A Quantum Walk-Driven Algorithm for the Minimum Spanning Tree Problem under a Maximal Degree Constraint

TL;DR

This paper introduces a quantum walk-based algorithm for solving the Minimum Spanning Tree problem with a maximal degree constraint, achieving efficient quantum resource use and competitive classical performance, especially for higher degree constraints.

Contribution

The paper presents Quantum Kruskal with MDC, a novel quantum algorithm that reduces quantum resource requirements while effectively solving constrained MST problems.

Findings

Effective for MDC > 4, yields optimal or near-optimal MSTs.

Uses only $ ext{O}( ext{log} N)$ qubits, reducing quantum resource needs.

Outperforms classical algorithms in large-scale graph tests.

Abstract

We present a novel quantum walk-based approach to solve the Minimum Spanning Tree (MST) problem under a maximal degree constraint (MDC). By recasting the classical MST problem as a quantum walk on a graph, where vertices are encoded as quantum states and edge weights are inverted to define a modified Hamiltonian, we demonstrate that the quantum evolution naturally selects the MST by maximizing the cumulative transition probability (and thus the Shannon entropy) over the spanning tree. Our method, termed Quantum Kruskal with MDC, significantly reduces the quantum resource requirement to $\mathcal{O}(\log N)$ qubits while retaining a competitive classical computational complexity. Numerical experiments on fully connected graphs up to $10^4$ vertices confirm that, particularly for MDC values exceeding $4$, the algorithm delivers MSTs with optimal or near-optimal total weights. When MDC values are less or equal to $4$, some instances achieve a suboptimal solution, still outperforming several established classical algorithms. These results open promising perspectives for hybrid quantum-classical solutions in large-scale graph optimization.