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

TL;DR

The paper analyzes MAX-CUT on bipartite graphs under quantum variational algorithms, proving a density-based upper bound for level-1 QAOA and showing that Recursive QAOA (RQAOA) generally outperforms QAOA but may fail to be exact on larger graphs. It introduces Modified RQAOA with a restricted QAOA parameter domain, and proves that this approach can exactly solve MAX-CUT on graphs with parity-signed weights, supported by numerical evidence that standard RQAOA improves over QAOA yet can be suboptimal. The key contribution is a cost-preserving reduction strategy combined with restricted-parameter optimization that enables exact solutions on a broad class of structured graphs, suggesting a practical pathway for NISQ-era optimization on graph problems. The results illuminate how preserving graph structure during recursion and constraining parameter searches can unlock exact quantum-assisted optimization for targeted instances.

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.

Figures (3)

• Figure 1: Plots illustrating the tightness of the bounds in Theorem \ref{['thm:bipar']} and Corollary \ref{['cor']}. In (a) and (b), bipartite graphs with two disjoint sets of $n,m$ vertices. Here, $n,m$ were randomly selected from $5$ to $51$ for $p=0.6$ and $p=0.9$, respectively, where $p$ represents the edge probability. The level-$1$ QAOA approximation ratio $\alpha_1$ was compared with two upper bounds provided in Theorem \ref{['thm:bipar']}. Compared to (a), the graph instances in (b) are denser, and the bounds are tighter. In (c) and (d), the level-$1$ QAOA approximation ratio $\alpha_1$ is compared with the bound provided in Corollary \ref{['cor']} for complete bipartite graphs $K_{n,n}$ and $K_{n,m}$, respectively. In this case, it is confirmed that the bounds are tight. The points plotted below the boundary appear to have achieved relatively low approximation ratios compared to the upper bound due to numerical optimization issues.
• Figure 2: Benchmark results comparing the level-1 QAOA, RQAOA, and RQAOA$^*$ (our modified RQAOA introduced in Algorithm \ref{['alg:modified_rqaoa']}) on weighted bipartite graphs $G_{64,64}^w$ and $G_{128,128}^w$ for various probabilities $p$ ranging from 0.1 to 1.0 in increments of 0.1. Here, $p$ represents the edge probability. Edge weights were sampled from a Gaussian distribution with a mean of 50 and a variance of 25, subsequently converted to integers. Each benchmark was performed on 10 random instances per $p$. The QAOA algorithm was optimized through a line search on $\gamma \in [0,\pi]$ with 20 samples, followed by gradient descent, with $\beta^*=\pi/8$. The RQAOA was optimized using a coarse grid search with $20 \times 20$ points, followed by gradient descent on the best-found point. The results indicate that RQAOA$^*$ consistently achieves an approximation ratio of 1 across all 200 instances, while RQAOA and QAOA typically yield an approximation ratio of approximately 0.8 and 0.55, respectively, for both graph sizes.
• Figure 3: A schematic diagram showing how the MAX-CUT cost function on $K^w_{4,3}$ and the weight structure of the graph change after one iteration of our modified RQAOA for the case when we identify (a) two connected vertices in the same partition with the same sign and (b) in the different partitions with different signs; The red/blue edges indicate the edges with the positive/negative weights, respectively. The bold lines represent the edges whose weights changed after identifying two vertices. After one iteration, the reduced graph remains to have the same weight structure as shown in the right graph for both cases.

Theorems & Definitions (11)

• Theorem 3: Bipartite graph
• proof
• Corollary 4: Complete bipartite graph
• proof
• Definition 5
• Remark 6
• Theorem 8
• Lemma 9
• Lemma 10
• proof : Proof of Theorem \ref{['thm:main']}