In this note, we describe a method for reconstructing matrix product states
from a small number of efficiently-implementable measurements. Our method is
exponentially faster than standard tomography, and it can also be used to
certify that the unknown state is an MPS. The basic idea is to use local
unitary operations to measure in the Schmidt basis, giving direct access to the
MPS representation. This compares favorably with recently and independently
proposed methods that recover the MPS tensors by performing a variational
minimization, which is computationally intractable in certain cases. Our method
also has the advantage of recovering any MPS, while other approaches were
limited to special classes of states that exclude important examples such as
GHZ and W states.
