  Quantum state tomography, the ability to deduce the state of a quantum system
from measured data, is the gold standard for verification and benchmarking of
quantum devices. It has been realized in systems with few components, but for
larger systems it becomes infeasible because the number of quantum measurements
and the amount of computation required to process them grows exponentially in
the system size. Here we show that we can do exponentially better than direct
state tomography for a wide range of quantum states, in particular those that
are well approximated by a matrix product state ansatz. We present two schemes
for tomography in 1-D quantum systems and touch on generalizations. One scheme
requires unitary operations on a constant number of subsystems, while the other
requires only local measurements together with more elaborate post-processing.
Both schemes rely only on a linear number of experimental operations and
classical postprocessing that is polynomial in the system size. A further
strength of the methods is that the accuracy of the reconstructed states can be
rigorously certified without any a priori assumptions.
