class of circuits cannot perform a certain natural task beyond just computing a function.

One such natural task, introduced by Viola [@Vi12], is that of sampling distributions. In this problem, for a given distribution ${{\mathcal D}}$ supported on ${\{0,1\}^{n}}$, we are looking for a function $f \colon {\{0,1\}^{m}} \to {\{0,1\}^{n}}$ that samples (or approximates) ${{\mathcal D}}$, that is, for a uniformly random $x\!~\sim~\!{\{0,1\}^{m}}$, the distribution $f(x)$ is equal (or close to) ${{\mathcal D}}$, and furthermore, each output bit $f_i$ of the function $f$ belongs to some low-level complexity class, such as ${\mathbf{AC^{0}}}$ or ${\mathbf{TC^{0}}}$.

As a concrete example, let $U_{\oplus}$ be the uniform distribution over the set $\{ (x,{{\sf parity}}(x)) \colon x \in {\{0,1\}^{n-1}} \} {\subseteq}{\{0,1\}^{n}}$. Note that although the ${{\sf parity}}$ function is not computable in ${\mathbf{AC^{0}}}$ (see [@H86] and references therein) there is a function $f \colon {\{0,1\}^{n-1}} \to {\{0,1\}^{n}}$ that samples $U_{\oplus}$, such that each output bit depends on only two input bits: $$f(x_1,\dots,x_{n-1}) = (x_1, x_1 + x_2, x_2 + x_3,\dots, x_{n-2} + x_{n-1},x_{n-1}).$$ Motivated by the foregoing somewhat surprising example, Viola [@Vi12] raised the following question.

\[challenge:main\] Exhibit an explicit map $b \colon {\{0,1\}^{n}}\to \{0,1\}$ such that for a uniformly sampled $x \sim {\{0,1\}^{n}}$, the 