  Learning unknown processes affecting a quantum system reveals underlying
physical mechanisms and enables suppression, mitigation, and correction of
unwanted effects. Describing a general quantum process requires an
exponentially large number of parameters. Measuring these parameters, when they
are encoded in incompatible observables, is constrained by the uncertainty
principle and requires exponentially many measurements. However, for Pauli
channels, having access to an ideal quantum memory and entangling operations
allows encoding parameters in commuting observables, thereby exponentially
reducing measurement complexity. In practice, though, quantum memory and
entangling operations are always noisy and introduce errors, making the
advantage of using noisy quantum memory unclear. To address these challenges we
introduce error-mitigated entanglement-enhanced learning and show, both
theoretically and experimentally, that even with noise, there is a separation
in efficiency between learning Pauli channels with and without entanglement
with noisy quantum memory. We demonstrate our protocol's efficacy in examples
including hypothesis testing with up to 64 qubits and learning inherent noise
processes in a layer of parallel gates using up to 16 qubits on a
superconducting quantum processor. Our protocol provides accurate and practical
information about the process, with an overhead factor of $1.33 \pm 0.05$ per
qubit, much smaller than the fundamental lower bound of 2 without entanglement
with quantum memory. Our study demonstrates that entanglement with auxiliary
noisy quantum memory combined with error mitigation considerably enhances the
learning of quantum processes.
