The exact cover problem is an NP-complete problem with broad applications. Studies show that although applying the Quantum Approximate Optimization Algorithm (QAOA) to this problem can yield improved solution quality with deeper circuit depth, it can limit the algorithm's applicability on noisy intermediate-scale quantum devices. To improve solution quality at shallow depth, we propose a Quantum-Assisted Recursive Algorithm (QARA) for solving the exact cover problem. QARA addresses the problem by alternately applying classical and quantum pruning. Classical pruning is a repeatable pre-processing step to simplify the problem. When the classical pruning cannot promote the problem simplification, quantum pruning is invoked. During quantum pruning, QARA extracts information from the QAOA's output state to identify the subset with the strongest selection bias. This subset is then used to prune the problem based on our problem-tailored reduction rules. Furthermore, QARA incorporates a local verification and rollback mechanism to assistively judge the effectiveness of the quantum simplification. After quantum pruning, classical pruning is applied again to the reduced problem if the remaining subsets and element set are not null. This alternating process repeats until the original problem is fully resolved. In our numerical simulations, we evaluate the performance of QARA at one-layer depth on 140 instances with subset sizes ranging from 8 to 20. Numerical results show that the probability of QARA in finding an exact solution is approximately 60\% higher than that of both QAOA and Recursive QAOA, highlighting its efficiency.
The exact cover problem is an NP-complete problem with broad applications. Studies show that although applying the Quantum Approximate Optimization Algorithm (QAOA) to this problem can yield improved solution quality with deeper circuit depth, it can limit the algorithm's applicability on noisy intermediate-scale quantum devices. To improve solution quality at shallow depth, we propose a Quantum-Assisted Recursive Algorithm (QARA) for solving the exact cover problem. QARA addresses the problem by alternately applying classical and quantum pruning. Classical pruning is a repeatable pre-processing step to simplify the problem. When the classical pruning cannot promote the problem simplification, quantum pruning is invoked. During quantum pruning, QARA extracts information from the QAOA's output state to identify the subset with the strongest selection bias. This subset is then used to prune the problem based on our problem-tailored reduction rules. Furthermore, QARA incorporates a local verification and rollback mechanism to assistively judge the effectiveness of the quantum simplification. After quantum pruning, classical pruning is applied again to the reduced problem if the remaining subsets and element set are not null. This alternating process repeats until the original problem is fully resolved. In our numerical simulations, we evaluate the performance of QARA at one-layer depth on 140 instances with subset sizes ranging from 8 to 20. Numerical results show that the probability of QARA in finding an exact solution is approximately 60% higher than that of both QAOA and Recursive QAOA, highlighting its efficiency.
In recent years, several quantum algorithms have been proposed for addressing combinatorial optimization problems. Among them, the Quantum Approximate Optimization Algorithm (QAOA) has become a widely used approach. However, reported limitations of QAOA have motivated the development of multiple algorithmic variants, including recursive hybrid methods such as the Recursive Quantum Approximate Optimization Algorithm (RQAOA), as well as the Quantum-Informed Recursive Optimization (QIRO) framework. In this work, we integrate the Quantum Alternating Operator Ansatz within the QIRO framework in order to improve its quantum inference stage. Both the original and the enhanced versions of QIRO are applied to the Minimum Vertex Cover problem, an NP-complete problem of practical relevance. Performance is evaluated on a benchmark of Erdös-Rényi graph instances with varying sizes, densities, and random seeds. The results show that the proposed modification leads to a higher number of successfully solved instances across the considered benchmark, indicating that refinements of the variational layer can improve the effectiveness of the QIRO framework.
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.
Variational quantum algorithms are believed to be promising for solving computationally hard problems and are often comprised of repeated layers of quantum gates. An example thereof is the quantum approximate optimization algorithm (QAOA), an approach to solve combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) systems. Gaining computational power from QAOA critically relies on the mitigation of errors during the execution of the algorithm, which for coherence-limited operations is achievable by reducing the gate count. Here, we demonstrate an improvement of up to a factor of 3 in algorithmic performance as measured by the success probability, by implementing a continuous hardware-efficient gate set using superconducting quantum circuits. This gate set allows us to perform the phase separation step in QAOA with a single physical gate for each pair of qubits instead of decomposing it into two CZ-gates and single-qubit gates. With this reduced number of physical gates, which scales with the number of layers employed in the algorithm, we experimentally investigate the circuit-depth-dependent performance of QAOA applied to exact-cover problem instances mapped onto three and seven qubits, using up to a total of 399 operations and up to 9 layers. Our results demonstrate that the use of continuous gate sets may be a key component in extending the impact of near-term quantum computers.
The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising approach for solving NP hard combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) hardware. However, its performance is critically dependent on the selection of the circuit depth a parameter that must be specified a priori without clear guidance. In this paper, we introduce a variant of QAOA called dynamic depth Quantum Approximate Optimization Algorithm (DDQAOA) that resolves the challenge of pre selecting a fixed circuit depth. Our method adaptively expands circuit depth, starting from p = 1 and progressing up to p = 10, by transferring learned parameters to deeper circuits based on convergence criteria. We tested this approach on 100 instances of the Constrained Shortest Path Problem (CSPP) at 10 qubit and 16 qubit scales. Our DDQAOA achieved superior approximation ratios and success probabilities with fewer CNOT gate evaluations than the standard QAOA for p = 3, 5, 10, and 15. In particular, while standard QAOA at p = 15 achieved results close to our approach, it used 217% and 159.3% more CNOT gates for 10 qubit and 16 qubit instances, respectively. This demonstrates the performance and practical applicability of DDQAOA to solve combinatorial optimization problems on near term devices.
The max-cut problem (MCP) is a classic problem in the field of combinatorial optimization and has important applications in various fields, including statistical physics and image processing. However, except for some special cases, the MCP still encounters a non-deterministic polynomial complete problem (an NP-complete problem), and there is currently no known efficient classical algorithm that can solve it in polynomial time. The quantum approximate optimization algorithm (QAOA), as a pivotal algorithm in the noisy intermediate-scale quantum (NISQ) computing era, has shown significant potential for solving the MCP. However, due to the lack of quantum error correction, the reliability of computations in NISQ systems sharply declines as the circuit depth of the algorithm increases. Therefore, designing an efficient, shallow-depth, and low-complexity QAOA for the MCP is a critical challenge in demonstrating the advantages of quantum computing in the NISQ era.In this paper, according to the standard QAOA algorithm, we introduce Pauli Y rotation gates into the target Hamiltonian circuit for the MCP. By enhancing the flexibility of quantum trial functions and improving the efficiency of Hilbert space exploration within a single iteration, we significantly improve the performance of QAOA on the MCP.We conduct extensive numerical simulations using the MindSpore quantum platform, and compare the proposed RY-layer-assisted QAOA with standard QAOA and its existing variants, including MA-QAOA and QAOA+. The experiments are performed on various graph types, including complete graphs, 3-regular graphs, 4-regular graphs, and random graphs with edge probabilities between 0.3 and 0.5. Our results show that the RY-layer-assisted QAOA achieves higher approximation ratios in all graph types, particularly in regular and random graphs, where traditional QAOA variants are difficult to implement. Moreover, the proposed method exhibits strong robustness as the graph size increases, and can maintain high performance even for larger graphs. Importantly, the RY-layer-assisted QAOA requires fewer CNOT gates and has a lower circuit depth than the standard QAOA and its variants, making it more suitable for NISQ devices with limited coherence times and high error rates.In conclusion, the RY-layer-assisted QAOA provides a promising approach for solving MCP in the NISQ era. By improving the approximation ratio while reducing circuit complexity, this method demonstrates significant potential for practical quantum computing applications, thus paving the way for developing more efficient and reliable quantum optimization algorithms.
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.