Learning quantum systems via out-of-time-order correlators

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 in-formation 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 learning advantage: (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. Motivated by these physical scenarios, we introduce several learning tasks-including Clifford tomography, and learning the connectivity of an unknown unitary-in which out-of-time-order experiments have a provable exponential advantage over any learning protocol involving only time-ordered operations.