We develop a framework for learning from noisy quantum experiments in which fault-tolerant devices access uncharacterized systems through noisy couplings. Introducing the complexity class $\textsf{NBQP}$ ("noisy BQP''), we model noisy fault-tolerant quantum computers that cannot generally error-correct the oracle systems they query. Using this class, we prove that while noise can eliminate the exponential quantum learning advantages of unphysical, noiseless learners, a superpolynomial gap remains between $\textsf{NISQ}$ and fault-tolerant devices. Turning to canonical learning tasks in noisy settings, we find that the exponential two-copy advantage for purity testing collapses under local depolarizing noise. Nevertheless, we identify a setting motivated by AdS/CFT in which noise-resilient physical structure restores this quantum learning advantage. We then analyze noisy Pauli shadow tomography, deriving lower bounds characterizing how instance size, quantum memory and noise jointly control sample complexity, and design algorithms with parametrically matching scalings. We study similar tradeoffs in quantum metrology, and show that the Heisenberg-limited sensitivity of existing error-correction-based protocols persists only up to a timescale inverse-polynomial in the error rate per probe qubit. Together, our results demonstrate that the primitives underlying quantum-enhanced experiments are fundamentally fragile to noise, and that realizing meaningful quantum advantages in future experiments will require interfacing noise-robust physical properties with available algorithmic techniques.

翻译：我们建立了一个从噪声量子实验中学习的理论框架，其中容错设备通过噪声耦合访问未表征系统。通过引入复杂度类$\textsf{NBQP}$（"噪声BQP"），我们对噪声容错量子计算机进行建模，这类计算机通常无法对它们所查询的预言机系统进行纠错。利用此类复杂度，我们证明：虽然噪声能够消除非物理、无噪声学习者的指数级量子学习优势，但$\textsf{NISQ}$设备与容错设备之间仍存在超多项式差距。针对噪声环境中的典型学习任务，我们发现纯度测试的指数级双拷贝优势在局域退极化噪声下会崩溃。尽管如此，我们识别出一个受AdS/CFT启发的场景，其中具有噪声弹性的物理结构能够恢复这种量子学习优势。随后我们分析噪声泡利影层析，推导出刻画实例规模、量子存储器与噪声如何共同控制样本复杂度的下界，并设计了参数匹配标度的算法。我们进一步研究量子计量学中的类似权衡关系，证明现有基于纠错协议的Heisenberg极限灵敏度仅能维持到与每个探针量子比特错误率成反多项式关系的时间尺度。综合而言，我们的结果表明：量子增强实验所依赖的基本原语对噪声具有本质脆弱性，未来实验要实现有意义的量子优势，必须将抗噪声的物理特性与现有算法技术相结合。