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Controllable arrays of ions and ultra-cold atoms can simulate complexmany-body phenomena and may provide insights into unsolved problems in modernscience. To this end, experimentally feasible protocols for quantifying thebuildup of quantum correlations and coherence are needed, as performing fullstate tomography does not scale favorably with the number of particles. Here wedevelop and experimentally demonstrate such a protocol, which uses timereversal of the many-body dynamics to measure out-of-time-order correlationfunctions (OTOCs) in a long-range Ising spin quantum simulator with more than100 ions in a Penning trap. By measuring a family of OTOCs as a function of atunable parameter we obtain fine-grained information about the state of thesystem encoded in the multiple quantum coherence spectrum, extract the quantumstate purity, and demonstrate the buildup of up to 8-body correlations. Futureapplications of this protocol could enable studies of many-body localization,quantum phase transitions, and tests of the holographic duality between quantumand gravitational systems.

Measuring out-of-time-order correlations and multiple quantum spectra in a trapped ion quantum magnet