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arXiv:2505.03911 · IEEE International Conference on Quantum Artificial Intelligence (QAI)

Explaining Anomalies with Tensor Networks

Hans Hohenfeld, Marius Beuerle, Elie Mounzer

Published Nov 2, 2025• 0 citations
arXivHTMLPDF

Abstract

Tensor networks, a class of variational quantum many-body wave functions have attracted considerable research interest across many disciplines, including classical machine learning. Recently, Aizpurua et al. demonstrated explainable anomaly detection with matrix product states on a discrete-valued cyber-security task, using quantum-inspired methods to gain insight into the learned model and detected anomalies. Here, we extend this framework to real-valued data domains. We furthermore introduce tree tensor networks for the task of explainable anomaly detection. We demonstrate these methods with three benchmark problems, show adequate predictive performance compared to several baseline models and both tensor network architectures' ability to explain anomalous samples. We thereby extend the application of tensor networks to a broader class of potential problems and open a pathway for future extensions to more complex tensor network architectures.

DOI
10.1109/qai63978.2025.00033

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Tensor networks, a class of variational quantum many-body wave functions have attracted considerable research interest across many disciplines, including classical machine learning. Recently, Aizpurua et al. demonstrated explainable anomaly detection with matrix product states on a discrete-valued cyber-security task, using quantum-inspired methods to gain insight into the learned model and detected anomalies. Here, we extend this framework to real-valued data domains. We furthermore introduce tree tensor networks for the task of explainable anomaly detection. We demonstrate these methods with three benchmark problems, show adequate predictive performance compared to several baseline models and both tensor network architectures' ability to explain anomalous samples. We thereby extend the application of tensor networks to a broader class of potential problems and open a pathway for future extensions to more complex tensor network architectures.
With the increasing adoption of machine learning tools like neural networks across several domains, interesting connections and comparisons to concepts from other domains are coming to light. In this work, we focus on the class of Tensor Networks, which has been a work horse for physicists in the last two decades to analyse quantum many-body systems. Building on the recent interest in tensor networks for machine learning, we extend the Matrix Product State tensor networks (which can be interpreted as linear classifiers operating in exponentially high dimensional spaces) to be useful in medical image analysis tasks. We focus on classification problems as a first step where we motivate the use of tensor networks and propose adaptions for 2D images using classical image domain concepts such as local orderlessness of images. With the proposed locally orderless tensor network model (LoTeNet), we show that tensor networks are capable of attaining performance that is comparable to state-of-the-art deep learning methods. We evaluate the model on two publicly available medical imaging datasets and show performance improvements with fewer model hyperparameters and lesser computational resources compared to relevant baseline methods.
Tensor networks can compactly represent high-dimensional data in the form of low-rank tensor decompositions, which helps capture complex structures and correlations within the data. However, their inherently multilinear nature limits their potential as universal approximators. To address this limitation, tensor networks are combined with classical neural network layers, introducing nonlinear elements to enhance the model’s expressive power. The Hybrid Tensor Networks (HTN) system encompasses two prevalent tensor network configurations, Matrix Product States and Tree Tensor Networks, which facilitates the adaptable design of models customized for particular machine learning applications. This framework has been validated on classification and image reconstruction tasks, demonstrating its potential to handle complex data structures and its applicability within the machine learning field. Employing this approach, the goal is to establish a foundation for future research on hybrid quantum-classical models, thereby opening up avenues for the realm of quantum machine learning.
Beyond their origin in modeling many-body quantum systems, tensor networks have emerged as a promising class of models for solving machine learning problems, notably in unsupervised generative learning. While possessing many desirable features arising from their quantum-inspired nature, tensor network generative models have previously been largely restricted to binary or categorical data, limiting their utility in real-world modeling problems. We overcome this by introducing a new family of tensor network generative models for continuous data, which are capable of learning from distributions containing continuous random variables. We develop our method in the setting of matrix product states, first deriving a universal expressivity theorem proving the ability of this model family to approximate any reasonably smooth probability density function with arbitrary precision. We then benchmark the performance of this model on several synthetic and real-world datasets, finding that the model learns and generalizes well on distributions of continuous and discrete variables. We develop methods for modeling different data domains, and introduce a trainable compression layer which is found to increase model performance given limited memory or computational resources. Overall, our methods give important theoretical and empirical evidence of the efficacy of quantum-inspired methods for the rapidly growing field of generative learning.
Tensor networks were developed in the context of many-body physics as compressed representations of multiparticle quantum states. These representations mitigate the exponential complexity of many-body systems by capturing only the most relevant dependencies. Due to the formal similarity between quantum entanglement and statistical correlations, tensor networks have recently been integrated in machine learning, operating both as alternative learning architectures and as decompositions of components of neural networks. The expectation is that the theoretical understanding of tensor networks developed within quantum many-body physics leads to novel methods that offer advantages in terms of computational efficiency, explainability, or privacy. Here we review the use of tensor networks in the context of machine learning, providing a critical assessment of the state of the art, the potential advantages, and the challenges that must be overcome.
Modeling the joint distribution of high-dimensional data is a central task in unsupervised machine learning. In recent years, many interests have been attracted to developing learning models based on tensor networks, which have the advantages of a principle understanding of the expressive power using entanglement properties, and as a bridge connecting classical computation and quantum computation. Despite the great potential, however, existing tensor network models for unsupervised machine learning only work as a proof of principle, as their performance is much worse than the standard models such as restricted Boltzmann machines and neural networks. In this Letter, we present autoregressive matrix product states (AMPS), a tensor network model combining matrix product states from quantum many-body physics and autoregressive modeling from machine learning. Our model enjoys the exact calculation of normalized probability and unbiased sampling. We demonstrate the performance of our model using two applications, generative modeling on synthetic and real-world data, and reinforcement learning in statistical physics. Using extensive numerical experiments, we show that the proposed model significantly outperforms the existing tensor network models and the restricted Boltzmann machines, and is competitive with state-of-the-art neural network models.
Machine learning (ML) has recently facilitated many advances in solving problems related to many-body physical systems. Given the intrinsic quantum nature of these problems, it is natural to speculate that quantum-enhanced machine learning will enable us to unveil even greater details than we currently have. With this motivation, this paper examines a quantum machine learning approach based on shallow variational ansatz inspired by tensor networks for supervised learning tasks. In particular, we first look at the standard image classification tasks using the Fashion-MNIST dataset and study the effect of repeating tensor network layers on ansatz's expressibility and performance. Finally, we use this strategy to tackle the problem of quantum phase recognition for the transverse-field Ising and Heisenberg spin models in one and two dimensions, where we were able to reach ≥98%\geq 98\%≥98% test-set accuracies with both multi-scale entanglement renormalization ansatz (MERA) and tree tensor network (TTN) inspired parametrized quantum circuits.