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

Abstract

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.

Figures (5)

• Figure 1: Diagram comparing VQAs and FQAs. Both can be formulated with the same objective function $J$ and use quantum circuits with an identical structure (center). The difference lies in how they minimize $J$: VQAs adjust all parameters $\vec{\theta}$ simultaneously through classical optimization (left), whereas FQAs eliminate this step and determine the parameters iteratively using a feedback law, layer by layer (right).
• Figure 2: Illustrative diagram of the FALQON PhysRevB.110.224422. The process begins with the state $\ket{\psi_0}$, and at each layer $k$, the unitary operators $e^{-iH_p \Delta t}$ and $e^{-iH_d \Delta t \beta_k}$ are applied sequentially. The parameter $\beta_k$ is adaptively adjusted at each iteration. This dynamic is repeated iteratively, guiding the evolution of the state $\ket{\psi}$ through the layers until the desired solution is reached.
• Figure 3: Illustrative diagram of the TR-FALQON qc91-5mj2. The process starts with the initial state $\ket{\psi_0}$. In each layer $k$, the unitary operators $e^{-iH_p \dot{f}(k\Delta \tau) \Delta \tau}$ and $e^{-iH_d \dot{f}(k\Delta \tau) \Delta \tau \tilde{\beta}_k}$ are applied sequentially, adaptively adjusting the parameter $\tilde{\beta}_k$.
• Figure 4: (a) Example of a graph constructed with randomly generated edge weights. (b) Minimum Spanning Tree derived from the graph in (a), where the highest-cost edges are removed while preserving connectivity among the nodes at minimal total cost.
• Figure 5: Numerical simulation results for the MST Hamiltonian formulated as a QUBO with time step $\Delta t=0.02$. (a) Convergence of the cost $J=\langle\Psi_k|H_{MST}|\Psi_k\rangle$ as a function of the number of layers $k$ for FALQON (one-drive), FALQON (multi-drive), and TR-FALQON (multi-drive); energy decreases with the number of layers, with TR-FALQON multi-drive achieving the lowest final energy and fastest convergence. (b) Probability distribution of basis states at the end of the protocol for FALQON one-drive, where the state encoding the MST does not appear among the most probable. (c) Analogous distribution for FALQON multi-drive, where the solution state appears with significant probability, indicating amplitude concentration in the ground state. (d) TR-FALQON multi-drive, showing higher probability concentration in the solution state and a less dispersed distribution, consistent with the lower final energy shown in (a).