Reconstructing Quantum States from Sparse Measurements

Abstract

Quantum state tomography (QST) is a central technique to fully characterize an unknown quantum state. However, standard QST requires an exponentially growing number of quantum measurements against the system size, which limits its application to smaller systems. Here, we explore the sparsity of underlying quantum state and propose a QST scheme that combines the matrix product states’ representation of the quantum state with a supervised machine learning algorithm. Our method could reconstruct the unknown sparse quantum states with very high precision using only a portion of the measurement data in a randomly selected basis set. In particular, we demonstrate that the Wolfgang states could be faithfully reconstructed using around 25% of the whole basis, and that the randomly generated quantum states, which could be efficiently represented as matrix product states, could be faithfully reconstructed using a number of bases that scales sub-exponentially against the system size.