  Quantum state tomography (QST) is the gold standard technique for obtaining
an estimate for the state of small quantum systems in the laboratory. Its
application to systems with more than a few constituents (e.g. particles) soon
becomes impractical as the effort required grows exponentially in the number of
constituents. Developing more efficient techniques is particularly pressing as
precisely-controllable quantum systems that are well beyond the reach of QST
are emerging in laboratories. Motivated by this, there is a considerable
ongoing effort to develop new characterisation tools for quantum many-body
systems. Here we demonstrate Matrix Product State (MPS) tomography, which is
theoretically proven to allow the states of a broad class of quantum systems to
be accurately estimated with an effort that increases efficiently with
constituent number. We first prove that this broad class includes the
out-of-equilbrium states produced by 1D systems with finite-range interactions,
up to any fixed point in time. We then use the technique to reconstruct the
dynamical state of a trapped-ion quantum simulator comprising up to 14
entangled spins (qubits): a size far beyond the reach of QST. Our results
reveal the dynamical growth of entanglement and description complexity as
correlations spread out during a quench: a necessary condition for future
beyond-classical performance. MPS tomography should find widespread use to
study large quantum many-body systems and to benchmark and verify quantum
simulators and computers.
