  Among recent insights into learning quantum states, online learning and
shadow tomography procedures are notable for their ability to accurately
predict expectation values even of adaptively chosen observables. In contrast
to the state case, quantum process learning tasks with a similarly adaptive
nature have received little attention. In this work, we investigate online
learning tasks for quantum processes. Whereas online learning is infeasible for
general quantum channels, we show that channels of bounded gate complexity as
well as Pauli channels can be online learned in the regret and mistake-bounded
models of online learning. In fact, we can online learn probabilistic mixtures
of any exponentially large set of known channels. We also provide a provably
sample-efficient shadow tomography procedure for Pauli channels. Our results
extend beyond quantum channels to non-Markovian multi-time processes, with
favorable regret and mistake bounds, as well as a shadow tomography procedure.
We complement our online learning upper bounds with mistake as well as
computational lower bounds. On the technical side, we make use of the
multiplicative weights update algorithm, classical adaptive data analysis, and
Bell sampling, as well as tools from the theory of quantum combs for multi-time
quantum processes. Our work initiates a study of online learning for classes of
quantum channels and, more generally, non-Markovian quantum processes. Given
the importance of online learning for state shadow tomography, this may serve
as a step towards quantum channel variants of adaptive shadow tomography.
