Tensor networks now offer an alternative for handling increasingly complex datasets where conventional machine learning techniques are failing. Gustav J L Jäger and colleagues at Deutsches Zentrum für Luft- und Raumfahrt and Universität Ulm have introduced a global normalisation condition into matrix product state networks, effectively ensuring they represent valid quantum states. They enhanced tensor networks, a machine learning technique inspired by quantum physics simulations, with a new method for optimising their structure.

This involved adding a ‘normalisation condition’ which ensures accurate representation of valid quantum states during training, improving strong performance when using standard optimisation techniques. The team adapted Density Matrix Renormalization Group, originally developed to simulate complex physical systems, creating an alternative approach to methods like gradient descent alone.

These networks, often structured as matrix product states or ‘tensor trains’, can be visualised like connected building blocks forming a larger structure representing complex data; optimising them traditionally relies on techniques such as gradient descent. By implementing the ‘normalisation condition, the team addressed a key challenge in these systems and ensured accurate representation of valid quantum states, analogous to confirming that rolling a six-sided dice always totals 100% probability across all faces.

Stabilised tensor network optimisations unlock improved MNIST dataset performance

A training loss of 0.35820 was achieved by scientists at Deutsches Zentrum für Luft- und Raumfahrt and Universität Ulm using their modified Density Matrix Renormalization Group (DMRG) approach on the MNIST dataset; this represents an improvement over initial unoptimized models which yielded a loss of 0.43646. Previously, attaining such accuracy proved difficult due to instabilities stemming from non-normalized matrix product states, interconnected building blocks representing data, hindering effective optimisation with techniques like gradient descent.

The introduction of a global normalisation condition ensured accurate representation of quantum states within these tensor networks, enabling robust optimisation alongside DMRG, initially developed for simulating complex physical systems. Further analysis revealed that conjugate gradient descent achieved a training loss of 0.06487 and a testing accuracy of 94.700% using the same subset of 5000 rescaled images. However, its resulting matrix product state exhibited an exceptionally high norm of 3.9x 10⁶.

High Accuracy Versus Norm Instability in Tensor Network Optimisation

Deutsches Zentrum für Luft- und Raumfahrt and Universität Ulm researchers are pioneering a bridge between quantum physics and machine learning through tensor networks, systems representing data with interconnected components to optimise performance on complex tasks. While their modified Density Matrix Renormalization Group (DMRG) approach reduces training loss compared to initial models, conjugate gradient descent encountered an unexpected issue: achieving impressive accuracy alongside an extraordinarily high norm within the resulting matrix product state.

This unexpectedly large value obtained during optimisation raises questions about efficiency; this high ‘norm’ suggests potential instability despite good accuracy figures. Adapting tensor networks for quantum machine learning applications constitutes a step forward in the field. By enforcing global normalisation conditions, ensuring accurate representation of valid quantum states, instabilities previously hindering effective training using standard techniques like gradient descent were overcome by scientists from Deutsches Zentrum für Luft- und Raumfahrt and Universität Ulm.

The adaptation of Density Matrix Renormalization Group, originally developed to simulate physical systems, offers an alternative pathway beyond conventional machine learning methods, potentially unlocking new capabilities within artificial intelligence.

Researchers demonstrated that applying a global normalisation condition to tensor network optimisation successfully represents quantum states. This allows for more robust training with DMRG, initially designed for simulating complex physics problems. The scientists suggest further investigation is needed to address this instability despite achieving good performance metrics.

👉 More information
🗞 Quantum Tensor Network Learning with DMRG
✍️ Gustav J L Jäger, Martin B Plenio and Hans-Martin Rieser
🧠 DOI: https://doi.org/10.14428/esann/2025.ES2025-157

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