  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.
