学习随机量子电路与伪随机性的涌现
Learning Random Quantum Circuits and the Emergence of Pseudorandomness
AI总结：
本文提出一种高效学习随机量子电路的算法，利用局部相关性准则和维度无关的反集中不等式，在ℓd=O(log n)时达到多项式时间，并阐明了伪随机性涌现的条件。
AI中文摘要：
我们给出了一种高效算法，仅利用将U作用于全零输入所得到的输出态的副本，即可学习k维砖墙随机量子电路。对于n个格点上、深度为d、由随机2ℓ量子比特门构成的电路，该算法在poly(n,2^{ℓd})时间内以高概率学习到原始电路U，适用于任意常数维晶格。特别地，只要ℓd=O(log n)，算法即为多项式时间。在一维情形下，这达到了伪随机性所暗示的自然边界：伪随机态要求ℓd=ω(log n)，而结构化密码学构造表明这一尺度可能从上方实现。在更高维度中，同样的ℓd=ω(log n)尺度仍可能是无辅助比特伪随机性的阈值，我们的结果有助于阐明在该设置下伪随机性得以出现的条件。我们算法的核心思想是一个局部相关性准则，它能够识别最后一层中的门而无需学习其整个后向光锥，从而避免了先前方法中的瓶颈。一个关键的技术要素是Carbery-Wright类型的反集中不等式，适用于Haar随机酉的低次多项式，其小球指数与矩阵维度无关。该与维度无关的指数对于处理不断增长的门的局域性至关重要。这一反集中结果也可能具有独立的研究价值。
英文摘要：
We give an efficient algorithm for learning $k$-dimensional brickwork random quantum circuits using only copies of the output state obtained by applying $U$ to the all-zero input. For a depth-$d$ circuit on $n$ sites with random $2\ell$-qubit gates, the algorithm learns the original circuit $U$ with high probability in $\text{poly}(n,2^{\ell d})$ time for every constant dimensional lattice. In particular, the algorithm is polynomial time as long as $\ell d = O(\log n)$. In one dimension, this reaches the natural boundary suggested by pseudorandomness: pseudorandom states require $\ell d=ω(\log n)$, and structured cryptographic constructions suggest that this scale may be achievable from above. In higher dimensions, it remains plausible that the same $\ell d=ω(\log n)$ scale remains the threshold for ancilla-free pseudorandomness, and our results help clarify the conditions under which pseudorandomness can arise in this setting. The main idea behind our algorithm is a local correlation criterion that identifies gates in the final layer without learning their entire backward light cones, avoiding a bottleneck in the previous approaches. A key technical ingredient is a Carbery-Wright type anticoncentration inequality for low-degree polynomials of Haar random unitaries whose small-ball exponent is independent of the matrix dimension. The dimension-independent exponent is crucial for handling gates of growing locality. This anticoncentration result may also be of independent interest.
