Qubit seriation: Improving data-model alignment using spectral ordering

Abstract

With the advent of quantum and quantum-inspired machine learning, adapting the structure of learning models to match the structure of target datasets has been shown to be crucial for obtaining high performance. Probabilistic models based on tensor networks (TNs) are prime candidates to benefit from data-dependent design considerations, owing to their bias towards correlations which are local with respect to the topology of the model. In this work, we use methods from spectral graph theory to search for optimal permutations of model sites which are adapted to the structure of an input dataset. Our method uses pairwise mutual information estimates from the target dataset to ensure that strongly correlated bits are placed closer to each other relative to the model's topology. We demonstrate the effectiveness of such preprocessing for probabilistic modeling tasks, finding substantial improvements in the performance of generative models based on matrix product states (MPS) across a variety of datasets. We also show how spectral embedding, a dimensionality reduction technique from spectral graph theory, can be used to gain further insights into the structure of datasets of interest.