  Learning about physical systems from quantum-enhanced experiments, relying on
a quantum memory and quantum processing, can outperform learning from
experiments in which only classical memory and processing are available.
Whereas quantum advantages have been established for a variety of state
learning tasks, quantum process learning allows for comparable advantages only
with a careful problem formulation and is less understood. We establish an
exponential quantum advantage for learning an unknown $n$-qubit quantum process
$\mathcal{N}$. We show that a quantum memory allows to efficiently solve the
following tasks: (a) learning the Pauli transfer matrix of an arbitrary
$\mathcal{N}$, (b) predicting expectation values of bounded Pauli-sparse
observables measured on the output of an arbitrary $\mathcal{N}$ upon input of
a Pauli-sparse state, and (c) predicting expectation values of arbitrary
bounded observables measured on the output of an unknown $\mathcal{N}$ with
sparse Pauli transfer matrix upon input of an arbitrary state. With quantum
memory, these tasks can be solved using linearly-in-$n$ many copies of the Choi
state of $\mathcal{N}$, and even time-efficiently in the case of (b). In
contrast, any learner without quantum memory requires exponentially-in-$n$ many
queries, even when querying $\mathcal{N}$ on subsystems of adaptively chosen
states and performing adaptively chosen measurements. In proving this
separation, we extend existing shadow tomography upper and lower bounds from
states to channels via the Choi-Jamiolkowski isomorphism. Moreover, we combine
Pauli transfer matrix learning with polynomial interpolation techniques to
develop a procedure for learning arbitrary Hamiltonians, which may have
non-local all-to-all interactions, from short-time dynamics. Our results
highlight the power of quantum-enhanced experiments for learning highly complex
quantum dynamics.
