Efficient tomography of a quantum many-body system

Quantum state tomography is the standard technique for estimating the quantum state of small systems(1). But its application to larger systems soon becomes impractical as the required resources scale exponentially with the size. Therefore, considerable effort is dedicated to the development of new characterization tools for quantum many-body states(2-11). Here we demonstrate matrix product state tomography(2), which is theoretically proven to allow for the efficient and accurate estimation of a broad class of quantum states. We use this technique to reconstruct the dynamical state of a trapped-ion quantum simulator comprising up to 14 entangled and individually controlled spins: a size far beyond the practical limits of quantum state tomography. Our results reveal the dynamical growth of entanglement and describe its complexity as correlations spread out during a quench: a necessary condition for future demonstrations of better-than-classical performance. Matrix product state tomography should therefore find widespread use in the study of large quantum many-body systems and the benchmarking and verification of quantum simulators and computers.