Interpretation of Unfair Sampling in Quantum Annealing by Node Centrality

TL;DR

This paper uses degenerate perturbation theory to analyze how quantum annealing samples degenerate ground states, revealing that node centrality in the ground state graph predicts sampling probabilities and suggesting ways to improve fairness.

Contribution

It introduces a novel theoretical framework linking ground state sampling bias in quantum annealing to eigenvector centrality, providing insights into the underlying mechanisms.

Findings

Eigenvector centrality predicts sampling probabilities.

Second-order weights encode local barrier information.

Connectivity and centrality heterogeneity affect sampling fairness.

Abstract

In applications where multiple optimal solutions are needed, transverse-field quantum annealing (QA) is known to sample degenerate ground states in a strongly biased manner. Despite extensive empirical observations, it remains unclear which features of degenerate ground states are preferentially sampled and why by QA. Here we analyze the final states using degenerate perturbation theory to characterize the preference among them. In this analysis, the adjacency matrix of the graph composed by the ground states naturally emerges, and we can predict the eigenvector centralities (one of the node centralities) are related to the probabilities of these states. We verify this prediction on toy models where degeneracy is lifted at first and second order, and we show that second-order weights encode local barrier information, relating sampling fairness to the flatness of the local energy landscape. Finally, this perspective suggests two practical routes toward fair sampling -- promoting connectivity of the graph and reducing heterogeneity of centralities -- and we illustrate consistency with higher-order drivers and minor-embedding transformations.