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. We introduce a distribution $D_{n}$ and give a constant-depth, $n$ qubit, quantum circuit that 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 $n + n^\delta$ uniformly random bits 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 which can't be reproduced by constant-depth bounded fan-in classical circuits, even up to additive error. The distribution $D_n$ and classical circuit lower bounds are based on 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最近的工作表明,存在一个持续深度量子电路可以解决的搜索问题,但任何固定深度的古典电路都无法解决,但任何固定深度的、带封扇形的古典电路都无法解决。 它们也提出了这样一个问题: 我们能否为投入独立的取样任务找到类似的分离证据? 在本文中, 我们显示对这一问题的答案是肯定的。 我们引入了一个分配 $D ⁇ n}, 并给出一个恒定深度的量子电路, 即从接近于$D ⁇ n 的分布中提取到总变化距离的量子电路。 对于任何以固定深度的粉丝 < 1 美元为底部的古典电路, 我们无条件地证明, 任何以连接扇门为输入 $ + n ⁇ delta$ 统一随机点, 并产生接近$D ⁇ n 美元 的输出。 我们无条件地证明, 定深度的量电路可以从分布中提取不易被持续深度粉红度的粉丝- 的粉丝路, 也无法在常规路流中复制。