Geometric Prototype Learning in Quantum Hilbert Space with Matrix Product States

TL;DR

This paper introduces a quantum-inspired prototype learning method using matrix product states in Hilbert space, demonstrating improved classification performance and interpretability over classical approaches.

Contribution

The authors propose a novel quantum Hilbert space-based prototype learning scheme with matrix product states, extending classical methods into quantum domain.

Findings

Outperforms classical prototype methods on benchmark datasets.

Remains competitive with standard neural networks.

Identifies an attraction effect in quantum prototypes.

Abstract

Quantum probability provides a novel framework for formulating machine-learning (ML) problems in Hilbert space. We introduce a prototype-based learning scheme where class representatives are encoded as generative matrix product states (MPS). Because these prototypes reside in the same Hilbert space as quantum-encoded data samples, various ML tasks such as classification and clustering can be performed through geometric measures of quantum states. This approach lifts prototype learning from classical feature space to quantum Hilbert space. Benchmarks on Fashion-MNIST and a real-world electrocardiogram dataset demonstrate that our method outperforms classical prototype approaches while remaining competitive with standard black-box neural networks. We also identify an ``attraction'' effect induced by the quantum-probabilistic prototypes and introduce a dimensionality-reduction scheme based on prototype distances. Our results establish quantum states as an explainable framework for prototype learning, opening new directions for designing ML algorithms in quantum Hilbert space.