Progressive Quantum Algorithm for Maximum Independent Set with Quantum Alternating Operator Ansatz

Hadfield et al. proposed a novel Quantum Alternating Operator Ansatz algorithm (QAOA+), and this algorithm has wide applications in solving constrained combinatorial optimization problems (CCOPs) because of the advantages of QAOA+ ansatz in constructing a feasible solution space. In this paper, we propose a Progressive Quantum Algorithm (PQA) with QAOA+ ansatz to solve the Maximum Independent Set (MIS) problem using fewer qubits. The core idea of PQA is to construct a subgraph that is likely to contain the MIS solution of the target graph and then solve the MIS problem on this subgraph to obtain an approximate solution. To construct such a subgraph, PQA starts with a small-scale initial subgraph and progressively expands its graph size utilizing heuristic expansion strategies. After each expansion, PQA solves the MIS problem on the newly generated subgraph. In each run, PQA repeats the expansion and solving process until a predefined stopping condition is reached. Simulation results demonstrate that to achieve an approximation ratio of 0.95, PQA requires only $5.565\%$ ($2.170\%$) of the qubits and $17.59\%$ ($6.430\%$) of the runtime compared with directly solving the original problem using QAOA+ on Erd\H{o}s-R\'enyi (3-regular) graphs, highlighting the efficiency of PQA.