Randomness-Efficient Sampling within NC

We construct a randomness-efficient averaging sampler that is computable by uniform constantdepth circuits with parity gates (i.e., in uniform AC0[⊕]). 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. For example, we obtain the following results: • Randomness-efficient error-reduction for uniform probabilistic NC, TC, AC0[⊕] and AC: Any function computable by uniform probabilistic circuits with error 1/3 using r random bits is computable by uniform probabilistic circuits with error δ using O(r+log(1/δ)) random bits. • An optimal explicit -biased generator in AC0[⊕]: There exists a 1/2-biased generator G : {0, 1}O(n) → {0, 1}2n for which poly(n)-size uniform AC0[⊕] circuits can compute G(s)i given (s, i) ∈ {0, 1}O(n) × {0, 1}n. This resolves a question raised by Gutfreund and Viola (Random 2004). • uniform BP · AC ⊆ uniform AC/O(n). Our sampler is based on the zig-zag graph product of Reingold, Vadhan and Wigderson (Annals of Math 2002) and as part of our analysis we give an elementary proof of a generalization of Gillman’s Chernoff Bound for Expander Walks (FOCS 1998).