Modified Recursive QAOA for Exact Max-Cut Solutions on Bipartite Graphs: Closing the Gap Beyond QAOA Limit

TL;DR

This paper analyzes the limitations of low-level QAOA for MAX-CUT on bipartite graphs, proves performance bounds, and introduces a modified recursive QAOA that guarantees exact solutions for bipartite and certain weighted graphs.

Contribution

It analytically establishes performance bounds for level-1 QAOA and proposes a modified recursive QAOA that always finds exact MAX-CUT solutions on bipartite graphs.

Findings

Level-1 QAOA has an upper bound on approximation ratio for bipartite graphs.

Recursive QAOA outperforms level-1 QAOA but has scalability limitations.

Modified RQAOA guarantees exact MAX-CUT solutions for bipartite and parity-signed weighted graphs.

Abstract

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 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. To this end, we derive an upper bound for the approximation ratio based on the average degree of bipartite graphs. Second, we demonstrate that Recursive QAOA (RQAOA), which recursively reduces graph size using QAOA as a subroutine, outperforms the level-1 QAOA. However, the performance of RQAOA exhibits limitations as the graph size increases. Finally, we show that RQAOA with a restricted parameter regime can fully address these limitations. Surprisingly, this modified RQAOA always finds the exact maximum cut for any bipartite graphs and even for a more general graph with parity-signed weights.