Recent results on the Quantum Approximate Optimization Algorithm (QAOA) have cast pessimism on its potential to exhibit practical quantum speedups. For instance, QAOA’s locality limits its performance on tasks such as coloring bipartite graphs—which is easy for classical methods. Motivated by these limitations, the Recursive QAOA was introduced to overcome the locality and symmetry of QAOA. Despite being more powerful than QAOA, RQAOA is fully classically simulable at level-1 depth (p = 1). We report results on RQAOA in this classically simulable regime, benchmarked on random Quantum Unconstrainted Binary Optimization (QUBO) problems with up to 100 variables. We find that RQAOA generally matches the performance of classical simulated annealing and significantly outperforms ordinary QAOA.
Recent results on the Quantum Approximate Optimization Algorithm (QAOA) have cast pessimism on its potential to exhibit practical quantum speedups. For instance, QAOA’s locality limits its performance on tasks such as coloring bipartite graphs—which is easy for classical methods. Motivated by these limitations, the Recursive QAOA was introduced to overcome the locality and symmetry of QAOA. Despite being more powerful than QAOA, RQAOA is fully classically simulable at level-1 depth (p = 1). We report results on RQAOA in this classically simulable regime, benchmarked on random Quantum Unconstrainted Binary Optimization (QUBO) problems with up to 100 variables. We find that RQAOA generally matches the performance of classical simulated annealing and significantly outperforms ordinary QAOA.
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 the standard 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 standard 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.
Among quantum algorithms, the quantum approximation optimization algorithm (QAOA) is an algorithm that finds approximate solutions. Recursive QAOA is proposed to overcome the limitation that QAOA has lower performance than Goemans-Williamson algorithm in the MAX-CUT problem [Physical Review Letter, 125, 260504, 2020]. In this paper, we introduce a method with lower complexity than RQAOA and compare its numerical results in the MAX-CUT problem. At the same level, the performance of the proposed method is lower than that of RQAOA, but the number of computations of quantum and classical parts is very small.
We show how to apply the recursive quantum approximate optimization algorithm (RQAOA) to MAX- k -CUT, the problem of finding an approximate k -vertex coloring of a graph. We compare this proposal to the best known classical and hybrid classical-quantum algorithms. First, we show that the standard (non-recursive) QAOA fails to solve this optimization problem for most regular bipartite graphs at any constant level p : the approximation ratio achieved by QAOA is hardly better than assigning colors to vertices at random. Second, we construct an efficient classical simulation algorithm which simulates level- 1 QAOA and level- 1 RQAOA for arbitrary graphs. In particular, these hybrid algorithms give rise to efficient classical algorithms, and no benefit arising from the use of quantum mechanics is to be expected. Nevertheless, they provide a suitable testbed for assessing the potential benefit of hybrid algorithm: We use the simulation algorithm to perform large-scale simulation of level- 1 QAOA and RQAOA with up to 300 qutrits applied to ensembles of randomly generated 3 -colorable constant-degree graphs. We find that level- 1 RQAOA is surprisingly competitive: for the ensembles considered, its approximation ratios are often higher than those achieved by the best known generic classical algorithm based on rounding an SDP relaxation. This suggests the intriguing possibility that higher-level RQAOA may be a potentially useful algorithm for NISQ devices.
This study presents a rigorous empirical evaluation of the Quantum Approximate Optimization Algorithm (QAOA) within a hybrid quantum-classical framework for solving the NP-hard Max-Cut problem. Using Python and Qiskit Aer statevector simulations, we benchmark QAOA (depths p = 2 and p = 3) against established classical algorithms, including simulated annealing, the Goemans–Williamson algorithm, and greedy heuristics, across Erdős–Rényi and regular graphs with 4 to 20 nodes (100 instances per size). Results demonstrate that low-depth QAOA suffers from significant performance degradation, achieving a 0.467 approximation ratio at 20 nodes compared to 0.856 for Goemans–Williamson, alongside a 210× runtime overhead. Convergence analysis further reveals low optimality probabilities and high iteration costs, highlighting limitations of variational optimization in NISQ-era devices. This work provides a reproducible benchmarking framework and contributes to a realistic assessment of near-term quantum advantage.
This study presents a rigorous empirical evaluation of the Quantum Approximate Optimization Algorithm (QAOA) within a hybrid quantum-classical framework for solving the NP-hard Max-Cut problem. Using Python and Qiskit Aer statevector simulations, we benchmark QAOA (depths p = 2 and p = 3) against established classical algorithms, including simulated annealing, the Goemans–Williamson algorithm, and greedy heuristics, across Erdős–Rényi and regular graphs with 4 to 20 nodes (100 instances per size). Results demonstrate that low-depth QAOA suffers from significant performance degradation, achieving a 0.467 approximation ratio at 20 nodes compared to 0.856 for Goemans–Williamson, alongside a 210× runtime overhead. Convergence analysis further reveals low optimality probabilities and high iteration costs, highlighting limitations of variational optimization in NISQ-era devices. This work provides a reproducible benchmarking framework and contributes to a realistic assessment of near-term quantum advantage.
In recent years, variational quantum algorithms such as the Quantum Approximation Optimization Algorithm (QAOA) have gained popularity as they provide the hope of using NISQ devices to tackle hard combinatorial optimization problems. It is, however, known that at low depth, certain locality constraints of QAOA limit its performance. To go beyond these limitations, a non-local variant of QAOA, namely recursive QAOA (RQAOA), was proposed to improve the quality of approximate solutions. The RQAOA has been studied comparatively less than QAOA, and it is less understood, for instance, for what family of instances it may fail to provide high-quality solutions. However, as we are tackling NP-hard problems (specifically, the Ising spin model), it is expected that RQAOA does fail, raising the question of designing even better quantum algorithms for combinatorial optimization. In this spirit, we identify and analyze cases where (depth-1) RQAOA fails and, based on this, propose a reinforcement learning enhanced RQAOA variant (RL-RQAOA) that improves upon RQAOA. We show that the performance of RL-RQAOA improves over RQAOA: RL-RQAOA is strictly better on these identified instances where RQAOA underperforms and is similarly performing on instances where RQAOA is near-optimal. Our work exemplifies the potentially beneficial synergy between reinforcement learning and quantum (inspired) optimization in the design of new, even better heuristics for complex problems.