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

In analogy with ε-biased sets over Z2 , we construct explicit ε-biased sets over nonabelian finite groups G. That is, we find sets S ⊂ G such that ‖Ex∈S ρ(x)‖ ≤ ε for any nontrivial irreducible representation ρ. Equivalently, such sets make G’s Cayley graph an expander with eigenvalue |λ| ≤ ε. The Alon-Roichman theorem shows that random sets of sizeO(log |G|/ε2) suffice. For groups of the form G = G1×· · ·×Gn, our construction has size poly(maxi |Gi|, n, ε), and we show that a set S ⊂ G considered by Meka and Zuckerman that fools read-once branching programs over G is also ε-biased in this sense. For solvable groups whose abelian quotients have constant exponent, we obtain ε-biased sets of size (log |G|)1+o(1) poly(ε). 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.