Tensor network (TN) has recently triggered extensive interests in developing machine-learning models in quantum many-body Hilbert space. Here we purpose a generative TN classification (GTNC) approach for supervised learning. The strategy is to train the generative TN for each class of the samples to construct the classifiers. The classification is implemented by comparing the distance in the many-body Hilbert space. The numerical experiments by GTNC show impressive performance on the MNIST and Fashion-MNIST dataset. The testing accuracy is competitive to the state-of-the-art convolutional neural network while higher than the naive Bayes classifier (a generative classifier) and support vector machine. Moreover, GTNC is more efficient than the existing TN models that are in general discriminative. By investigating the distances in the manybody Hilbert space, we find that (a) the samples are naturally clustering in such a space; and (b) bounding the bond dimensions of the TN's to finite values corresponds to removing redundant information in the image recognition. These two characters make GTNC an adaptive and universal model of excellent performance.

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 | 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 | , where | can be acquired by partially projecting |. 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 |. 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.

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Non-Abelian anyons are exotic quasiparticle excitations hosted by certain topological phases of matter. They break the fermion-boson dichotomy and obey non-Abelian braiding statistics: their interchanges yield unitary operations, rather than merely a phase factor, in a space spanned by topologically degenerate wavefunctions. They are the building blocks of topological quantum computing. However, experimental observation of non-Abelian anyons and their characterizing braiding statistics is notoriously challenging and has remained elusive hitherto, in spite of various theoretical proposals. Here, we report an experimental quantum digital simulation of projective non-Abelian anyons and their braiding statistics with up to 68 programmable superconducting qubits arranged on a two-dimensional lattice. By implementing the ground states of the toric-code model with twists through quantum circuits, we demonstrate that twists exchange electric and magnetic charges and behave as a particular type of non-Abelian anyons—the Ising anyons. In particular, we show experimentally that these twists follow the fusion rules and non-Abelian braiding statistics of the Ising type, and can be explored to encode topological logical qubits. Furthermore, we demonstrate how to implement both single- and two-qubit logic gates through applying a sequence of elementary Pauli gates on the underlying physical qubits. Our results demonstrate a versatile quantum digital approach for simulating non-Abelian anyons, offering a new lens into the study of such peculiar quasiparticles.