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arXiv:2403.12969

Entangling Machine Learning with Quantum Tensor Networks

Constantijn van der Poel, Dan Zhao

Published Jan 9, 2024• 0 citations
arXivHTMLPDF

Abstract

This paper examines the use of tensor networks, which can efficiently represent high-dimensional quantum states, in language modeling. It is a distillation and continuation of the work done in (van der Poel, 2023). To do so, we will abstract the problem down to modeling Motzkin spin chains, which exhibit long-range correlations reminiscent of those found in language. The Matrix Product State (MPS), also known as the tensor train, has a bond dimension which scales as the length of the sequence it models. To combat this, we use the factored core MPS, whose bond dimension scales sub-linearly. We find that the tensor models reach near perfect classifying ability, and maintain a stable level of performance as the number of valid training examples is decreased.

DOI
10.48550/arxiv.2403.12969

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This paper examines the use of tensor networks, which can efficiently represent high-dimensional quantum states, in language modeling. It is a distillation and continuation of the work done in (van der Poel, 2023). To do so, we will abstract the problem down to modeling Motzkin spin chains, which exhibit long-range correlations reminiscent of those found in language. The Matrix Product State (MPS), also known as the tensor train, has a bond dimension which scales as the length of the sequence it models. To combat this, we use the factored core MPS, whose bond dimension scales sub-linearly. We find that the tensor models reach near perfect classifying ability, and maintain a stable level of performance as the number of valid training examples is decreased.
Matrix Product States (MPS), also known as Tensor Train (TT) decomposition in mathematics, has been proposed originally for describing an (especially one-dimensional) quantum system, and recently has found applications in various applications such as compressing high-dimensional data, supervised kernel linear classifier, and unsupervised generative modeling. However, when applied to systems which are not defined on one-dimensional lattices, a serious drawback of the MPS is the exponential decay of the correlations, which limits its power in capturing long-range dependences among variables in the system. To alleviate this problem, we propose to introduce long-range interactions, which act as shortcuts, to MPS, resulting in a new model \textit{ Shortcut Matrix Product States} (SMPS). When chosen properly, the shortcuts can decrease significantly the correlation length of the MPS, while preserving the computational efficiency. We develop efficient training methods of SMPS for various tasks, establish some of their mathematical properties, and show how to find a good location to add shortcuts. Finally, using extensive numerical experiments we evaluate its performance in a variety of applications, including function fitting, partition function calculation of 2−2-2−d Ising model, and unsupervised generative modeling of handwritten digits, to illustrate its advantages over vanilla matrix product states.
Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.
Tensor networks are a powerful modeling framework developed for computational many-body physics, which have only recently been applied within machine learning. In this work we utilize a uniform matrix product state (u-MPS) model for probabilistic modeling of sequence data. We first show that u-MPS enable sequence-level parallelism, with length-n sequences able to be evaluated in depth O(log n). We then introduce a novel generative algorithm giving trained u-MPS the ability to efficiently sample from a wide variety of conditional distributions, each one defined by a regular expression. Special cases of this algorithm correspond to autoregressive and fill-in-the-blank sampling, but more complex regular expressions permit the generation of richly structured text in a manner that has no direct analogue in current generative models. Experiments on synthetic text data find u-MPS outperforming LSTM baselines in several sampling tasks, and demonstrate strong generalization in the presence of limited data.
Tensor network methods have been a key ingredient of advances in condensed matter physics and have recently sparked interest in the machine learning community for their ability to compactly represent very high-dimensional objects. Tensor network methods can for example be used to efficiently learn linear models in exponentially large feature spaces [Stoudenmire and Schwab, 2016]. In this work, we derive upper and lower bounds on the VC dimension and pseudo-dimension of a large class of tensor network models for classification, regression and completion. Our upper bounds hold for linear models parameterized by arbitrary tensor network structures, and we derive lower bounds for common tensor decomposition models~(CP, Tensor Train, Tensor Ring and Tucker) showing the tightness of our general upper bound. These results are used to derive a generalization bound which can be applied to classification with low rank matrices as well as linear classifiers based on any of the commonly used tensor decomposition models. As a corollary of our results, we obtain a bound on the VC dimension of the matrix product state classifier introduced in [Stoudenmire and Schwab, 2016] as a function of the so-called bond dimension~(i.e. tensor train rank), which answers an open problem listed by Cirac, Garre-Rubio and Pérez-García in [Cirac et al., 2019].
Abstract Quantum machine learning (QML) is a rapidly expanding field that merges the principles of quantum computing with the techniques of machine learning. One of the powerful mathematical frameworks in this domain is tensor networks. These networks are used to approximate high-order tensors by contracting tensors with lower ranks. Originally developed for simulating quantum systems, tensor networks have become integral to quantum computing and, by extension, to QML. Their ability to efficiently represent and manipulate complex, high-dimensional data makes them suitable for various machine learning tasks within the quantum realm. Here, we present a matrix product state (MPS) model, where the MPS functions as both a classifier and a generator. The dual functionality of this novel MPS model permits a strategy that enhances the traditional training of supervised MPS models. This framework is inspired by generative adversarial networks and is geared towards generating more realistic samples by reducing outliers. Additionally, our contributions offer insights into the mechanics of tensor network methods for generation tasks. Specifically, we discuss alternative embedding functions and a new sampling method from non-normalized MPSs.
These notes provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS). They develop the basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation. The main MPS algorithms are then introduced, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods. The notes also briefly discuss projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction. Finally, tensor-network representations of mixed states, quantum channels, and Lindblad dynamics are presented, with applications to thermal states and open quantum systems. The presentation is accompanied by short Julia code examples based on ITensor, ITensorMPS, and TensorMixedStates. These notes were written for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems, held in June 2026.