Recent work by Bravyi et al. constructs a relation problem that a noisy constant-depth quantum circuit (QNC$^0$) can solve with near certainty (probability $1 - o(1)$), but that any bounded fan-in constant-depth classical circuit (NC$^0$) fails with some constant probability. We show that this robustness to noise can be achieved in the other low-depth quantum/classical circuit separations in this area. In particular, we show a general strategy for adding noise tolerance to the interactive protocols of Grier and Schaeffer. As a consequence, we obtain an unconditional separation between noisy QNC$^0$ circuits and AC$^0[p]$ circuits for all primes $p \geq 2$, and a conditional separation between noisy QNC$^0$ circuits and log-space classical machines under a plausible complexity-theoretic conjecture. A key component of this reduction is showing average-case hardness for the classical simulation tasks -- that is, showing that a classical simulation of the quantum interactive task is still powerful even if it is allowed to err with constant probability over a uniformly random input. We show that is true even for quantum tasks which are $\oplus$L-hard to simulate. To do this, we borrow techniques from randomized encodings used in cryptography.

中文翻译：

---

具有嘈杂的浅克利福德电路的交互式量子优势

Bravyi等人的最新工作。构造一个关系问题，一个嘈杂的恒定深度量子电路（QNC $ ^ 0 $）可以几乎确定地解决（概率$ 1-o（1）$），但是任何有界扇入式恒定深度经典电路（NC $ ^ 0 $）以一定的概率失败。我们表明，可以在该区域的其他低深度量子/经典电路分离中实现这种对噪声的鲁棒性。特别是，我们展示了为Grier和Schaeffer的交互协议增加抗噪能力的一般策略。结果，对于所有素数$ p \ geq 2 $，我们得到了嘈杂的QNC $ ^ 0 $电路和AC $ ^ 0 [p] $电路之间的无条件分离，以及嘈杂的QNC $ ^ 0 $电路与在合理的复杂性理论猜想下的对数空间经典机器。这种减少的一个关键组成部分是显示经典模拟任务的平均情况硬度-也就是说，即使允许在均匀随机输入上以恒定概率犯错误，量子交互任务的经典模拟仍然具有强大的功能。 。我们证明即使对于难以模拟$ \ oplus $ L的量子任务也是如此。为此，我们借鉴了加密技术中使用的随机编码技术。