Quantum Tensor Network Learning with DMRG

Gustav J L Jäger; Martin B Plenio; Hans-Martin Rieser

quant-phAug 19, 2026

Quantum Tensor Network Learning with DMRG

Authors: Gustav J L Jäger, Martin B Plenio, Hans-Martin Rieser

Organizations: Deutsches Zentrum für Luft- und Raumfahrt e.V. – Institut für KI-Sicherheit Wilhelm-Runge-Straße 10, 89081 Ulm, Germany · Universität Ulm – Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, 89081 Ulm, Germany

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Architecture of the underlying tensor network. The trainable part of the setup is the MPS, here in blue. The encoded input data of datapoint $i$ of the dataset is represented by the orange tensors. The yellow tensor represents the single site which serves as the output of the label.

Figure 2: The loss is displayed over the iterative steps that are given by three sweeps. The modified DMRG method, and the gradient descent with normalization are compared with conjugate gradient descent.

method	training loss	test loss	training acc.	testing acc.
initialization	$0.43646$	$0.43984$	$67.725\%$	$63.300\%$
conjugate grad. desc.	$0.06487$	$0.08680$	$96.950\%$	$94.700\%$
norm. grad. desc.	$0.35820$	$0.36256$	$75.750\%$	$73.000\%$
modified Lanczos	$0.35820$	$0.36257$	$75.775\%$	$73.100\%$

Table 1: Overview of accuracy and loss of the different methods.

Tensor Networks

Tensor networks represent high-dimensional data and quantum states as interconnected low-rank tensors to enable efficient computation, learning, and simulation.

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Apr 15, 2026J/K move · Enter open · S save

Quantum-inspired tensor networks in machine learning models

Guillermo Valverde, Igor García-Olaizola, Giannicola Scarpa, Alejandro Pozas-Kerstjens

Vicomtech Foundation, Basque Research and Technology Alliance (BRTA), Donostia-San Sebastián 20009, Spain. · University of Deusto, Avda. de las Universidades, 24, Bilbao 48007, Spain. · Escuela Técnica Superior de Ingeniería de Sistemas Informáticos, Universidad Politécnica de Madrid, Madrid 28031, Spain.+1

Tensor Networks Quantum-Inspired ML

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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.

Abstract

Tensor Networks

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