Artificial Neural Networks (ANNs) are widely used in machine learning, inspired by biological neural networks. Recently, connections between machine learning and quantum physics have been identified, leading to quantum-inspired algorithms. Tensor Networks (TNs), numerical tools for simulating quantum many-body systems, have shown effectiveness in machine learning tasks. TNs offer a compact representation of high-order tensors, enabling efficient computation and the extraction of quantities like quantum correlations and entanglement entropy. This paper investigates the use of TTNs for the challenging problem of b-jet classification in high-energy physics. B-jets, originating from b-quarks, are difficult to classify due to the complex fragmentation process. The study aims to demonstrate the effectiveness of TTNs, particularly in feature selection and network optimization for both high precision and fast response times, which are crucial for real-time LHCb event classification at MHz scales. The paper introduces two protocols: Quantum-Information Post-learning feature Selection (QuIPS) and Quantum-Information Adaptive Network Optimisation (QIANO), for enhancing efficiency and interpretability.

The paper reviews the existing literature on ANNs and their applications in machine learning and high-energy physics. It highlights the increasing interest in deep connections between machine learning and quantum physics, noting that ANNs have been used to describe quantum systems, and conversely, quantum-inspired techniques are being applied to solve machine learning problems. The use of TNs in quantum physics is discussed, emphasizing their ability to represent quantum wavefunctions efficiently. Recent research applying TN methods to machine learning tasks, showing comparable results to ANNs, is also reviewed. The authors emphasize the unique benefits of TNs, such as easily computing quantum correlations and entanglement entropy which provide insights into the learned data, moving towards explainable AI. Existing methods for b-jet classification in LHCb, such as boosted decision tree classifiers and muon tagging are mentioned, highlighting the challenges of b-quark charge identification and the need for more efficient methods.

The study employs a TTN to represent the weight tensor in a supervised learning framework. Each sample is encoded by a feature map, and classification is based on the overlap between the input data and the TTN. The authors use a specific feature map based on cosine and sine functions of rescaled feature values. The TTN is structured hierarchically, enabling the computation of entanglement entropy and correlations. QuIPS is introduced to select features based on information content, using entanglement entropy as a measure. Features with low entanglement entropy are deemed less important and discarded. Correlation functions are also used to identify redundant features. The LHCb dataset used is described, including the 16 features considered: momentum, charge, and distance from the jet axis for five particle types (muon, kaon, pion, electron, and proton), plus the total jet charge. A DNN with three hidden layers of 96 nodes is also trained for comparison. The datasets are split into training (60%) and testing (40%) sets. Both the TTN and DNN output the probability of a jet originating from a b-quark. A threshold is introduced to optimize the tagging power. The QIANO method is introduced to optimize the TTN for speed without retraining, achieved by reducing the bond dimension after training using singular value decomposition. The performance is evaluated using tagging power, accuracy, prediction time and ROC curves.

The TTN and DNN achieve similar performance in terms of raw prediction accuracy and tagging power. Both significantly outperform the muon tagging approach across a range of jet transverse momenta. However, the prediction confidence distributions differ. The DNN produces more conservative predictions with lower confidences, while the TTN exhibits a flatter distribution with more high-confidence predictions, particularly linked to muon presence. Despite this difference in confidence distributions, the Pearson correlation coefficient between the TTN and DNN outputs is high (r=0.97), suggesting they capture similar information. The QuIPS feature selection protocol, using entanglement entropy and correlations, identified eight key features that yielded comparable tagging power to the model using all 16 features. The model with the eight least important features performed far worse, underlining the effectiveness of QuIPS. The QIANO method effectively reduces prediction time by truncating the TTN after training, with minimal loss in accuracy. The prediction time is reduced from 345 µs to 19 µs for the optimized model, which is compatible with real-time classification rates.

The results demonstrate the competitive performance of TTNs for b-jet classification in high-energy physics. While DNNs offer comparable performance, TTNs provide advantages in interpretability, allowing the identification of important features and the understanding of how the classifier operates. The QuIPS and QIANO protocols enable significant improvements in both efficiency and speed. The ability to optimize for speed without retraining is a key advantage of TTNs over DNNs. The achieved prediction time of 19 µs is suitable for real-time applications, and further improvements are anticipated through parallelization on GPUs. The study lays the groundwork for future real-time applications at LHCb and similar experiments.

This research successfully applied TTNs to a challenging high-energy physics problem, achieving comparable performance to DNNs while offering distinct advantages. The QuIPS and QIANO protocols demonstrate the ability to optimize TTNs for both accuracy and speed, making them suitable for real-time applications. Future work could explore more sophisticated optimization techniques and adaptation to other high-energy physics problems, such as differentiating b-jets, c-jets, and light flavor jets.

The study uses a specific feature map and optimization technique. Exploring alternative feature maps and optimization algorithms could potentially improve performance. The availability of the TTN code is currently restricted, limiting reproducibility. The analysis is based on simulated data, and further validation with real experimental data is needed.