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

TL;DR

This paper constructs explicit epsilon-biased sets over nonabelian groups, derandomizing the Alon-Roichman theorem, and introduces techniques like derandomized squaring and a Chernoff-like bound for operator norms.

Contribution

It provides explicit constructions of epsilon-biased sets for nonabelian groups, extending prior random-based results and introducing new derandomization techniques.

Findings

Constructed epsilon-biased sets for nonabelian groups.

Size of sets for product groups is polynomial in group parameters.

Derived a Chernoff-like bound for norms of random operator products.

Abstract

In analogy with epsilon-biased sets over Z_2^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| / epsilon^2) suffice. For groups of the form G = G_1 x ... x G_n, our construction has size poly(max_i |G_i|, n, epsilon^{-1}), and we show that a set S \subset G^n 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.