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.

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