  Learning the properties of dynamical quantum systems underlies applications
ranging from nuclear magnetic resonance spectroscopy to quantum device
characterization. A central challenge in this pursuit is the learning of
strongly-interacting systems, where conventional observables decay quickly in
time and space, limiting the information that can be learned from their
measurement. In this work, we introduce a new class of observables into the
context of quantum learning -- the out-of-time-order correlator -- which we
show can substantially improve the learnability of strongly-interacting systems
by virtue of displaying informative physics at large times and distances. We
identify two general scenarios in which out-of-time-order correlators provide a
significant advantage for learning tasks in locally-interacting systems: (i)
when experimental access to the system is spatially-restricted, for example via
a single "probe" degree of freedom, and (ii) when one desires to characterize
weak interactions whose strength is much less than the typical interaction
strength. We numerically characterize these advantages across a variety of
learning problems, and find that they are robust to both read-out error and
decoherence. Finally, we introduce a binary classification task that can be
accomplished in constant time with out-of-time-order measurements. In a
companion paper, we prove that this task is exponentially hard with any
adaptive learning protocol that only involves time-ordered operations.
