Randomness-Efficient Sampling within NC1

. We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth circuits with parity gates (i.e., in uniform AC 0 [⊕]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC 1 . For example, we obtain the following results: Randomness-efficient error-reduction for uniform probabilistic NC 1 , TC 0 , AC 0 [⊕] and AC 0 : Any function computable by uniform probabilistic circuits with error 1/3 using r random bits is computable by circuits of the same type with error δ using r + O (log(1/δ)) random bits. An optimal bitwise ϵ-biased generator in AC 0 [⊕]: There exists a 1/2 Ω( n ) -biased generator G : {0, 1} O ( n ) → {0, 1} 2 n for which poly( n )-size uniform AC 0 [⊕] circuits can compute G ( s ) i given ( s, i ) ∈ {0, 1} O ( n ) × {0, 1} n . This resolves question raised by Gutfreund and Viola ( Random 2004 ). uniform BP · AC 0 ⊆ uniform AC 0 / O ( n ). Our sampler is based on the zig-zag graph product of Reingold, Vadhan & Wigderson ( Annals of Math 2002 ) and as part of our analysis we givean elementary proof of a generalization of Gillman’s Chernoff Bound for Expander Walks ( SIAM Journal on Computing 1998 ).