An emerging direction of quantum computing is to establish meaningful quantum applications in various fields of artificial intelligence, including natural language processing (NLP). Although some efforts based on syntactic analysis have opened the door to research in Quantum NLP (QNLP), limitations such as heavy syntactic preprocessing and syntax-dependent network architecture make them impracticable on larger and real-world data sets. In this paper, we propose a new simple network architecture, called the quantum self-attention neural network (QSANN), which can compensate for these limitations. Specifically, we introduce the self-attention mechanism into quantum neural networks and then utilize a Gaussian projected quantum self-attention serving as a sensible quantum version of self-attention. As a result, QSANN is effective and scalable on larger data sets and has the desirable property of being implementable on near-term quantum devices. In particular, our QSANN outperforms the best existing QNLP model based on syntactic analysis as well as a simple classical self-attention neural network in numerical experiments of text classification tasks on public data sets. We further show that our method exhibits robustness to low-level quantum noises and showcases resilience to quantum neural network architectures.

Tensor networks (TNs) and neural networks (NNs) are two fundamental data modeling approaches. TNs were introduced to solve the curse of dimensionality in large-scale tensors by converting an exponential number of dimensions to polynomial complexity. As a result, they have attracted significant attention in the fields of quantum physics and machine learning. Meanwhile, NNs have displayed exceptional performance in various applications, e.g., computer vision, natural language processing, and robotics research. Interestingly, although these two types of networks originate from different observations, they are inherently linked through the common multilinearity structure underlying both TNs and NNs, thereby motivating a significant number of intellectual developments regarding combinations of TNs and NNs. In this paper, we refer to these combinations as tensorial neural networks (TNNs), and present an introduction to TNNs in three primary aspects: network compression, information fusion, and quantum circuit simulation. Furthermore, this survey also explores methods for improving TNNs, examines flexible toolboxes for implementing TNNs, and documents TNN development while highlighting potential future directions. To the best of our knowledge, this is the first comprehensive survey that bridges the connections among NNs, TNs, and quantum circuits. We provide a curated list of TNNs at \url{https://github.com/tnbar/awesome-tensorial-neural-networks}.

Variational quantum algorithms have been acknowledged as a leading strategy to realize near-term quantum advantages in meaningful tasks, including machine learning and combinatorial optimization. When applied to tasks involving classical data, such algorithms generally begin with quantum circuits for data encoding and then train quantum neural networks (QNNs) to minimize target functions. Although QNNs have been widely studied to improve these algorithms' performance on practical tasks, there is a gap in systematically understanding the influence of data encoding on the eventual performance. In this paper, we make progress in filling this gap by considering the common data encoding strategies based on parameterized quantum circuits. We prove that, under reasonable assumptions, the distance between the average encoded state and the maximally mixed state could be explicitly upper-bounded with respect to the width and depth of the encoding circuit. This result in particular implies that the average encoded state will concentrate on the maximally mixed state at an exponential speed on depth. Such concentration seriously limits the capabilities of quantum classifiers, and strictly restricts the distinguishability of encoded states from a quantum information perspective. We further support our findings by numerically verifying these results on both synthetic and public data sets. Our results highlight the significance of quantum data encoding in machine learning tasks and may shed light on future encoding strategies.

Hamiltonian learning is crucial to the certification of quantum devices and quantum simulators. In this paper, we propose a hybrid quantum-classical Hamiltonian learning algorithm to find the coefficients of the Pauli operator components of the Hamiltonian. Its main subroutine is the practical log-partition function estimation algorithm, which is based on the minimization of the free energy of the system. Concretely, we devise a stochastic variational quantum eigensolver (SVQE) to diagonalize the Hamiltonians and then exploit the obtained eigenvalues to compute the free energy's global minimum using convex optimization. Our approach not only avoids the challenge of estimating von Neumann entropy in free energy minimization, but also reduces the quantum resources via importance sampling in Hamiltonian diagonalization, facilitating the implementation of our method on near-term quantum devices. Finally, we demonstrate our approach's validity by conducting numerical experiments with Hamiltonians of interest in quantum many-body physics.

Classification of quantum data is essential for quantum machine learning and near-term quantum technologies. In this paper, we propose a new hybrid quantum-classical framework for supervised quantum learning, which we call Variational Shadow Quantum Learning (VSQL). Our method in particular utilizes the classical shadows of quantum data, which fundamentally represent the side information of quantum data with respect to certain physical observables. Specifically, we first use variational shadow quantum circuits to extract classical features in a convolution way and then utilize a fully-connected neural network to complete the classification task. We show that this method could sharply reduce the number of parameters and thus better facilitate quantum circuit training. Simultaneously, less noise will be introduced since fewer quantum gates are employed in such shadow circuits. Moreover, we show that the Barren Plateau issue, a significant gradient vanishing problem in quantum machine learning, could be avoided in VSQL. Finally, we demonstrate the efficiency of VSQL in quantum classification via numerical experiments on the classification of quantum states and the recognition of multi-labeled handwritten digits. In particular, our VSQL approach outperforms existing variational quantum classifiers in the test accuracy in the binary case of handwritten digit recognition and notably requires much fewer parameters.

Recently, deep neural networks (DNNs) have been regarded as the state-of-the-art classification methods in a wide range of applications, especially in image classification. Despite the success, the huge number of parameters blocks its deployment to situations with light computing resources. Researchers resort to the redundancy in the weights of DNNs and attempt to find how fewer parameters can be chosen while preserving the accuracy at the same time. Although several promising results have been shown along this research line, most existing methods either fail to significantly compress a well-trained deep network or require a heavy fine-tuning process for the compressed network to regain the original performance. In this paper, we propose the \textit{Block Term} networks (BT-nets) in which the commonly used fully-connected layers (FC-layers) are replaced with block term layers (BT-layers). In BT-layers, the inputs and the outputs are reshaped into two low-dimensional high-order tensors, then block-term decomposition is applied as tensor operators to connect them. We conduct extensive experiments on benchmark datasets to demonstrate that BT-layers can achieve a very large compression ratio on the number of parameters while preserving the representation power of the original FC-layers as much as possible. Specifically, we can get a higher performance while requiring fewer parameters compared with the tensor train method.

Recurrent Neural Networks (RNNs) are powerful sequence modeling tools. However, when dealing with high dimensional inputs, the training of RNNs becomes computational expensive due to the large number of model parameters. This hinders RNNs from solving many important computer vision tasks, such as Action Recognition in Videos and Image Captioning. To overcome this problem, we propose a compact and flexible structure, namely Block-Term tensor decomposition, which greatly reduces the parameters of RNNs and improves their training efficiency. Compared with alternative low-rank approximations, such as tensor-train RNN (TT-RNN), our method, Block-Term RNN (BT-RNN), is not only more concise (when using the same rank), but also able to attain a better approximation to the original RNNs with much fewer parameters. On three challenging tasks, including Action Recognition in Videos, Image Captioning and Image Generation, BT-RNN outperforms TT-RNN and the standard RNN in terms of both prediction accuracy and convergence rate. Specifically, BT-LSTM utilizes 17,388 times fewer parameters than the standard LSTM to achieve an accuracy improvement over 15.6\% in the Action Recognition task on the UCF11 dataset.

Probabilistic Temporal Tensor Factorization (PTTF) is an effective algorithm to model the temporal tensor data. It leverages a time constraint to capture the evolving properties of tensor data. Nowadays the exploding dataset demands a large scale PTTF analysis, and a parallel solution is critical to accommodate the trend. Whereas, the parallelization of PTTF still remains unexplored. In this paper, we propose a simple yet efficient Parallel Probabilistic Temporal Tensor Factorization, referred to as P$^2$T$^2$F, to provide a scalable PTTF solution. P$^2$T$^2$F is fundamentally disparate from existing parallel tensor factorizations by considering the probabilistic decomposition and the temporal effects of tensor data. It adopts a new tensor data split strategy to subdivide a large tensor into independent sub-tensors, the computation of which is inherently parallel. We train P$^2$T$^2$F with an efficient algorithm of stochastic Alternating Direction Method of Multipliers, and show that the convergence is guaranteed. Experiments on several real-word tensor datasets demonstrate that P$^2$T$^2$F is a highly effective and efficiently scalable algorithm dedicated for large scale probabilistic temporal tensor analysis.

Complex adaptive systems (CAS) promise to be useful in modeling and understanding real-world phenomena, but remain difficult to validate and verify. The authors present an adaptive, tool-chain-based approach to continuous validation and verification that allows the subject matter experts (SMEs) and modelers to interact in a useful manner. A CAS simulation of the ICU at the Mayo Clinic is used as a working example to illustrate the method and its benefits.