Small-Bias Sets for Nonabelian Groups

Abstract

In analogy with ε-biased sets over ${\mathbb Z}_2^n$, we construct explicit ε-biased sets over nonabelian finite groups G. That is, we find sets S ⊂ G such that $\parallel{{\mathbb E}_{x \in S} \rho(x)} \parallel \leq \epsilon$ 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 size O(log|G| / ε ²) suffice. For groups of the form G = G ₁ × ⋯ ×G _n , our construction has size poly( max _i |G _i |, n, ε ^− 1), and we show that a specific set S ⊂ G ⁿ 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 ^− 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.