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 (Chen et al., 2022) 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 qubits 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.

翻译：本文重新审视了量子设备噪声结构表征中的一项原型任务：估计一个n量子比特泡利噪声信道的每个特征值，误差不超过ϵ。先前的工作（Chen等，2022）证明，在实际的限制量子记忆量的情况下，该任务存在不可能性定理。例如，任何使用不超过0.99n个辅助量子比特量子记忆的非串联协议都必须进行指数数量的测量。此类协议仅能通过重复制备一个态、将其通过信道并立即测量来与信道交互。这留下了一个自然的问题：下界是否对一般协议也成立，即在测量之前将多次信道查询与任意数据处理信道交织在一起？令人惊讶的是，本文展示了相反的结果：存在一个协议，仅需O(log n/ϵ^2)个辅助量子比特和Õ(n^2/ϵ^2)次测量，即可估计泡利信道特征值至误差ϵ。相比之下，我们证明任何零辅助量子比特的协议（即使是串联协议）都必须进行Ω(2^n/ϵ^2)次测量，该下界是紧的。据我们所知，我们的结果揭示了首个量子学习任务，其中对数数量的量子记忆量子比特足以实现指数级的统计优势。