Abstract
We propose tensor-network compressed sensing (TNCS) by combining the ideas of compressed sensing, tensor network (TN), and machine learning, which permits novel and efficient quantum communications of realistic data. The strategy is to use the unsupervised TN machine learning algorithm to obtain the entangled state $|\Psi \rangle$ that describes the probability distribution of a huge amount of classical information considered to be communicated. To transfer a specific piece of information with $|\Psi \rangle$, our proposal is to encode such information in the separable state with the minimal distance to the measured state $|\Phi \rangle$ that is obtained by partially measuring on $|\Psi \rangle$ in a designed way. To this end, a measuring protocol analogous to the compressed sensing with neural-network machine learning is suggested, where the measurements are designed to minimize uncertainty of information from the probability distribution given by $|\Phi \rangle$. In this way, those who have $|\Phi \rangle$ can reliably access the information by simply measuring on $|\Phi \rangle$. We propose q-sparsity to characterize the sparsity of quantum states and the efficiency of the quantum communications by TNCS. The high q-sparsity is essentially due to the fact that the TN states describing nicely the probability distribution obey the area law of entanglement entropy. Testing on realistic datasets (hand-written digits and fashion images), TNCS is shown to possess high efficiency and accuracy, where the security of communications is guaranteed by the fundamental quantum principles.
Highlights
Hybridizing the ideas and techniques in information theories and quantum physics has given birth to significant and fruitful achievements
To characterize the efficiency of tensor-network compressed sensing (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
While random ordering (RO) works well for TNCS, in the following, we will propose to improve the performance by incorporating with a quantum-inspired sampling protocol based on entanglement and the post-selections of measurements
Summary
Hybridizing the ideas and techniques in information theories and quantum physics has given birth to significant and fruitful achievements. We here consider to combine the ideas of compressed sensing [8], quantum communication [14], and unsupervised tensor network (TN) machine learning [15]. The main idea is to encode and communicate the information by implementing designed projections on the state | ( called Born machine [35]) that is trained by the unsupervised TN machine learning algorithm [15]. For TNCS, q sparsity characterizes how fast the Shannon entropy of the probability distribution will decrease by projecting the Born machine | and how efficient the compressed sampling can be via |.
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