Derandomizing the AW matrix-valued Chernoff bound using pessimistic estimators and applications

Ahlswede and Winter [AW02] 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 [Rag88]). As a consequence, we derandomize a construction of Alon and Roichman [AR94] (see also [LR04, LS04]) to efficiently construct an expanding Cayley graph of logarithmic degree on any (possibly non-abelian) group. This also gives an optimal solution to the homomorphism testing problem of Shpilka and Wigderson [SW04]. We also apply these pessimistic estimators to the problem of solving semi-definite covering problems, thus giving a deterministic algorithm for the quantum hypergraph cover problem of [AW02]. The results above appear as theorems in the paper [WX05a], as consequences to the main theorem of that paper: a randomness efficient sampler for matrix valued functions via expander walks. However, we discovered an error in the proof of that main theorem (which we briefly describe in the appendix). One purpose of the current paper is to show that the applications in that paper hold true despite this error.