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arXiv:1711.03357 · British Machine Vision Conference (BMVC)

Compact Neural Networks based on the Multiscale Entanglement Renormalization Ansatz

Andrew Hallam, Edward Grant, Vid Stojevic, Simone Severini B, Andrew G. Green

Published Nov 9, 2017• 6 citations
arXivHTMLPDF

Abstract

This paper demonstrates a method for tensorizing neural networks based upon an efficient way of approximating scale invariant quantum states, the Multi-scale Entanglement Renormalization Ansatz (MERA). We employ MERA as a replacement for the fully connected layers in a convolutional neural network and test this implementation on the CIFAR-10 and CIFAR-100 datasets. The proposed method outperforms factorization using tensor trains, providing greater compression for the same level of accuracy and greater accuracy for the same level of compression. We demonstrate MERA layers with 14000 times fewer parameters and a reduction in accuracy of less than 1% compared to the equivalent fully connected layers, scaling like O(N).

DOI
10.48550/arxiv.1711.03357

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This paper demonstrates a method for tensorizing neural networks based upon an efficient way of approximating scale invariant quantum states, the Multi-scale Entanglement Renormalization Ansatz (MERA). We employ MERA as a replacement for the fully connected layers in a convolutional neural network and test this implementation on the CIFAR-10 and CIFAR-100 datasets. The proposed method outperforms factorization using tensor trains, providing greater compression for the same level of accuracy and greater accuracy for the same level of compression. We demonstrate MERA layers with 14000 times fewer parameters and a reduction in accuracy of less than 1% compared to the equivalent fully connected layers, scaling like O(N).
Understanding the limiting capabilities of classical methods in simulating complex quantum systems is of paramount importance for quantum technologies. Although many advanced approaches have been proposed and recently used to challenge quantum advantage experiments, novel efficient methods for the approximate simulation of complex quantum systems are still in high demand. Here, we propose a scalable technique for approximate simulations of intermediate-size quantum circuits on the basis of the multi-scale entanglement renormalization ansatz (MERA) and Riemannian optimization. The MERA is a tensor network, whose geometry together with orthogonality constraints imposed on its tensors allow approximating many-body quantum states lying beyond the area-law scaling of the entanglement entropy. We benchmark the proposed technique for brick-wall quantum circuits of up to 243 qubits with various depths up to 20 layers. Our approach paves a way to exploring efficient simulation techniques for quantum many-body systems.
The multiscale entanglement renormalization ansatz (MERA) provides a constructive algorithm for realizing wave functions that are inherently scale invariant. Unlike conformally invariant partition functions, however, the finite bond dimension $\ensuremath{\chi}$ of the MERA provides a cutoff in the fields that can be realized. In this paper, we demonstrate that this cutoff is equivalent to the one obtained when approximating a thermal state of a critical Hamiltonian with a matrix product operator (MPO) of finite bond dimension $\ensuremath{\chi}$. This is achieved by constructing an explicit mapping between the isometries of the MERA and the local tensors of the MPO. In terms of energy scales, our results show that a finite bond dimension MERA is equivalent to introducing both an infrared and an ultraviolet scale, characterizing relevant and irrelevant perturbations on the underlying conformal field theory.
We present an algorithm for supervised learning using tensor networks, employing a step of preprocessing the data by coarse-graining through a sequence of wavelet transformations. We represent these transformations as a set of tensor network layers identical to those in a multi-scale entanglement renormalization ansatz (MERA) tensor network, and perform supervised learning and regression tasks through a model based on a matrix product state (MPS) tensor network acting on the coarse-grained data. Because the entire model consists of tensor contractions (apart from the initial non-linear feature map), we can adaptively fine-grain the optimized MPS model backwards through the layers with essentially no loss in performance. The MPS itself is trained using an adaptive algorithm based on the density matrix renormalization group (DMRG) algorithm. We test our methods by performing a classification task on audio data and a regression task on temperature time-series data, studying the dependence of training accuracy on the number of coarse-graining layers and showing how fine-graining through the network may be used to initialize models with access to finer-scale features.
The convolutional neural network (CNN) is a widely recognized model in computer vision due to its capability to utilize data correlation effectively. However, CNNs often face challenges when training on large datasets or dimensions. To address this, the Quantum Convolutional Neural Network (QCNN) introduces an innovative solution for CNN-based problems, specifically within quantum computing frameworks. By integrating CNN architecture into quantum environments, QCNN enhances the model's overall efficiency and functionality. This approach marks an initial effort to propose a solution for classification challenges in quantum physics and chemistry. Furthermore, the study highlights a computational method utilizing the Multi-scale Entanglement Renormalization Ansatz (MERA) with logarithmic depth O(log(n)). Experiments conducted using the TensorFlow Quantum platform on the MNIST dataset demonstrate that QCNNs can achieve faster learning compared to conventional CNNs.
We use TensorNetwork [C. Roberts et al., arXiv: 1905.01330], a recently developed API for performing tensor network contractions using accelerated backends such as TensorFlow, to implement an optimization algorithm for the Multi-scale Entanglement Renormalization Ansatz (MERA). We use the MERA to approximate the ground state wave function of the infinite, one-dimensional transverse field Ising model at criticality, and extract conformal data from the optimized ansatz. Comparing run times of the optimization on CPUs vs. GPU, we report a very significant speed-up, up to a factor of 200, of the optimization algorithm when run on a GPU.
Abstract In this article, two new algorithms are presented that convert a given data tensor train into either a Tucker decomposition with orthogonal matrix factors or a multi-scale entanglement renormalization ansatz (MERA). The Tucker core tensor is never explicitly computed but stored as a tensor train instead, resulting in both computationally and storage efficient algorithms. Both the multilinear Tucker-ranks as well as the MERA-ranks are automatically determined by the algorithm for a given upper bound on the relative approximation error. In addition, an iterative algorithm with low computational complexity based on solving an orthogonal Procrustes problem is proposed for the first time to retrieve optimal rank-lowering disentangler tensors, which are a crucial component in the construction of a low-rank MERA. Numerical experiments demonstrate the effectiveness of the proposed algorithms together with the potential storage benefit of a low-rank MERA over a tensor train.