Recent work by Bravyi, Gosset, and Koenig showed that there exists a search problem that a constant-depth quantum circuit can solve, but that any constant-depth classical circuit with bounded fan-in cannot. They also pose the question: Can we achieve a similar proof of separation for an input-independent sampling task? In this paper, we show that the answer to this question is yes when the number of random input bits given to the classical circuit is bounded. We introduce a distribution $D_{n}$ over $\{0,1\}^n$ and construct a constant-depth uniform quantum circuit family $\{C_n\}_n$ such that $C_n$ samples from a distribution close to $D_{n}$ in total variation distance. For any $\delta < 1$ we also prove, unconditionally, that any classical circuit with bounded fan-in gates that takes as input $kn + n^\delta$ i.i.d. Bernouli random variables with entropy $1/k$ and produces output close to $D_{n}$ in total variation distance has depth $\Omega(\log \log n)$. This gives an unconditional proof that constant-depth quantum circuits can sample from distributions that can't be reproduced by constant-depth bounded fan-in classical circuits, even up to additive error. We also show a similar separation between constant-depth quantum circuits with advice and classical circuits with bounded fan-in and fan-out, but access to an unbounded number of i.i.d random inputs. The distribution $D_n$ and classical circuit lower bounds are inspired by work of Viola, in which he shows a different (but related) distribution cannot be sampled from approximately by constant-depth bounded fan-in classical circuits.

翻译：Bravyi、Gosset和Koenig近期研究表明，存在一个搜索问题可由恒定深度量子电路解决，而任何恒定深度的有界扇入经典电路都无法解决。他们还提出一个问题：对于与输入无关的采样任务，我们能否实现类似的分离证明？在本文中，我们证明当经典电路获得的随机输入比特数量受限时，这个问题的答案是肯定的。我们引入一个在$\{0,1\}^n$上的分布$D_{n}$，并构造一个恒定深度的均匀量子电路族$\{C_n\}_n$，使得$C_n$以总变差距离接近$D_{n}$的分布进行采样。对于任意$\delta < 1$，我们还无条件证明，任何采用有界扇入门、输入为$kn + n^\delta$个独立同分布伯努利随机变量（熵为$1/k$）且输出在总变差距离上接近$D_{n}$的经典电路，其深度为$\Omega(\log \log n)$。这提供了一个无条件证明：恒定深度量子电路可以从恒定深度有界扇入经典电路无法复现的分布中采样，即使允许加性误差。我们还展示了在辅助量子比特辅助下的恒定深度量子电路与有界扇入扇出但可访问无限数量独立同分布随机输入的经典电路之间的类似分离。分布$D_n$及经典电路下界受Viola工作的启发，他在其中证明了一个不同但相关的分布无法由恒定深度有界扇入经典电路近似采样。