A quantum feasibility preserving modeling for the min cut problem

TL;DR

This paper explores a quantum approach to the minimum cut problem that maintains feasible solutions throughout the computation, potentially improving scalability and solution quality in quantum network optimization.

Contribution

It introduces a quantum model with a ring structured XY mixer that preserves feasibility without penalty terms, and proposes an iterative strategy for larger problem instances.

Findings

Feasible solutions are maintained throughout quantum evolution.

The initial probability distribution can be controlled for better solutions.

An iterative decomposition approach improves scalability.

Abstract

We study the minimum cut problem in weighted undirected graphs using variational quantum algorithms in which only feasible cut configurations are explored. Although minimum cut admits efficient classical solutions, it is a fundamental component of more complex network optimization problems such as multicut and network interdiction. Our objective is to examine quantum models in which feasibility is preserved by the mixer dynamics, without introducing penalty terms in the cost Hamiltonian. We employ a ring structured XY mixer that restricts the quantum evolution to the subspace of valid cut configurations, ensuring that all sampled states correspond to feasible solutions. To address scalability limitations, we suggest an iterative metaheuristic strategy that decomposes large instances into smaller subproblems solved sequentially using the same quantum model. The results obtained using the mixer indicate that the initial probability distribution can be systematically controlled, thereby enabling the development of warm start techniques within variational quantum based algorithms.