Reconstructing quantum states efficiently

Quantum state tomography has been realized in systems with few components but for larger systems it becomes rapidly infeasible because the number of quantum measurements and computational resources required to process them grow exponentially in the system size. We show that one can do better, gaining an exponential advantage over direct state tomography for quantum states typically realized in Nature. The method is based on matrix product states and makes use of singular value thresholding from compressed sensing. Our results suggest that state tomography of large quantum systems may be much more feasible than the exponential size of state space suggests.