Tensor Network Research Articles

2500 vibronic eigenstates of the NO3 radical.

The nitrate radical NO3 plays an important role in atmospheric chemistry, yet many aspects of its coupled and anharmonic vibronic structure remain elusive. Here, using an accurate, coupled full-dimensional diabatic potential that includes five electronic states, we revisit the vibronic spectrum associated with the electronic state. Using recently developed tensor network state methods, we are able to compute more than 2500 vibronic states, thereby increasing the number of computed full-dimensional states by a factor of 50, compared to previous work. While we obtain good agreement with experiment for most of the assigned vibronic levels, for several others, we observe striking disagreement. Further, for the antisymmetric bending motion we find remarkably large symmetry-induced level splittings that are larger than the zero-order reference. We discuss non-negligible nonadiabatic effects and show that the Born-Oppenheimer approximation leads to significant errors in the spectrum.

Fully-connected tensor network decomposition for robust tensor completion problem

Fully-connected tensor network decomposition for robust tensor completion problem

Striking the right balance of encoding electron correlation in the Hamiltonian and the wavefunction ansatz.

Multi-configurational electronic structure theory delivers the most versatile approximations to many-electron wavefunctions, flexible enough to deal with all sorts of transformations, ranging from electronic excitations, to open-shell molecules and chemical reactions. Multi-configurational models are therefore essential to establish universally applicable, predictive ab initio methods for chemistry. Here, we present a discussion of explicit correlation approaches which address the nagging problem of dealing with static and dynamic electron correlation in multi-configurational active-space approaches. We review the latest developments and then point to their key obstacles. Our discussion is supported by new data obtained with tensor network methods. We argue in favor of simple electron-only correlator expressions that may allow one to define transcorrelated models in which the correlator does not bear a dependence on molecular structure.

Spectral Tensor Layers for Communication-Free Distributed Deep Learning.

In this article, we propose a novel spectral tensor layer for communication-free distributed deep learning. The overall framework is as follows: first, we represent the data in tensor form (instead of vector form) and replace the matrix product in conventional neural networks with the tensor product, which in effect imposes certain transformed-induced structure on the original weight matrices, e.g., a block-circulant structure; then, we apply a linear transform along a certain dimension to split the original dataset into multiple spectral subdatasets; as a result, the proposed spectral tensor network consists of parallel branches where each branch is a conventional neural network trained on a spectral subdataset with ZERO communication cost. The parallel branches are directly ensembled (i.e., the weighted sum of their outputs) to generate an overall network with substantially stronger generalization capability than that of each branch. Moreover, the proposed method enjoys a byproduct of decentralization gain in terms of memory and computation, compared with traditional networks. It is a natural yet elegant solution for heterogeneous data in federated learning (FL), where data at different nodes have different resolutions. Finally, we evaluate the proposed spectral tensor networks on the MNIST, CIFAR-10, ImageNet-1K, and ImageNet-21K datasets, respectively, to verify that they simultaneously achieve communication-free distributed learning, distributed storage reduction, parallel computation speedup, and learning with multiresolution data.

Nested Fully-Connected Tensor Network Decomposition for Multi-Dimensional Visual Data Recovery

Nested Fully-Connected Tensor Network Decomposition for Multi-Dimensional Visual Data Recovery

Hyperspectral Image Classification Based on Spectral–Spatial Attention Tensor Network

Hyperspectral Image Classification Based on Spectral–Spatial Attention Tensor Network

Fully-connected tensor network decomposition and group sparsity for multitemporal images cloud removal

Fully-connected tensor network decomposition and group sparsity for multitemporal images cloud removal

Theoretical Advancement of Xi Jinping’s Socialist Thought with Chinese Characteristics in the New Era under the Construction of Knowledge Map

Abstract This paper discusses the theory and practice of constructing knowledge graph under the guidance of Xi Jinping’s socialist thought with Chinese characteristics in the new era. The study first adopts the Transformer technique based on the BERT model for effectively extracting entity relationships. Then, the article proposes an entity alignment method based on Neural Tensor Network, which enhances the semantic and structural information utilization of entity embedding vectors through optimized NTN and joint knowledge representation learning. In addition, the study employs a knowledge graph link prediction method that combines semantics and paths to fill in missing relations, and applies probabilistic soft logic to solve the inference problem of non-deterministic knowledge. Ultimately, the effectiveness of the constructed knowledge graph is demonstrated through empirical analysis of Xi Jinping’s theoretical dataset of socialist thought with Chinese characteristics in the new era. The results show that the method is more effective in dealing with polysemous words and participle errors than traditional methods, improving the data structuring level. For example, the accuracy of entity alignment is significantly better than the baseline algorithm on different datasets, e.g., on the DBP-WE dataset, its Hits@1, Hits@10 and MRR values are higher than those of the baseline by 4.01-29.49, 1.99-31.05, and 0.036-0.289, respectively. The method of this paper is remarkably effective in improving the performance of the entity alignment task, which is helpful in understanding and applying the Xi Jinping’s thought of socialism with Chinese characteristics in the new era provides new theoretical perspectives and practical methods.

Finite-Entanglement Scaling of 2D Metals.

We extend the study of finite-entanglement scaling from one-dimensional gapless models to two-dimensional systems with a Fermi surface. In particular, we show that the entanglement entropy of a contractible spatial region with linear size L scales as S∼Llog[ξf(L/ξ)] in the optimal tensor network, and hence area-law entangled, state approximation to a metallic state, where f(x) is a scaling function which depends on the shape of the Fermi surface and ξ is a finite correlation length induced by the restricted entanglement. Crucially, the scaling regime can be realized with numerically tractable bond dimensions. We also discuss the implications of the Lieb-Schultz-Mattis theorem at fractional filling for tensor network state approximations of metallic states.

Infrared Dim and Small Target Detection Based on Superpixel Segmentation and Spatiotemporal Cluster 4D Fully-Connected Tensor Network Decomposition

The detection of infrared dim and small targets in complex backgrounds is very challenging because of the low signal-to-noise ratio of targets and the drastic change in background. Low-rank sparse decomposition based on the structural characteristics of infrared images has attracted the attention of many scholars because of its good interpretability. In order to improve the sensitivity of sliding window size, insufficient utilization of time series information, and inaccurate tensor rank in existing methods, a four-dimensional tensor model based on superpixel segmentation and statistical clustering is proposed for infrared dim and small target detection (ISTD). First, the idea of superpixel segmentation is proposed to eliminate the dependence of the algorithm on selection of the sliding window size. Second, based on the improved structure tensor theory, the image pixels are statistically clustered into three types: corner region, flat region, and edge region, and are assigned different weights to reduce the influence of background edges. Next, in order to better use spatiotemporal correlation, a Four-Dimensional Fully-Connected Tensor Network (4D-FCTN) model is proposed in which 3D patches with the same feature types are rearranged into the same group to form a four-dimensional tensor. Finally, the FCTN decomposition method is used to decompose the clustered tensor into low-dimensional tensors, with the alternating direction multiplier method (ADMM) used to decompose the low-rank background part and sparse target part. We validated our model across various datasets, employing cutting-edge methodologies to assess its effectiveness in terms of detection precision and reduction of background interference. A comparative analysis corroborated the superiority of our proposed approach over prevailing techniques.

QTN-VQC: an end-to-end learning framework for quantum neural networks

This work focuses on investigating an end-to-end learning approach for quantum neural networks (QNN) on noisy intermediate-scale quantum devices. The proposed model combines a quantum tensor network (QTN) with a variational quantum circuit (VQC), resulting in a QTN-VQC architecture. This architecture integrates a QTN with a horizontal or vertical structure related to the implementation of quantum circuits for a tensor-train network. The study provides theoretical insights into the quantum advantages of the end-to-end learning pipeline based on QTN-VQC from two perspectives. The first perspective refers to the theoretical understanding of QTN-VQC with upper bounds on the empirical error, examining its learnability and generalization powers; The second perspective focuses on using the QTN-VQC architecture to alleviate the Barren Plateau problem in the training stage. Our experimental simulation on CPU/GPUs is performed on a handwritten digit classification dataset to corroborate our proposed methods in this work.

Synergistic pretraining of parametrized quantum circuits via tensor networks

Parametrized quantum circuits (PQCs) represent a promising framework for using present-day quantum hardware to solve diverse problems in materials science, quantum chemistry, and machine learning. We introduce a “synergistic” approach that addresses two prominent issues with these models: the prevalence of barren plateaus in PQC optimization landscapes, and the difficulty to outperform state-of-the-art classical algorithms. This framework first uses classical resources to compute a tensor network encoding a high-quality solution, and then converts this classical output into a PQC which can be further improved using quantum resources. We provide numerical evidence that this framework effectively mitigates barren plateaus in systems of up to 100 qubits using only moderate classical resources, with overall performance improving as more classical or quantum resources are employed. We believe our results highlight that classical simulation methods are not an obstacle to overcome in demonstrating practically useful quantum advantage, but rather can help quantum methods find their way.

Random singlets and permutation symmetry in the disordered spin-2 Heisenberg chain: A tensor network renormalization group study

Random singlets and permutation symmetry in the disordered spin-2 Heisenberg chain: A tensor network renormalization group study

Sampling diverse near-optimal solutions via algorithmic quantum annealing.

Sampling a diverse set of high-quality solutions for hard optimization problems is of great practical relevance in many scientific disciplines and applications, such as artificial intelligence and operations research. One of the main open problems is the lack of ergodicity, or mode collapse, for typical stochastic solvers based on Monte Carlo techniques leading to poor generalization or lack of robustness to uncertainties. Currently, there is no universal metric to quantify such performance deficiencies across various solvers. Here, we introduce a new diversity measure for quantifying the number of independent approximate solutions for NP-hard optimization problems. Among others, it allows benchmarking solver performance by a required time-to-diversity (TTD), a generalization of often used time-to-solution (TTS). We illustrate this metric by comparing the sampling power of various quantum annealing strategies. In particular, we show that the inhomogeneous quantum annealing schedules can redistribute and suppress the emergence of topological defects by controlling space-time separated critical fronts, leading to an advantage over standard quantum annealing schedules with respect to both TTS and TTD for finding rare solutions. Using path-integral Monte Carlo simulations for up to 1600 qubits, we demonstrate that nonequilibrium driving of quantum fluctuations, guided by efficient approximate tensor network contractions, can significantly reduce the fraction of hard instances for random frustrated 2D spin glasses with local fields. Specifically, we observe that by creating a class of algorithmic quantum phase transitions, the diversity of solutions can be enhanced by up to 40% with the fraction of hard-to-sample instances reducing by more than 25%.

Tensor network reduced order models for wall-bounded flows

Tensor network reduced order models for wall-bounded flows

Studies of nonadiabatic dynamics in the singlet fission processes of pentacene dimer via tensor network method

Singlet fission (SF) is a very significant photophysical phenomenon and possesses potential applications. In this work, we try to give a rather detailed theoretical investigation of the SF process in the stacked polyacene dimer by combining the high-level quantum chemistry calculations and the quantum dynamics simulations based on the tensor network method. Starting with the construction of the linear vibronic coupling model, we explore the pure electronic dynamics and the vibronic dynamics in the SF processes. The role of vibrational modes in nonadiabatic dynamics is addressed. The results show that the super-exchange mechanism mediated by the charge-transfer state is found in both pure electronic dynamics and the nonadiabatic dynamics. Particularly the vibrational modes with the frequencies resonance with the adiabatic energy gap play very import roles in the SF dynamics. This work not only provides a deep and detailed understanding of the SF process but also verifies the efficiency of the tensor network method with the train structure that can serve as the reference dynamics method to explore the dynamics behaviors of complex systems.

Quantum correlation functions through tensor network path integral.

Tensor networks have historically proven to be of great utility in providing compressed representations of wave functions that can be used for the calculation of eigenstates. Recently, it has been shown that a variety of these networks can be leveraged to make real time non-equilibrium simulations of dynamics involving the Feynman-Vernon influence functional more efficient. In this work, a tensor network is developed for non-perturbatively calculating the equilibrium correlation function for open quantum systems using the path integral methodology. These correlation functions are of fundamental importance in calculations of rates of reactions, simulations of response functions and susceptibilities, spectra of systems, etc. The influence of the solvent on the quantum system is incorporated through an influence functional, whose unconventional structure motivates the design of a new optimal matrix product-like operator that can be applied to the so-called path amplitude matrix product state. This complex-time tensor network path integral approach provides an exceptionally efficient representation of the path integral, enabling simulations for larger systems strongly interacting with baths and at lower temperatures out to longer time. The derivation, design, and implementation of this method are discussed along with a wide range of illustrations ranging from rate theory and symmetrized spin correlation functions to simulation of response of the Fenna-Matthews-Olson complex to light.

Effective quantum volume, fidelity and computational cost of noisy quantum processing experiments

Today’s experimental noisy quantum processors can compete with and surpass all known algorithms on state-of-the-art supercomputers for the computational benchmark task of Random Circuit Sampling (Boixo et al., 2018, Arute et al., 2019, Wu et al., 2021, Zhu et al., 2022, Morvan et al., 2023). Additionally, a circuit-based quantum simulation of quantum information scrambling (Mi et al., 2021), which measures a local observable, has already outperformed standard full wave function simulation algorithms, e.g., exact Schrodinger evolution and Matrix Product States (MPS). However, this experiment has not yet surpassed tensor network contraction for computing the value of the observable. Based on those studies, we provide a unified framework that utilizes the underlying effective circuit volume to explain the tradeoff between the experimentally achievable signal-to-noise ratio for a specific observable, and the corresponding computational cost. We apply this framework to recent quantum processor experiments of Random Circuit Sampling (Morvan et al., 2023), quantum information scrambling (Mi et al., 2021), and a Floquet circuit unitary (Kim et al., 2023). This allows us to reproduce the results of Ref. (Kim et al., 2023) in less than one second per data point using one GPU.

General tensor network theory for frustrated classical spin models in two dimensions

General tensor network theory for frustrated classical spin models in two dimensions

Emergent symmetry in quantum phase transition: From deconfined quantum critical point to gapless quantum spin liquid

The emergence of exotic quantum phenomena in frustrated magnets is rapidly driving the development of quantum many-body physics, raising fundamental questions on the nature of quantum phase transitions. Here we unveil the behaviour of emergent symmetry involving two extraordinarily representative phenomena, i.e., the deconfined quantum critical point (DQCP) and the quantum spin liquid (QSL) state. Via large-scale tensor network simulations, we study a spatially anisotropic spin-1/2 square-lattice frustrated antiferromagnetic (AFM) model, namely the J1x-J1y-J2 model, which contains anisotropic nearest-neighbor couplings J1x,J1y and the next nearest neighbor coupling J2. For small J1y/J1x, by tuning J2, a direct continuous transition between the AFM and valence bond solid phase is observed. With growing J1y/J1x, a gapless QSL phase gradually emerges between the AFM and VBS phases. We observe an emergent O(4) symmetry along the AFM–VBS transition line, which is consistent with the prediction of DQCP theory. Most surprisingly, we find that such an emergent O(4) symmetry holds for the whole QSL–VBS transition line as well. These findings reveal the intrinsic relationship between the QSL and DQCP from categorical symmetry point of view, and strongly constrain the quantum field theory description of the QSL phase. The phase diagram and critical exponents presented in this paper are of direct relevance to future experiments on frustrated magnets and cold atom systems.