  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.
