Randomness-Efficient Sampling Within NC 1

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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 uniform probabilistic circuits with error δ using r + O(log(1/δ)) random bits. Optimal explicit ε-biased generator in AC 0[⊕]: There is a 1/2Ω( n)-biased generator \(G:{0, 1}^{O(n)} \to {0, 1}^{2^n}\) 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 a question raised by Gutfreund & 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 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 1994).