The Quantum Approximate Optimization Algorithm (QAOA) is a well-known hybrid quantum-classical algorithm for combinatorial optimization problems. Improving QAOA involves enhancing its approximation ratio while addressing practical constraints of Noisy Intermediate Scale Quantum (NISQ) devices, such as minimizing the number of two-qubit gates and reducing circuit depth. Although existing research has optimized designs for phase and mixer operators to improve performance, challenges remain, particularly concerning the excessive use of two-qubit gates in the construction of phase operators. To address these issues, we introduce a Dynamic Adaptive Phase Operator (DAPO) algorithm, which dynamically constructs phase operators based on the output of previous layers and neighborhood search approach, optimizing the problem Hamiltonian more efficiently. By using solutions generated by QAOA itself to simplify the problem Hamiltonian at each layer, the algorithm captures the problem's structural properties more effectively, progressively steering the solution closer to the optimal target. Experimental results on MaxCut and NAE3SAT problems show that DAPO achieves higher approximation ratios and significantly reduces two-qubit RZZ gates, especially in dense graphs. Compared to vanilla QAOA, DAPO uses only 66% of RZZ gates at the same depth while delivering better results, demonstrating its potential for efficient combinatorial optimization in the NISQ era.
The Quantum Approximate Optimization Algorithm (QAOA) is a well-known hybrid quantum-classical algorithm for combinatorial optimization problems. Improving QAOA involves enhancing its approximation ratio while addressing practical constraints of Noisy Intermediate Scale Quantum (NISQ) devices, such as minimizing the number of two-qubit gates and reducing circuit depth. Although existing research has optimized designs for phase and mixer operators to improve performance, challenges remain, particularly concerning the excessive use of two-qubit gates in the construction of phase operators. To address these issues, we introduce a Dynamic Adaptive Phase Operator (DAPO) algorithm, which dynamically constructs phase operators based on the output of previous layers and neighborhood search approach, optimizing the problem Hamiltonian more efficiently. By using solutions generated by QAOA itself to simplify the problem Hamiltonian at each layer, the algorithm captures the problem's structural properties more effectively, progressively steering the solution closer to the optimal target. Experimental results on MaxCut and NAE3SAT problems show that DAPO achieves higher approximation ratios and significantly reduces two-qubit RZZ gates, especially in dense graphs. Compared to vanilla QAOA, DAPO uses only 66% of RZZ gates at the same depth while delivering better results, demonstrating its potential for efficient combinatorial optimization in the NISQ era.
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.
Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA-QAOA+. Unlike QAOA+, AMA-QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA-QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA-QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by 5.30% on ER random graphs and 5.41% on 3-regular graphs. Moreover, AMA-QAOA+ significantly reduces the CNOT gate consumption, requiring only 15.30% and 25.18% of the CNOT gates used by QAOA+ on ER and 3-regular random graphs, respectively. These results demonstrate that AMA-QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource-efficient quantum circuits.
The quantum approximate optimization algorithm (QAOA) is a promising quantum algorithm that can be used to approximately solve combinatorial optimization problems. The usual QAOA ansatz consists of an alternating application of the cost and mixer Hamiltonians. In this work, we study how using Hamiltonians other than the usual cost Hamiltonian, dubbed custom phase operators, can affect the performance of the QAOA. We derive an expected value formula for the QAOA with custom phase operators at p=1 and show numerically that some of these custom phase operators can achieve higher approximation ratios than the original algorithm implementation. Out of all the graphs tested at p=1, 0.036% of the random custom phase operators, 75.9% of the subgraph custom phase operators, 95.1% of the triangle-removed custom phase operators, and 93.9% of the maximal degree edge-removed custom phase operators have a higher approximation ratio than the original QAOA implementation. Furthermore, we numerically simulate these phase operators for p=2 and p=3 levels of QAOA and find that there exist a large number of subgraph, triangle-removed, and maximal degree edge-removed custom phase operators that have a higher approximation ratio than the QAOA at the same depth. These findings open up the question of whether better phase operators can be designed to further improve the performance of the QAOA.
ABSTRACT Recently, Hadfield et al. proposed the quantum alternating operator ansatz algorithm (QAOA+), an extension of the quantum approximate optimization algorithm (QAOA), to solve constrained combinatorial optimization problems (CCOPs). Compared with QAOA, QAOA+ enables the search for optimal solutions within a feasible solution space by encoding problem constraints into the mixer Hamiltonian, thereby reducing the search space and eliminating the possibility of yielding infeasible solutions. However, QAOA+ may incur high overall gate costs when the mixer is applied to all qubits in each layer, and each mixer is costly to implement. To address this challenge, an adaptive mixer allocation strategy is tailored for QAOA+. The resulting algorithm, which integrates this strategy into the original QAOA+ framework, is referred to as AMA‐QAOA+. Unlike QAOA+, AMA‐QAOA+ adaptively applies the mixer to a subset of qubits in each layer of the mixer unitary operator based on an evaluation function. The performance of AMA‐QAOA+ is evaluated on the maximum independent set problem. Numerical simulation results show that, under the same number of optimization runs, AMA‐QAOA+ achieves better solution quality than QAOA+, with the optimal approximation ratio improved by on Erdős–Rényi random graphs and on 3‐regular graphs. Moreover, AMA‐QAOA+ significantly reduces the CNOT gate consumption, requiring only and of the CNOT gates used by QAOA+ on Erdős–Rényi and 3‐regular random graphs, respectively. These results demonstrate that AMA‐QAOA+ enhances solution quality and computational efficiency, enabling the design of more compact and resource‐efficient quantum circuits.
The quantum approximate optimization algorithm (QAOA) is an appealing proposal to solve NP problems on noisy intermediate-scale quantum (NISQ) hardware. Making NISQ implementations of the QAOA resilient to noise requires short ansatz circuits with as few controlled- () gates as possible. Here we present the dynamic adaptive quantum approximate optimization algorithm (Dynamic-ADAPT-QAOA). Our algorithm significantly reduces the circuit depth and the count of standard ADAPT-QAOA, a leading proposal for near-term implementations of the QAOA. Throughout our algorithm, the decision to apply -intensive operations is made dynamically, based on algorithmic benefits. Using density-matrix simulations, we benchmark the noise resilience of ADAPT-QAOA and Dynamic-ADAPT-QAOA. We compute the gate-error probability p gate ★ below which these algorithms provide, on average, more accurate solutions than the classical, polynomial-time approximation algorithm by Goemans and Williamson. For small systems with six to ten qubits, we show that p gate ★ > 10 − 3 for Dynamic-ADAPT-QAOA. Compared to standard ADAPT-QAOA, this constitutes an order-of-magnitude improvement in noise resilience. This improvement should make Dynamic-ADAPT-QAOA viable for implementations on superconducting NISQ hardware, even in the absence of error mitigation. Published by the American Physical Society 2024
The solution of large-scale combinatorial optimization problems, which are exorbitantly complex and resource-consuming, is one of the most significant challenges that computing systems face. Traditional algorithms often provide suboptimal results when confronted with bigger issues. This research aims to bring attention to these problems by introducing the Extending Quantum Approximate Optimization Algorithm (EQAOA) framework, which improves scalability, parameter optimization, and error resilience. Fundamental to the EQAOA methodology is repeatedly optimizing solution states via adaptive-depth, parameter-tuned quantum circuits. EQAOA also employs the method of converting combinatorial problems into cost Hamiltonians. Similar problems are shown in the Max-Cut and Graph Coloring programs. The experiments were conducted on several benchmark circumstances using real NISQ hardware and quantum simulators. The enhanced EQAOA outperforms baseline versions in terms of convergence time and solution quality. The efficient integration of dynamic circuit depth with hybrid classical-quantum parameter approaches applied to increasingly complicated problem scenarios and yielded good results. According to the study's findings, applying the proposed QAOA extension to practical combinatorial contexts yields a quantum advantage. The future of this field is bright, with researchers aiming to improve its interaction with traditional optimizers and extend its usage to other practical optimization areas.