Quantum-inspired tensor networks in machine learning models
Authors: Guillermo Valverde, Igor García-Olaizola, Giannicola Scarpa, Alejandro Pozas-Kerstjens
Organizations: 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. · Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland.
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.
Figures & tables
Figure 1: Penrose notation for a scalar, vector, matrix, and a order-3 tensor.
Figure 2: Tensor network example. The matrices B and C , the order-3 tensor D , and the order-5 tensor A are multiplied along the relevant dimensions (i.e., contracted) to produce the order-4 tensor T .
Figure 3: Fundamental tensor network operations: (a) reshaping, (b) tensor contraction, (c) SVD decomposition of a high-dimensional matrix, and (d) tensorization into a linear tensor network.
Figure 4: Gauge freedom and canonical forms: (a) inserting GG−1 on a virtual bond leaves the global tensor invariant; (b) the three canonical forms of an MPS; (c) isometry property A†A=I .
Figure 5: A matrix product state of five sites.
Figure 6: A matrix product operator of five sites.
Figure 7: PEPS lattice of size three.
Figure 8: Tree tensor network with eight leaf nodes.
Figure 9: MERA network. In contrast to TTNs, the disentanglers (blue rectangles) break short-range correlations so that the bond dimensions can accommodate long-range correlations.
Figure 10: Graphical representations of common tensor decompositions.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 11: Topic-author graph for the works analyzed. Green nodes correspond to topics, while pink nodes denote individual authors. Two nodes are connected if an author has published a work in said topic. An interactive version of this graph, that allows for accessing the relevant manuscripts, can be found in Valverde (2026) .
Tensor networks represent high-dimensional data and quantum states as interconnected low-rank tensors to enable efficient computation, learning, and simulation.
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.
Gustav J L Jäger, Martin B Plenio, Hans-Martin Rieser
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
While tensor networks have their traditional application in simulating quantum systems, in the recent decade they have gathered interest as machine learning models. We combine the experience from both fields and derive how quantum constraints placed on a tensor network manifest a change in capabilities. To this end, we employ a method of inference of classical tensor networks on a quantum computer to define a hybrid architecture. This hybrid tensor network is a practical unified framework for it's classical and quantum tensor network edge cases. We identify post-selection as the important property on which this interpolation hinges. The amount of post-selection corresponds to the level to which quantum constraints are enforced on the tensor network. On this basis, we propose a new hyperparameter which controls the transition between the hybrid and the quantum tensor network. In the comparison of classical and quantum tensor networks it complements the bond dimension. Quantum machine learning is improved by using the hyperparameter to allocate the practically limited post-selection to the quantum model in a trainable manner.
Gustav J L Jäger, Krzysztof Bieniasz, Martin B Plenio +1
Institut für KI-Sicherheit, Deutsches Zentrum für Luft- und Raumfahrt e.V., Wilhelm-Runge-Straße 10, Ulm, 89081, Germany. · Institut für Theoretische Physik and IQST, Universität Ulm,2026 Albert-Einstein-Allee 11, Ulm, 89081, Germany.
Tensor networks, which are originally developed for characterizing complex quantum many-body systems, have recently emerged as a powerful framework for capturing high-dimensional probability distributions with strong physical interpretability. This paper systematically studies matrix product states (MPS) for generative modeling and shows that unitary MPS, which is a tensor-network architecture that is both simple and expressive, offers clear benefits for unsupervised learning by reducing ambiguity in parameter updates and improving efficiency. To overcome the inefficiency of standard gradient-based MPS training, we develop a Riemannian optimization approach that casts probabilistic modeling as an optimization problem with manifold constraints, and further derive an efficient space-decoupling algorithm. Experiments on Bars-and-Stripes and EMNIST datasets demonstrate fast adaptation to data structure, stable updates, and strong performance while maintaining the efficiency and expressive power of MPS.
Haotong Duan, Zhongming Chen, Ngai Wong
Department of Mathematics, School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China. · Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong.