Small-Bias Sets for Nonabelian Groups: Derandomizing the Alon-Roichman Theorem

In analogy with epsilon-biased sets over Z2^n, we construct explicit epsilon-biased sets over nonabelian finite groups G. That is, we find sets S subset G such that | Exp{x in S} rho(x)| <= epsilon for any nontrivial irreducible representation rho. Equivalently, such sets make G's Cayley graph an expander with eigenvalue |lambda| <= epsilon. The Alon-Roichman theorem shows that random sets of size O(log |G| / epsilon²⁾ suffice. For groups of the form G = G1 x ... x Gn, our construction has size poly(maxi |Gi|, n, epsilon^{-1}), and we show that a set S \subset Gⁿ considered by Meka and Zuckerman that fools read-once branching programs over G is also epsilon-biased in this sense. For solvable groups whose abelian quotients have constant exponent, we obtain epsilon-biased sets of size (log |G|)^{1+o(1)} poly(epsilon^{-1}). Our techniques include derandomized squaring (in both the matrix product and tensor product senses) and a Chernoff-like bound on the expected norm of the product of independently random operators that may be of independent interest.