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Active learning with quantics tensor networks

Krzysztof Bieniasz, Hans-Martin Rieser

Published Nov 18, 2025• 0 citations
OpenAlex

Abstract

Tensor networks are a computational technique originating in condensed matter physics, however in recent years they have been increasingly applied to machine learning, in particular in the context of more efficient classical models and model compression, as well as a natural language for quantum machine learning. Here we present a novel active learning algorithm for constructing models of black-box functions from very few samples, which is based on a DMRG-like local optimization of tensor networks, but utilizing the tensor cross-interpolation which bypasses the need for defining an energy. We underscore the potential of this method for scientific computing, particularly in the context of fast differentiation/integration and Fourier transforms. We illustrate these features using examples of simple mathematical functions.

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Tensor networks are a computational technique originating in condensed matter physics, however in recent years they have been increasingly applied to machine learning, in particular in the context of more efficient classical models and model compression, as well as a natural language for quantum machine learning. Here we present a novel active learning algorithm for constructing models of black-box functions from very few samples, which is based on a DMRG-like local optimization of tensor networks, but utilizing the tensor cross-interpolation which bypasses the need for defining an energy. We underscore the potential of this method for scientific computing, particularly in the context of fast differentiation/integration and Fourier transforms. We illustrate these features using examples of simple mathematical functions.
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.
Tensor network algorithms seek to minimize correlations to compress the classical data representing quantum states. Tensor network algorithms and similar tools—called tensor network methods—form the backbone of modern numerical methods used to simulate many-body physics and have a further range of applications in machine learning. Finding and contracting tensor network states is a computational task, which may be accelerated by quantum computing. We present a quantum algorithm that returns a classical description of a rank-r tensor network state satisfying an area law and approximating an eigenvector given black-box access to a unitary matrix. Our work creates a bridge between several contemporary approaches, including tensor networks, the variational quantum eigensolver (VQE), quantum approximate optimization algorithm (QAOA), and quantum computation.
Tensor cross interpolation (TCI) is a powerful technique for learning a tensor train (TT) by adaptively sampling a target tensor based on an interpolation formula. However, when the tensor evaluations contain random noise, optimizing the TT is more advantageous than interpolating the noise. Here, we propose a new method that starts with an initial guess of TT and optimizes it using non-linear least-squares by fitting it to measured points obtained from TCI. We use quantics TCI (QTCI) in this method and demonstrate its effectiveness on sine and two-time correlation functions, with each evaluated with random noise. The resulting QTT exhibits increased robustness against noise compared to the QTCI method. Furthermore, we employ this optimized QTT of the correlation function in quantum simulation based on pseudo-imaginary-time evolution, resulting in ground-state energy with higher accuracy than the QTCI or Monte Carlo methods.
Tensor decomposition techniques have shown great successes in machine learning and data science by extending classical algorithms based on matrix factorization to multi-modal and multi-way data. However, there exist many tensor decomposition models~(CP, Tucker, Tensor Train, etc.) and the rank of such a decomposition is typically a collection of integers rather than a unique number, making model and hyper-parameter selection a tedious and costly task. At the same time, tensor network methods are powerful tools developed in the physics community which have recently shown their potential for machine learning applications and offer a unifying view of the various tensor decomposition models. In this paper, we leverage the tensor network formalism to develop a generic and efficient adaptive algorithm for tensor learning. Our method is based on a simple greedy approach optimizing a differentiable loss function starting from a rank one tensor and successively identifying the most promising tensor network edges for small rank increments. Our algorithm can adaptively identify tensor network structures with small number of parameters that effectively optimize the objective function from data. The framework we introduce is very broad and encompasses many common tensor optimization problems. Experiments on tensor decomposition and tensor completion tasks with both synthetic and real world data demonstrate the effectiveness of the proposed algorithm.
It is a critical challenge to simultaneously achieve high interpretability and high efficiency with the current schemes of deep machine learning (ML). The tensor network (TN), a well-established mathematical tool originating from quantum mechanics, has shown its unique advantages in developing efficient “white-box” ML schemes. Here, we provide a brief review of the inspiring progress in TN-based ML. On the one hand, the interpretability of TN ML can be accommodated by a solid theoretical foundation based on quantum information and many-body physics. On the other hand, high efficiency can be obtained from powerful TN representations and the advanced computational techniques developed in quantum many-body physics. Keeping pace with the rapid development of quantum computers, TNs are expected to produce novel schemes runnable on quantum hardware in the direction of “quantum artificial intelligence” in the near future.
Tensor networks have found a wide use in a variety of applications in physics and computer science, recently leading to both theoretical insights as well as practical algorithms in machine learning. In this work we explore the connection between tensor networks and probabilistic graphical models, and show that it motivates the definition of generalized tensor networks where information from a tensor can be copied and reused in other parts of the network. We discuss the relationship between generalized tensor network architectures used in quantum physics, such as string-bond states, and architectures commonly used in machine learning. We provide an algorithm to train these networks in a supervised-learning context and show that they overcome the limitations of regular tensor networks in higher dimensions, while keeping the computation efficient. A method to combine neural networks and tensor networks as part of a common deep learning architecture is also introduced. We benchmark our algorithm for several generalized tensor network architectures on the task of classifying images and sounds, and show that they outperform previously introduced tensor-network algorithms. The models we consider also have a natural implementation on a quantum computer and may guide the development of near-term quantum machine learning architectures.