In this paper we revisit one of the prototypical tasks for characterizing the structure of noise in quantum devices, estimating the eigenvalues of an $n$-qubit Pauli noise channel. Prior work (Chen et al., 2022) established exponential lower bounds for this task for algorithms with limited quantum memory. We first improve upon their lower bounds and show: (1) Any algorithm without quantum memory must make $\Omega(2^n/\epsilon^2)$ measurements to estimate each eigenvalue within error $\epsilon$. This is tight and implies the randomized benchmarking protocol is optimal, resolving an open question of (Flammia and Wallman, 2020). (2) Any algorithm with $\le k$ ancilla qubits of quantum memory must make $\Omega(2^{(n-k)/3})$ queries to the unknown channel. Crucially, unlike in (Chen et al., 2022), our bound holds even if arbitrary adaptive control and channel concatenation are allowed. In fact these lower bounds, like those of (Chen et al., 2022), hold even for the easier hypothesis testing problem of determining whether the underlying channel is completely depolarizing or has exactly one other nontrivial eigenvalue. Surprisingly, we show that: (3) With only $k=2$ ancilla qubits of quantum memory, there is an algorithm that solves this hypothesis testing task with high probability using a single measurement. Note that (3) does not contradict (2) as the protocol concatenates exponentially many queries to the channel before the measurement. This result suggests a novel mechanism by which channel concatenation and $O(1)$ qubits of quantum memory could work in tandem to yield striking speedups for quantum process learning that are not possible for quantum state learning.

翻译：本文重新审视了量子设备噪声结构表征中的典型任务之一——估计$n$量子比特Pauli噪声信道的特征值。此前工作（Chen等，2022）对量子内存有限的算法建立了该任务的指数下界。我们首先改进了其下界并证明：（1）任何无量子内存的算法必须进行$\Omega(2^n/\epsilon^2)$次测量才能将每个特征值估计至误差$\epsilon$内。该下界是紧的，表明随机基准测试协议是最优的，解决了（Flammia和Wallman，2020）的一个开放问题。（2）任何使用$\le k$个辅助比特量子内存的算法必须对未知信道进行$\Omega(2^{(n-k)/3})$次查询。关键的是，与（Chen等，2022）不同，我们的下界即使允许任意自适应控制和信道级联也成立。事实上，这些下界（如同Chen等（2022））甚至适用于更简单的假设检验问题：判断底层信道是完全退极化信道还是恰好存在另一个非平凡特征值。令人惊讶的是，我们证明：（3）仅使用$k=2$个辅助比特量子内存，存在一种算法通过单次测量就能以高概率解决该假设检验任务。注意（3）并不与（2）矛盾，因为该协议在测量前级联了指数量级的信道查询。该结果揭示了一种新颖机制：信道级联与$O(1)$量子比特量子内存协同作用，可在量子过程学习领域实现量子态学习中无法实现的显著加速。