Randomness-Efficient Sampling within NC1

Abstract.We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth 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 NC1. For example, we obtain the following results: Randomness-efficient error-reduction for uniform probabilistic NC1, TC0, AC0[⊕] and AC0: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 AC0[⊕]: There exists a 1/2Ω(n)-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 question raised by Gutfreund and Viola (Random 2004).uniform BP · AC0 ⊆ uniform AC0/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).