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arXiv:2211.14657 · Scientific Reports

Variational tensor neural networks for deep learning

Saeed S. Jahromi, Román Orús

Published Aug 16, 2024• 10 citations
arXivHTMLPDF

Abstract

Deep neural networks (NNs) encounter scalability limitations when confronted with a vast array of neurons, thereby constraining their achievable network depth. To address this challenge, we propose an integration of tensor networks (TN) into NN frameworks, combined with a variational DMRG-inspired training technique. This in turn, results in a scalable tensor neural network (TNN) architecture capable of efficient training over a large parameter space. Our variational algorithm utilizes a local gradient-descent technique, enabling manual or automatic computation of tensor gradients, facilitating design of hybrid TNN models with combined dense and tensor layers. Our training algorithm further provides insight on the entanglement structure of the tensorized trainable weights and correlation among the model parameters. We validate the accuracy and efficiency of our method by designing TNN models and providing benchmark results for linear and non-linear regressions, data classification and image recognition on MNIST handwritten digits.

DOI
10.1038/s41598-024-69366-8

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Deep neural networks (NNs) encounter scalability limitations when confronted with a vast array of neurons, thereby constraining their achievable network depth. To address this challenge, we propose an integration of tensor networks (TN) into NN frameworks, combined with a variational DMRG-inspired training technique. This in turn, results in a scalable tensor neural network (TNN) architecture capable of efficient training over a large parameter space. Our variational algorithm utilizes a local gradient-descent technique, enabling manual or automatic computation of tensor gradients, facilitating design of hybrid TNN models with combined dense and tensor layers. Our training algorithm further provides insight on the entanglement structure of the tensorized trainable weights and correlation among the model parameters. We validate the accuracy and efficiency of our method by designing TNN models and providing benchmark results for linear and non-linear regressions, data classification and image recognition on MNIST handwritten digits.
Abstract Quantum machine learning researchers often rely on incorporating tensor networks (TN) into deep neural networks (DNN) and variational optimization. However, the standard optimization techniques used for training the contracted trainable weights of each model layer suffer from the correlations and entanglement structure between the model parameters in classical implementations. To address this issue, a multi-layer design of a tensor ring optimized variational quantum learning classifier (Quan-TR) comprising cascading entangling gates replacing the fully connected (dense) layers of a TN is proposed, and it is referred to as tensor ring optimized quantum-enhanced tensor neural networks (TR-QNet). TR-QNet parameters are optimized using the stochastic gradient descent algorithm on qubit measurements. The proposed TR-QNet is evaluated on three distinct datasets, namely Iris, MNIST, and CIFAR-10, to demonstrate the enhanced precision achieved for binary classification. In quantum simulations, the proposed TR-QNet achieves promising precision of 94.5%94.5\%94.5% 94.5 % , 86.16%86.16\%86.16% 86.16 % , and 83.54%83.54\%83.54% 83.54 % on the Iris, MNIST, and CIFAR-10 datasets. Benchmark studies have been conducted on state-of-the-art quantum and classical implementations of TN models to show the efficacy of the proposed TR-QNet. Moreover, the scalability of TR-QNet highlights its potential for exhibiting in deep learning applications on a large scale. The PyTorch implementation of TR-QNet is available on Github https://github.com/konar1987/TR-QNet/ .
Tensor networks (TN) have found a wide use in machine learning, and in particular, TN and deep learning bear striking similarities. In this work, we propose the quantum-classical hybrid tensor networks (HTN) which combine tensor networks with classical neural networks in a uniform deep learning framework to overcome the limitations of regular tensor networks in machine learning. We first analyze the limitations of regular tensor networks in the applications of machine learning involving the representation power and architecture scalability. We conclude that in fact the regular tensor networks are not competent to be the basic building blocks of deep learning. Then, we discuss the performance of HTN which overcome all the deficiency of regular tensor networks for machine learning. In this sense, we are able to train HTN in the deep learning way which is the standard combination of algorithms such as Back Propagation and Stochastic Gradient Descent. We finally provide two applicable cases to show the potential applications of HTN, including quantum states classification and quantum-classical autoencoder. These cases also demonstrate the great potentiality to design various HTN in deep learning way.
The formalism of Tensor Network (TN) provides a compact way to approximate many-body quantum states with 1D chain of tensors. The 1D chain of tensors is found to be efficient in capturing the local correlations between neighboring subsystems, and machine learning approaches have been proposed using artificial neural networks (NN) of similar structure. However, a long chain of tensors is difficult to train due to exploding and vanishing gradients. In this paper, we propose methods to decompose the long-chain TN into short chains, which could improve the convergence property of the training algorithm by allowing stable stochastic gradient descent (SGD). In addition, the short-chain methods are robust to network initializations. Numerical experiments show that the short-chain TN achieves almost the same classification accuracy on MNIST dataset as LeNet-5 with less trainable network parameters and connections.
Neural networks have achieved state of the art results in many areas, supposedly due to parameter sharing, locality, and depth. Tensor networks (TNs) are linear algebraic representations of quantum many-body states based on their entanglement structure. TNs have found use in machine learning. We devise a novel TN based model called Deep convolutional tensor network (DCTN) for image classification, which has parameter sharing, locality, and depth. It is based on the Entangled plaquette states (EPS) TN. We show how EPS can be implemented as a backpropagatable layer. We test DCTN on MNIST, FashionMNIST, and CIFAR10 datasets. A shallow DCTN performs well on MNIST and FashionMNIST and has a small parameter count. Unfortunately, depth increases overfitting and thus decreases test accuracy. Also, DCTN of any depth performs badly on CIFAR10 due to overfitting. It is to be determined why. We discuss how the hyperparameters of DCTN affect its training and overfitting.
This paper proposes a Quantum Tensor Network Neural Architecture that synergistically integrates quantum and classic frameworks in order to advance quantum machine-learning. The architecture increases efficiency by constraining model weights structurally. It does this using partial symmetric decompositions of tensors that are integrated into variational quanta circuits. The symmetry constraint increases tensor reconstruction accuracy, and reduces errors compared to non-constrained approaches. A hybrid quantum-classical method of training accelerates convergence, stabilizes parameter updates and establishes robust and noise-resistant performance on simulated quanta devices. Stability and scaling of the model are supported by its theoretical underpinnings including algebraic and spectrum analyses. QTNNA is a promising framework that can process high-dimensional information effectively. This approach tackles the curse of dimensionality in quantum computing, by utilizing tensor network to manage exponential complexity.
We address the problem of implementing bottleneck layers from classical pretrained neural networks on a quantum computer, with the goal of exploring intrinsically quantum ansatz for representing large linear layers within hybrid classical-quantum models. Our approach begins with a compression step in which the target linear layer is represented as an effective matrix product operator (MPO) without degrading model performance. The MPO is then further disentangled into a more compact form. This enables a hybrid classical-quantum execution scheme, where the disentangling circuits are deployed on a quantum computer while the remainder of the network—including the disentangled MPO—runs on classical hardware. We introduce two complementary algorithms for MPO disentangling: (1) an disentangling variational method leveraging standard tensor network optimization techniques and (2) an disentangling gradient-descent-based approach. We validate these methods through a proof-of-concept translation of simple classical neural networks for MNIST and CIFAR-10 image classification into a hybrid classical-quantum form.