Improved Recursive QAOA for Solving MAX-CUT on Bipartite Graphs

Quantum Approximate Optimization Algorithm (QAOA) is a quantum-classical hybrid algorithm proposed with the goal of approximately solving combinatorial optimization problems such as the MAX-CUT problem. It has been considered a potential candidate for achieving quantum advantage in the Noisy Intermediate-Scale Quantum (NISQ) era and has been extensively studied. However, the performance limitations of low-level QAOA have also been demonstrated across various instances. In this work, we first analytically prove the performance limitations of level-1 QAOA in solving the MAX-CUT problem on bipartite graphs. We derive an upper bound for the approximation ratio based on the average degree of bipartite graphs. Second, we show through numerical results that solving the same problem using level-1 Recursive QAOA (RQAOA), which is one of the variants of QAOA that uses QAOA as a subroutine to reduce the graph size recursively, outperforms the original QAOA but still exhibits limitations as the graph size increases. To address these limitations, we propose a modified RQAOA that reduces the parameter regime optimized in the QAOA subroutine. While reducing the optimization space could generally make it more challenging to find the true optimal parameters, interestingly, we prove that this modified RQAOA can find the exact maximum cut for a parity partitioned graph which includes all weighted bipartite graphs.