Here we revisit one of the prototypical tasks for characterizing the structure of noise in quantum devices: estimating every eigenvalue of an $n$-qubit Pauli noise channel to error $\epsilon$. Prior work [14] proved no-go theorems for this task in the practical regime where one has a limited amount of quantum memory, e.g. any protocol with $\le 0.99n$ ancilla qubits of quantum memory must make exponentially many measurements, provided it is non-concatenating. Such protocols can only interact with the channel by repeatedly preparing a state, passing it through the channel, and measuring immediately afterward. This left open a natural question: does the lower bound hold even for general protocols, i.e. ones which chain together many queries to the channel, interleaved with arbitrary data-processing channels, before measuring? Surprisingly, in this work we show the opposite: there is a protocol that can estimate the eigenvalues of a Pauli channel to error $\epsilon$ using only $O(\log n/\epsilon^2)$ ancilla and $\tilde{O}(n^2/\epsilon^2)$ measurements. In contrast, we show that any protocol with zero ancilla, even a concatenating one, must make $\Omega(2^n/\epsilon^2)$ measurements, which is tight. Our results imply, to our knowledge, the first quantum learning task where logarithmically many qubits of quantum memory suffice for an exponential statistical advantage. Our protocol can be naturally extended to a protocol that learns the eigenvalues of Pauli terms within any subset $A$ of a Pauli channel with $O(\log\log(|A|)/\epsilon^2)$ ancilla and $\tilde{O}(n^2/\epsilon^2)$ measurements.

翻译：本文重新审视了表征量子设备噪声结构的一项典型任务：以误差$\epsilon$估计一个$n$量子比特泡利噪声信道的所有特征值。先前研究[14]在量子内存受限的实际场景中证明了该任务的不可行性定理，例如任何使用$\le 0.99n$个辅助量子比特内存且非级联的协议必然需要指数级测量次数。此类协议只能通过重复制备态、使其通过信道并立即测量的方式与信道交互。这留下了一个自然问题：对于通用协议（即多次级联调用信道、其间穿插任意数据处理信道再进行测量的协议），该下界是否仍然成立？令人惊讶的是，本研究表明相反结论：存在一种协议仅需$O(\log n/\epsilon^2)$个辅助量子比特和$\tilde{O}(n^2/\epsilon^2)$次测量即可实现泡利信道特征值的$\epsilon$精度估计。与之对比，我们证明任何零辅助量子比特的协议（即使是级联协议）必须进行$\Omega(2^n/\epsilon^2)$次测量，该下界是紧致的。据我们所知，我们的研究结果首次揭示了量子学习任务中，对数级量子比特内存即可实现指数级统计优势。本协议可自然扩展至学习泡利信道任意子集$A$中泡利项特征值的方案，仅需$O(\log\log(|A|)/\epsilon^2)$个辅助量子比特和$\tilde{O}(n^2/\epsilon^2)$次测量。