Interpretation of Unfair Sampling in Quantum Annealing by Node Centrality

TL;DR

This work probes why transverse-field quantum annealing exhibits unfair sampling of degenerate ground states. By applying degenerate perturbation theory around the final time, it derives ground-state adjacency matrices $A^{(1)}$ and $A^{(2)}$ that define a solution graph, where ground-state probabilities align with the principal eigenvector of $A^{(1)}$ and, in higher-order cases, with energy-barrier information captured by $A^{(2)}$. The study introduces the notions of energy-flatness $EF_i$ and relative energy-flatness $REF_i$ to quantify local landscape flatness and shows that central states with higher centrality are more likely to be sampled. It demonstrates these ideas on toy models (e.g., $N$-spin chain, triangular lattice, Matsuda five-spin, N-Queens) and discusses practical routes to fairness via higher-order drivers and problem embedding, including energy-landscape transformations (ELTIP). The framework provides a graph-based, centrality-driven interpretation of sampling bias with implications for hardware design and problem transformations to achieve more uniform sampling.

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.

Figures (7)

• Figure 1: Example of adjacency matrix (left) and solution graph (right) in a model with degenerate ground states: $|\uparrow \uparrow \uparrow \rangle, |\uparrow \downarrow \downarrow \rangle, |\downarrow \downarrow \downarrow \rangle$, using a transverse-field driver.
• Figure 2: (a) $N$-spin chain model in $N=4$. (b) Solution graph using a transverse-field driver. (c) Ground-state probabilities $P_{\mathrm{GS}}$ (bars) compared with the eigenvector-centrality (line).
• Figure 3: (a) Three-spin triangle model. (b) Solution graph in a transverse field. (c) For each ground state, the probability $P_{\mathrm{GS}}$ (solid) and the eigenvector centrality (dashed) are shown as functions of the relative energy flatness (horizontal axis) when sweeping $b \in (0, 2)$. We compute centrality using the full weighted matrix $A^{(2)}$, while the weights are omitted from the graph in (b) for clarity.
• Figure 4: (a) Solution graphs using a transverse-field $V_1$ (left) and second-order driver $V_2$ (right). (b) Ground-state probabilities $P_{\mathrm{GS}}$ (bars) compared with the eigenvector centrality (line). Weights used in the centrality calculations are omitted from the graph drawings for clarity.
• Figure 5: (a) Five-spin Matsuda models before and after embedding. (b) Solution graphs in the original and embedded models using a transverse-field driver. (c) Ground-state probabilities $P_{\mathrm{GS}}$ (bars) compared with the eigenvector centrality (line). In the embedded model, we set the chain strength $J_F = 0.5, 1.0, 1.5$. Weights used in the centrality calculations are omitted from the graph drawings for clarity.