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We compare and contrast the statistical physics and quantum physics inspired approaches for unsupervised generative modeling of classical data. The two approaches represent probabilities of observed data using energy-based models and quantum states, respectively. Classical and quantum information patterns of the target datasets therefore provide principled guidelines for structural design and learning in these two approaches. Taking the Restricted Boltzmann Machines (RBM) as an example, we analyze the information theoretical bounds of the two approaches. We also estimate the classical mutual information of the standard MNIST datasets and the quantum Rényi entropy of corresponding Matrix Product States (MPS) representations. Both information measures are much smaller compared to their theoretical upper bound and exhibit similar patterns, which imply a common inductive bias of low information complexity. By comparing the performance of RBM with various architectures on the standard MNIST datasets, we found that the RBM with local sparse connection exhibit high learning efficiency, which supports the application of tensor network states in machine learning problems.

We characterize the variational power of quantum circuit tensor networks in the representation of physical many-body ground states. Such tensor networks are formed by replacing the dense block unitaries and isometries in standard tensor networks by local quantum circuits. We explore both quantum circuit matrix product states and the quantum circuit multiscale entanglement renormalization Ansatz, and introduce an adaptive method to optimize the resulting circuits to high fidelity with more than104parameters. We benchmark their expressiveness against standard tensor networks, as well as other common circuit architectures, for the 1D and 2D Heisenberg and 1D Fermi-Hubbard models. We find quantum circuit tensor networks to be substantially more expressive than other quantum circuits for these problems, and that they can even be more compact than standard tensor networks. Extrapolating to circuit depths which can no longer be emulated classically, this suggests a region of advantage in quantum expressiveness in the representation of physical ground states.

In this paper, we aim to improve traditional DNN x-vector language identification performance by employing wide residual networks (WRN) as a powerful feature extractor which we combine with a novel frequency attention network. Compared with conventional time attention, our method learns discriminative weights for different frequency bands to generate weighted means and standard deviations for utterance-level classification. This mechanism enables the architecture to direct attention to important frequency bands rather than important time frames, as in traditional time attention methods. Furthermore, we then introduce a cross-layer frequency attention tensor network (CLF-ATN) which exploits information from different layers to recapture frame-level language characteristics that have been dropped by aggressive frequency pooling in lower layers. This effectively restores fine-grained discriminative language details. Finally, we explore the joint fusion of frame-level and frequency-band attention in a time–frequency attention network. Experimental results show that firstly, WRN can significantly outperform a traditional DNN x-vector implementation; secondly, the proposed frequency attention method is more effective than time attention; and thirdly, frequency–time score fusion can yield further improvement. Finally, extensive experiments on CLF-ATN demonstrate that it is able to improve discrimination by regaining dropped fine-grained frequency information, particularly for low-dimension frequency features.

Tensor network techniques have proved to be powerful tools that can be employed to explore the large scale dynamics of lattice systems. Nonetheless, the redundancy of degrees of freedom in lattice gauge theories (and related models) poses a challenge for standard tensor network algorithms. We accommodate for such systems by introducing an additional structure decorating the tensor network. This allows to explicitly preserve the gauge symmetry of the system under coarse graining and straightforwardly interpret the fixed point tensors. We propose and test (for models with finite Abelian groups) a coarse graining algorithm for lattice gauge theories based on decorated tensor networks. We also point out that decorated tensor networks are applicable to other models as well, where they provide the advantage to give immediate access to certain expectation values and correlation functions.

The resemblance between the methods used in quantum-many body physics and in machine learning has drawn considerable attention. In particular, tensor networks (TNs) and deep learning architectures bear striking similarities to the extent that TNs can be used for machine learning. Previous results used one-dimensional TNs in image recognition, showing limited scalability and flexibilities. In this work, we train two-dimensional hierarchical TNs to solve image recognition problems, using a training algorithm derived from the multi-scale entanglement renormalization ansatz. This approach introduces mathematical connections among quantum many-body physics, quantum information theory, and machine learning. While keeping the TN unitary in the training phase, TN states are defined, which encode classes of images into quantum many-body states. We study the quantum features of the TN states, including quantum entanglement and fidelity. We find these quantities could be properties that characterize the image classes, as well as the machine learning tasks.

We introduce a hybrid model combining a quantum-inspired tensor network and a variational quantum circuit to perform supervised learning tasks. This architecture allows for the classical and quantum parts of the model to be trained simultaneously, providing an end-to-end training framework. We show that compared to the principal component analysis, a tensor network based on the matrix product state with low bond dimensions performs better as a feature extractor for the input data of the variational quantum circuit in the binary and ternary classification of MNIST and Fashion-MNIST datasets. The architecture is highly adaptable and the classical-quantum boundary can be adjusted according to the availability of the quantum resource by exploiting the correspondence between tensor networks and quantum circuits.

We develop an efficient algorithm for determining optimal adaptive quantum estimation protocols with arbitrary quantum control operations between subsequent uses of a probed channel.We introduce a tensor network representation of an estimation strategy, which drastically reduces the time and memory consumption of the algorithm, and allows us to analyze metrological protocols involving up to N  = 50 qubit channel uses, whereas the state-of-the-art approaches are limited to N  < 5. The method is applied to study the performance of the optimal adaptive metrological protocols in presence of various noise types, including correlated noise.

We investigate tensor network simulations of the two-dimensional (2D) Hubbard model by mapping the lattice onto a one-dimensional chain using space-filling curves. In particular, we focus on the Hilbert curve, whose locality-preserving structure minimizes the range of effective interactions in the mapped model. This enables a more compact matrix product state representation compared to conventional snake mapping. Through systematic benchmarks, we show that the Hilbert curve consistently yields lower ground-state energies at fixed bond dimension, with the advantage increasing for larger system sizes and in physically relevant interaction regimes. Our implementation reaches clusters up to 32 × 32 sites with open and periodic boundary conditions, delivering reliable ground-state energies and correlation functions in agreement with established results, but at significantly reduced computational cost. These findings establish space-filling curve mappings, particularly the Hilbert curve, as a powerful tool for extending tensor-network studies of strongly correlated 2D quantum systems beyond the limits accessible with standard approaches.

It is commonly believed that area laws for entanglement entropies imply that a quantum many-body state can be faithfully represented by efficient tensor network states-a conjecture frequently stated in the context of numerical simulations and analytical considerations. In this work, we show that this is in general not the case, except in one-dimension. We prove that the set of quantum many-body states that satisfy an area law for all Renyi entropies contains a subspace of exponential dimension. We then show that there are states satisfying area laws for all Renyi entropies but cannot be approximated by states with a classical description of small Kolmogorov complexity, including polynomial projected entangled pair states or states of multi-scale entanglement renormalisation. Not even a quantum computer with post-selection can efficiently prepare all quantum states fulfilling an area law, and we show that not all area law states can be eigenstates of local Hamiltonians. We also prove translationally and rotationally invariant instances of these results, and show a variation with decaying correlations using quantum error-correcting codes.

We benchmark recently proposed tensor network based finite-size scaling analysis in ( Phys. Rev. B 2023 107 205123) against two-dimensional classical 3-state clock model. Due to the higher complexity of the model, more complicated crossover behavior is observed. We advocate that the crossover behavior can be understood from the perspective of finite bond dimension inducing relevant perturbation. This leads to a general strategy to best estimate the critical properties for a given set of control parameters. For the critical temperature T c , the relative error at the order of 10 −7 can be reached with bond dimension D  = 70. On the other hand, with bond dimension D  = 60, the relative errors of the critical exponents ν , β , α are at the order of 10 −2 . Increasing the bond dimension to D  = 90, these relative errors can be reduced at least to the order of 10 −3 . In all cases our results indicate that the errors can be systematically reduced by increasing the bond dimension and the stacking number.