We propose the tensor-network compressed sensing (TNCS) by incorporating the ideas of compressed sensing, tensor network (TN), and machine learning. The primary idea is to compress and communicate the real-life information through the generative TN state and by making projective measurements in a designed way. First, the state $|\mathrm{\Psi }\rangle$ is obtained by the unsupervised learning of TN, and then the data to be communicated are encoded in the separable state with the minimal distance to the projected state $|\mathrm{\Phi }\rangle$, where $|\mathrm{\Phi }\rangle$ can be acquired by partially projecting $|\mathrm{\Psi }\rangle$. A protocol analogous to the compressed sensing assisted by neural-network machine learning is thus suggested, where the projections are designed to rapidly minimize the uncertainty of information in $|\mathrm{\Phi }\rangle$. To characterize the efficiency of TNCS, we propose a quantity named as q sparsity to describe the sparsity of quantum states, which is analogous to the sparsity of the signals required in the standard compressed sensing. The need of the q sparsity in TNCS is essentially due to the fact that the TN states obey the area law of entanglement entropy. The tests on the real-life data (handwritten digits and fashion images) show that the TNCS has competitive efficiency and accuracy.

• Received 15 October 2019
• Revised 20 April 2020
• Accepted 3 August 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033293

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Published by the American Physical Society