[Paper Review] Classical versus Quantum: comparing Tensor Network-based Quantum Circuits on LHC data

Jack Y. Araz, Michael Spannowsky|arXiv (Cornell University)|Feb 21, 2022

Parallel Computing and Optimization Techniques4 citations

TL;DR

This paper compares classical Tensor Networks (TNs) and TN-inspired quantum circuits for machine learning on simulated Large Hadron Collider (LHC) data, demonstrating that classical TNs require exponentially larger bond dimensions and higher Hilbert-space mappings to match quantum TN performance, leading to flatter loss landscapes and less efficient optimization. Quantum TNs achieve comparable accuracy with fewer parameters and better optimization behavior due to superior entanglement representation via quantum superposition and entanglement.

ABSTRACT

Tensor Networks (TN) are approximations of high-dimensional tensors designed to represent locally entangled quantum many-body systems efficiently. This study provides a comprehensive comparison between classical TNs and TN-inspired quantum circuits in the context of Machine Learning on highly complex, simulated LHC data. We show that classical TNs require exponentially large bond dimensions and higher Hilbert-space mapping to perform comparably to their quantum counterparts. While such an expansion in the dimensionality allows better performance, we observe that, with increased dimensionality, classical TNs lead to a highly flat loss landscape, rendering the usage of gradient-based optimization methods highly challenging. Furthermore, by employing quantitative metrics, such as the Fisher information and effective dimensions, we show that classical TNs require a more extensive training sample to represent the data as efficiently as TN-inspired quantum circuits. We also engage with the idea of hybrid classical-quantum TNs and show possible architectures to employ a larger phase-space from the data. We offer our results using three main TN ansatz: Tree Tensor Networks, Matrix Product States, and Multi-scale Entanglement Renormalisation Ansatz.

Motivation & Objective

• To evaluate the performance gap between classical Tensor Networks (TNs) and TN-inspired quantum circuits in classifying top jets from QCD backgrounds in LHC calorimeter images.
• To investigate the scalability and optimization efficiency of classical TNs versus quantum TNs (QTNs) under increasing data complexity and qubit count.
• To quantify the data efficiency and representational capacity of classical TNs versus QTNs using metrics like Fisher information and effective dimension.
• To explore hybrid classical-quantum TN architectures that leverage classical data mapping with quantum classification layers to extend phase space and improve training.
• To assess the feasibility of using TN-based quantum circuits for near-term NISQ devices in high-energy physics applications.

Proposed method

• Employed three TN ansatz: Tree Tensor Networks (TTN), Matrix Product States (MPS), and Multi-scale Entanglement Renormalisation Ansatz (MERA) as both classical and quantum circuit architectures.
• Mapped high-dimensional LHC calorimeter images into 1D input sequences for TN processing, preserving spatial correlations through tensor decomposition.
• Implemented variational quantum circuits with TN-inspired gate structures (QTNs) to exploit quantum entanglement for richer feature representation.
• Used stochastic gradient descent (SGD) for classical TN optimization and compared convergence behavior, loss landscape flatness, and generalization performance.
• Applied quantitative metrics: Fisher information and effective dimension to assess data efficiency and model capacity.
• Explored hybrid classical-quantum TN architectures by combining classical TN layers for feature embedding with quantum TN layers for classification.

Experimental results

Research questions

• RQ1 How do classical TNs and TN-inspired quantum circuits compare in performance on LHC jet classification tasks?
• RQ2 What is the scaling behavior of bond dimensions and Hilbert-space mappings required for classical TNs to match the performance of quantum TNs?
• RQ3 How does the loss landscape of classical TNs compare to that of quantum TNs, and what impact does this have on optimization efficiency?
• RQ4 To what extent do classical TNs require larger training samples to achieve the same data representation efficiency as quantum TNs?
• RQ5 What are viable hybrid classical-quantum TN architectures that can exploit both classical data processing and quantum entanglement for improved performance?

Key findings

• Classical TNs require exponentially larger bond dimensions than quantum TNs to capture equivalent entanglement structures, making them computationally prohibitive for large-scale ML tasks.
• The loss landscape of classical TNs becomes highly flat with increasing dimensionality, severely hampering the effectiveness of stochastic gradient descent for optimization.
• Classical TNs require significantly larger training samples to represent the same data as efficiently as quantum TNs, as quantified by lower Fisher information and reduced effective dimension.
• Quantum TNs achieve state-of-the-art performance on top-jet discrimination with fewer parameters and superior optimization convergence due to quantum entanglement.
• Hybrid classical-quantum TN architectures are feasible and can extend the phase space of the network, enabling end-to-end training with classical data mapping and quantum classification layers.
• MPS-based quantum circuits achieved comparable accuracy to convolutional neural networks on top-tagging tasks using only 54% of the original pixel data, demonstrating high data efficiency.