Machine learning by unitary tensor network of hierarchical tree structure

Abstract

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.

Parameterized quantum circuits as machine learning models

“…By looking only at this small corner of the Hilbert space, one lowers the computational cost to a polynomial dependence on sys-tem size. This language has been explored for generative and classification tasks using hierarchical representations such as matrix product states (MPS), tree tensor networks (TTN), and the multi-scale entanglement renormalization ansatz (MERA) [17,43,44].…”

“…One of their simplest models is efficiently trained classically and then deployed on the IBM Q5 Tenerife quantum computer with significant resilience to noise. Stoudenmire et al [43] train a 2D TTN to perform pairwise classification of the MNIST image data. Although a fully classical experiment, they use quantum fidelity to measure the inherent difficulty to distinguish two classes, and entanglement entropy as quantifying the amount of information about one part of an image that can be gained by knowing the rest.…”

Tensor network compressed sensing with unsupervised machine learning

“…[41]). As shown in the previous works [15,25,26,[28][29][30][31][32], TN models (including MPS) possess remarkable generalization power that is competitive to neural networks. Notably, TN models surpass neural networks as they possess high interpretability and allow us to implement quantum processes.…”

“…Here, we take | in the form of MPS [40]. Note that TNCS is a general scheme, where one may also choose other TN forms to represent | , such as tree TN or MERA [26,31,32], or simply a quantum state without a specific entanglement structure.…”

“…Recently, TN [20][21][22][23][24] is rapidly developed into a powerful quantum-inspired computational tool for machine learning, which brings new possibilities and wide perspectives to process real-life data, such as images and texts, in the quantum processes based on many-qubit (or many-body) states representing the probability distribution of the data that Alice considers to send, (2) encode the specific piece of information to be sent by projecting | , (3) decode the information as a generative process by the projected Born machine. [15,[25][26][27][28][29][30][31][32]. High efficiencies have been demonstrated at least for the classical simulations of these quantum processes.…”