In this work, we show that learning the output distributions of brickwork random quantum circuits is average-case hard in the statistical query model. This learning model is widely used as an abstract computational model for most generic learning algorithms. In particular, for brickwork random quantum circuits on $n$ qubits of depth $d$, we show three main results: - At super logarithmic circuit depth $d=\omega(\log(n))$, any learning algorithm requires super polynomially many queries to achieve a constant probability of success over the randomly drawn instance. - There exists a $d=O(n)$, such that any learning algorithm requires $\Omega(2^n)$ queries to achieve a $O(2^{-n})$ probability of success over the randomly drawn instance. - At infinite circuit depth $d\to\infty$, any learning algorithm requires $2^{2^{\Omega(n)}}$ many queries to achieve a $2^{-2^{\Omega(n)}}$ probability of success over the randomly drawn instance. As an auxiliary result of independent interest, we show that the output distribution of a brickwork random quantum circuit is constantly far from any fixed distribution in total variation distance with probability $1-O(2^{-n})$, which confirms a variant of a conjecture by Aaronson and Chen.

翻译：在这项工作中，我们证明了在统计查询模型下，学习砖墙随机量子电路的输出分布是平均情况困难的。该学习模型被广泛用作大多数通用学习算法的抽象计算模型。具体而言，对于 $n$ 个量子比特、深度为 $d$ 的砖墙随机量子电路，我们展示了三个主要结果：- 在超对数电路深度 $d=\omega(\log(n))$ 时，任何学习算法都需要超多项式次查询才能以恒定概率在随机抽取的实例上成功。- 存在一个 $d=O(n)$，使得任何学习算法需要 $\Omega(2^n)$ 次查询才能以 $O(2^{-n})$ 的概率在随机抽取的实例上成功。- 在无限电路深度 $d\to\infty$ 时，任何学习算法需要 $2^{2^{\Omega(n)}}$ 次查询才能以 $2^{-2^{\Omega(n)}}$ 的概率在随机抽取的实例上成功。作为一个具有独立兴趣的辅助结果，我们证明了砖墙随机量子电路的输出分布以概率 $1-O(2^{-n})$ 在总变差距离上始终远离任何固定分布，这证实了Aaronson和Chen猜想的一个变体。