Bounded-Depth Circuits Cannot Sample Good Codes

Abstract

We study a variant of the classical circuit-lower-bound problems: proving lower bounds for sampling distributions given random bits. We prove a lower bound of 1 − 1/n ^Ω(1) on the statistical distance between (i) the output distribution of any small constant-depth (a.k.a. AC⁰) circuit f : {0, 1}^poly(n) → {0, 1}ⁿ, and (ii) the uniform distribution over any code ${\mathcal{C} \subseteq \{0, 1\}^n}$ that is “good,” that is, has relative distance and rate both Ω(1). This seems to be the first lower bound of this kind.

We give two simple applications of this result: (1) any data structure for storing codewords of a good code ${\mathcal{C} \subseteq \{0, 1\}^n}$ requires redundancy Ω(log n), if each bit of the codeword can be retrieved by a small AC⁰ circuit; and (2) for some choice of the underlying combinatorial designs, the output distribution of Nisan’s pseudorandom generator against AC⁰ circuits of depth d cannot be sampled by small AC⁰ circuits of depth less than d.