Variational Quantum Algorithm-Preserving Feasible Space for Solving the Uncapacitated Facility Location Problem

The Quantum Alternating Operator Ansatz (QAOA+) is one of the Variational Quantum Algorithm (VQA) specifically developed to tackle combinatorial optimization problems by exploring the feasible space in search of a target solution. For the Constrained Binary Optimization with Unconstrained Variables Problems (CBO-UVPs), the mixed operators in the QAOA+ circuit are applied to the constrained variables, while the single-qubit rotating gates operate on the unconstrained variables. The expressibility of this circuit is limited by the shortage of two-qubit gates and the parameter sharing in the single-qubit rotating gates, which consequently impacts the performance of QAOA+ for solving CBO-UVPs. Therefore, it is crucial to develop a suitable ansatz for CBO-UVPs. In this paper, the Variational Quantum Algorithm-Preserving Feasible Space (VQA-PFS) ansatz is proposed, exemplified by the Uncapacitated Facility Location Problem (UFLP), that applies mixed operators on constrained variables while employing Hardware-Efficient Ansatz (HEA) on unconstrained variables. The numerical results demonstrate that VQA-PFS significantly enhances the probability of success and exhibits faster convergence than QAOA+, Quantum Approximation Optimization Algorithm (QAOA), and HEA. Furthermore, VQA-PFS reduces the circuit depth dramatically compared to QAOA+ and QAOA. The algorithm is general and instructive in tackling CBO-UVPs.