AI 中文总结
该研究证明测量反馈可让自适应浅电路实现量子优势，固定相干深度下增加反馈结果能扩大可访问函数类，多项式反馈可将素域离散对数问题编码为单量子比特期望值，经典估计该期望值是困难的。
AI 中文摘要
量子优势通常被认为需要足够深的量子电路，这样关联和全局计算结构才能增长到超出经典高效模拟的范围。对于带有局域读出的常深度电路，这种预期尤为明显：任何固定局域可观测量的期望值都处于有界后向光锥内，因此在经典上是可处理的。本文证明测量反馈改变了这一情况，我们建立了计算能力的严格层级：在固定相干深度下，增加反馈结果的数量会严格扩大通过局域期望值可访问的函数类。该层级的两端呈现出不同的计算机制：使用对数级反馈时，乘积态输入的局域期望值是经典可高效模拟的；相比之下，多项式级反馈可使一类显式自适应浅电路将素域离散对数问题（DLP）编码为固定单量子比特期望值。假设DLP具有标准最坏情况经典困难性，估计该期望值在经典上是困难的。这些结果揭示了一种反馈驱动的复杂度相变，对DLP的资源下界以及面积律纠缠下局域可观测量估计的复杂度具有进一步启示，为用浅量子电路实现量子优势开辟了新途径。
英文摘要
Quantum advantage is widely expected to require sufficiently deep circuits, where correlations and global computational structure can grow beyond the reach of efficient classical simulation. This expectation is especially stark for constant-depth circuits with local readout: the expectation value of any fixed local observable lies within a bounded backward lightcone and is therefore classically tractable. Here we show that measurement feedback changes this picture. We establish a strict hierarchy of computational power: at fixed coherent depth, increasing the number of feedback outcomes strictly enlarges the class of functions accessible through a local expectation value. The two ends of this hierarchy exhibit distinct computational regimes. With logarithmic feedback, local expectation values for product-state inputs are efficiently classically simulable. Polynomial feedback, by contrast, enables an explicit family of adaptive shallow circuits to encode prime-field discrete logarithm problem~(DLP) into a fixed single-qubit expectation. Assuming the standard worst-case classical hardness of DLP, estimating this expectation value is classically hard. These results reveal a feedback-driven complexity transition, with further implications for resource lower bounds on DLP and the complexity of local-observable estimation under area-law entanglement. Our results open a new route to quantum advantage with shallow quantum circuits.
