State reconstruction from few measurements

Reconstructing a density matrix from measurement data is highly involved both from the experimental and the post-processing point of view: The number of measurements and computational resources needed to process them are exponentially large in the system size. If one knows that the system is in thermal equilibrium, one can do much better: We introduce a scheme that relies only on a linear number of correlation measurements. The computational resources needed to reconstruct the state with high fidelity from these measurements is only polynomial in the system size. The algorithm combines singular value thresholding and matrix product state methods.