MiS Preprint Repository

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 $|\Psi \rangle$ 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 $|\Phi \rangle$, where $|\Phi \rangle$ can be acquired by partially projecting $|\Psi \rangle$. 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 $|\Phi \rangle$. 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 (hand-written digits and fashion images) show that the TNCS has competitive efficiency and accuracy.