  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.
