuan Zhou

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We study the computational complexity of approximating the 2-to-q 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 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 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.

Original languageEnglish (US)
Title of host publicationSTOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing
Number of pages20
StatePublished - Jun 26 2012
Externally publishedYes
Event44th Annual ACM Symposium on Theory of Computing, STOC '12 - New York, NY, United States
Duration: May 19 2012May 22 2012

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Other44th Annual ACM Symposium on Theory of Computing, STOC '12
Country/TerritoryUnited States
CityNew York, NY


  • hypercontractive
  • injective tensor norm
  • semidefinite programming
  • unique games conjecture

ASJC Scopus subject areas

  • Software


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