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Acta Physica Sinica

Design of quantum approximation optimization algorithm for the maximum cut problem

Yun-Jiang Wang, XI Huiming, XIAO Zhuoyan, Wang Zengbin, Sha SHI

Published Jan 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.7498/aps.74.20241223

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•Jan 1, 2026•0 citations
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.
Maximum cut (Max-Cut) problem is one of the most important combinatorial optimization problems because of its various applications in real life, and recently Quantum Approximate Optimization Algorithm (QAOA) has been widely employed to solve it. However, as the size of the problem increases, the number of qubits required will become larger. With the aim of saving qubits, we propose a coupling framework for designing QAOA circuits to solve larger-scale Max-Cut problem. This framework relies on a classical algorithm that approximately solves a certain variant of Max-Cut, and we derive an approximation guarantee theoretically, assuming the approximation ratio of the classical algorithm and QAOA. Furthermore we design a heuristic approach that fits in our framework and perform sufficient numerical experiments, where we solve Max-Cut on various 242424-vertex Erdős-Rényi graphs. Our framework only consumes 181818 qubits and achieves 0.99500.99500.9950 approximation ratio on average, which outperforms the previous methods showing 0.97780.97780.9778 (quantum algorithm using the same number of qubits) and 0.96430.96430.9643 (classical algorithm). The experimental results indicate our well-designed quantum-classical coupling framework gives satisfactory approximation ratio while reduces the qubit cost, which sheds light on more potential computing power of NISQ devices.
To address the limited expressive capability of the standard quantum approximate optimization algorithm (QAOA) in shallow circuits and the increased noise sensitivity caused by deeper quantum circuits, an improved QAOA, termed MA-RY-QAOA, was proposed by integrating parameter multi-angleization and an RY-assisted structure. In the proposed algorithm, edge-parameter multi-angleization and RY expansion were incorporated into the problem layer, while qubit-parameter multi-angleization was introduced into the mixing layer. Simulation experiments were conducted on the Max-cut problem of random graphs, complete graphs, 2-regular graphs, and 4-regular graphs with circuit depth ( p=1 <?fx-imagestate width="7.45066643" height="2.87866688"?> <?fx-imagestate width="7.45066643" height="2.87866688"?> ) and node sizes ranging from 4 to 12. The average approximation ratio was adopted as the evaluation metric. The results showed that the average approximation ratios achieved by MA-RY-QAOA and MA-QAOA were generally superior to those obtained by the standard QAOA, RY-QAOA, and QAOA+ algorithms. In particular, MA-RY-QAOA exhibited better performance on 2-regular and 4-regular graphs and achieved higher approximation ratios for certain scales of random graphs. These findings indicate that MA-RY-QAOA improves optimization performance while maintaining a shallow circuit structure and provides a useful reference for the design of QAOA variants on noisy intermediate-scale quantum (NISQ) devices.
Abstract In the noisy intermediate-scale quantum (NISQ) era, quantum approximate optimisation algorithms are important tools for solving NP-hard problems such as combinatorial optimisation. However, most QAOA implementations ignore the inherent noise in quantum circuit environments. This study proposes a noisy quantum approximate optimisation algorithm (NQAOA) and analyses its performance under noisy conditions through numerical simulations. We compare the approximate ratios (AR) of various graphs under different optimisation strategies and noise conditions, and assess the impact of circuit depth, node count, edge count, and noise coefficient on the maximum cut (MaxCut). Experimental results show that noise leads to overfitting, and the approximate ratio ( AR ) exhibits fluctuating changes as the edge count e increases. The AR values under ideal conditions and the expected results are both superior to those under noisy conditions. This study is the first to systematically integrate multiple types of quantum noise into the QAOA framework, quantitatively analysing the synergistic effects of noise type, optimiser performance, and graph structure. Through theoretical derivations and numerical simulations, we established linear/exponential theoretical boundaries for AR decay related to noise coefficients, providing a critical reference for designing noise-resistant quantum optimisation algorithms on NISQ devices.
A novel quantum algorithm named the Quantum Approximate Shift-Phase Algorithm (QASPA) is proposed for the approximate solution of the Max-Cut problem. The Max-Cut problem consists in partitioning the set of vertices into two subsets in such a way that the total weight of the edges connecting vertices from different subsets is maximized. This problem is known to be NP-complete and represents a combinatorial challenge. Classical solution algorithms require excessive computational time, rendering them inefficient for medium and large graphs. Approximate solution methods offer guaranteed approximation ratios but still face significant limitations in terms of accuracy and performance on larger graphs. The proposed algorithm features a simple scheme and a minimal number of optimized parameters. The input graph is represented by an adjacency matrix, after which the edge weights are linearly normalized to fixed phase angles. The quantum circuit begins by applying a Hadamard transform to each qubit, thereby creating an equal superposition of all possible partitions. Subsequently, for each pair of adjacent vertices, a sequence comprising a controlled NOT gate, a single-qubit rotation around the Y-axis by an angle proportional to the edge weight, and a second controlled NOT gate is applied. This encodes the phase information about the edge weights into the quantum state of the system. After circuit execution, the qubits are measured in the standard computational basis, and the resulting probability distribution allows the selection of the most probable partition as an approximate solution to the problem. Experimental studies conducted on quantum processor simulators have demonstrated that the proposed algorithm achieves accuracy comparable to the Variational Quantum Approximate Optimization Algorithm (QAOA) while significantly reducing computation time due to the absence of iterative parameter optimization. Moreover, as the graph size increases, the algorithm's runtime grows considerably slower compared to classical brute-force approaches and QAOA, confirming its potential for solving medium-sized Max-Cut problems.
Quantum computing has emerged as a promising paradigm to tackle computationally intensive problems that classical computers struggle with. In this study, we explore the application of the Edge-based quantum approximate optimization algorithm (QAOA) to the MAX-CUT problem, a well-known combinatorial optimization challenge. MAX-CUT aims to partition the vertices of a graph into two subsets such that the number of edges between the subsets is maximized. We define the edge-based MAX-CUT problem and propose a method for applying QAOA specifically tailored to this formulation. We conduct simulations using IBM’s Qiskit framework, examining both vertex-based and edge-based QAOA implementations across various graph structures. Our results highlight the comparative performance of these approaches in terms of solution quality and computational efficiency. Specifically, we analyze the impact of different graph sizes and edge densities on the complexity and CNOT gate counts of the proposed algorithms. This analysis provides insights into leveraging quantum computing for combinatorial optimization tasks, particularly focusing on the implications for practical applications and future research directions.
The standard quantum approximate optimization algorithm (QAOA) has a limited expressive power in shallow circuits, whereas naively increasing the number of layers leads to greater quantum circuit depth and higher noise sensitivity. To address this trade-off, based on the ideas of the multi-angle quantum approximate optimization algorithm (MA-QAOA) and the RY-layer-assisted quantum approximate optimization algorithm (RY-QAOA), this paper proposes an improved algorithm, i.e., MA-RY-QAOA, which integrates parameter multangularization and RY-assisted architecture. In MA-RY-QAOA, the problem block incorporates edge parameter multangularization together with RY extension, while the mixer block implements qubit parameter multangularization. For layer number p=1 and node size n=4~12 , we conduct 50 simulation trials on maximum cut problem respectively for random graph with edge probability of 0.8, complete graph, 2-regular graph and 4-regular graph, by using the average approximation ratio as the evaluation metric. The simulation trial results show that, across these four class graphs, the average approximation ratio of MA-RY-QAOA and MA-QAOA significantly outperforms standard QAOA, RY-QAOA, and QAOA+ for overwhelming majority of node sizes; with respect to complete graph, the average approximation ratio of MA-QAOA outperforms that of MA-RY-QAOA for all node sizes; with regard to 2-regular graph, the average approximation ratio of MA-RY-QAOA exceeds that of MA-QAOA for all node sizes; with regard to 4-regular graph, the average approximation ratio of MA-RY-QAOA takes advantage over that of MA-QAOA for overwhelming majority of node sizes; concerning random graph, the average approximation ratio of MA-RY-QAOA is superior to that of MA-QAOA for both small node sizes and large node sizes, but is inferior to that of MA-QAOA for medium node sizes. Therefore, MA-RY-QAOA provides a useful reference for the structure design of shallow-layer QAOA on noisy intermediate-scale quantum (NISQ) devices.