FALQON-MST: A Fully Quantum Framework for Graph Optimization in Vision Systems

Finding the minimum spanning tree (MST) of a graph is an important task in computer vision, as it enables a sparse and low-cost representation of connectivity among elements (such as superpixels, points, or regions), which is useful for tasks such as segmentation, reconstruction, and clustering. In this work, we propose and evaluate a fully quantum pipeline for computing MSTs using the FALQON algorithm, a feedback-based quantum optimization method that does not require classical optimizers. We construct a Hamiltonian formulation whose ground-state energy encodes the MST of a graph and compare different FALQON strategies: (i) time rescaling (TR-FALQON) and (ii) multi-driver configurations. To avoid domain-specific biases, we adopt graphs with random weights and show that the FALQON variants exhibit significant differences in ground-state fidelity. We discuss the relevance of this approach for computer vision problems that naturally yield graph representations, and experimental results on synthetic instances together with a small demonstrative study on image segmentation illustrate both the potential and the current limitations of the method. Our numerical simulations on randomly weighted graphs show that standard one drive FALQON, although it reduces the expected energy, fails to concentrate amplitude in the MST solution. The multi drive variant succeeds in redistributing probability mass toward the ground state so that the MST appears among the most probable outcomes, and TR FALQON applied over multi drive produces the best results with faster convergence, lower final energy, and the highest solution state probability or fidelity in our tested instances. These improvements were observed on small synthetic graphs, underscoring both the promise of multi drive controls with temporal rescaling and the need for further scaling and hardware validation.