Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators, and applications

Ahlswede and Winter [IEEE Trans. Inf. Th. 2002] introduced a Chernoff bound for matrix-valued random variables, which is a non-trivial generalization of the usual Chernoff bound for real-valued random variables. We present an efficient derandomization of their bound using the method of pessimistic estimators (see Raghavan [JCSS 1988]). As a consequence, we derandomize an efficient construction by Alon and Roichman [RSA 1994] of an expanding Cayley graph of logarithmic degree on any (possibly non-abelian) group. This gives an optimal solution to the homomorphism testing problem of Shpilka and Wigderson [STOC 2004]. We also apply these pessimistic estimators to the problem of solving semidefinite covering problems, thus giving a deterministic algorithm for the quantum hypergraph cover problem of Ahslwede and Winter. ∗Partially supported by NSF grant CCR-0324906 †Supported by an NDSEG Graduate Fellowship and a NSF Graduate Fellowship ACM Classification: G.3, G.2.2, F.2.1, F.1.2 AMS Classification: 68W20, 68R10, 60F10, 20D60, 81P68, 15A18