Abstract Graph cut, a combinatorial optimization, is explored in quantum computing, seeking optimal bi-partitions of a directed or undirected, weighted or unweighted graph. The cut models, such as max-cut and min-cut, are applied in domains including VLSI design, circuit partitioning, data clustering, and network analysis. This paper explores the Ising modeling of max-cut and min-cut in a Laplacian-based QAOA, including benchmarking, spectral analysis, and a detailed analysis of the results. It aims to discover how Ising cut modeling and QAOA perform on a structured graph class, such as a weighted chordal graph. Chordal graphs offer decomposition properties favorable to QUBO and Ising models. Experimental results show that the proposed QAOA framework yields optimal or near-optimal cut values with high and stable approximation ratios across circuit depths. Also, hardware-aware considerations are introduced through superconducting qubit layout design and eigenmode simulations in Qiskit Metal and Ansys EPR HFSS. Further analysis reveals challenges related to convergence and performance in denser topologies. The study gives the modeling and the comprehensive evaluation of QAOA for graph-cuts, including a conceptual hardware-aware analysis.
Abstract Graph cut, a combinatorial optimization, is explored in quantum computing, seeking optimal bi-partitions of a directed or undirected, weighted or unweighted graph. The cut models, such as max-cut and min-cut, are applied in domains including VLSI design, circuit partitioning, data clustering, and network analysis. This paper explores the Ising modeling of max-cut and min-cut in a Laplacian-based QAOA, including benchmarking, spectral analysis, and a detailed analysis of the results. It aims to discover how Ising cut modeling and QAOA perform on a structured graph class, such as a weighted chordal graph. Chordal graphs offer decomposition properties favorable to QUBO and Ising models. Experimental results show that the proposed QAOA framework yields optimal or near-optimal cut values with high and stable approximation ratios across circuit depths. Also, hardware-aware considerations are introduced through superconducting qubit layout design and eigenmode simulations in Qiskit Metal and Ansys EPR HFSS. Further analysis reveals challenges related to convergence and performance in denser topologies. The study gives the modeling and the comprehensive evaluation of QAOA for graph-cuts, including a conceptual hardware-aware analysis.
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 weighted MAX k -CUT problem involves partitioning a weighted undirected graph into k subsets, or colors, to maximize the sum of the weights of edges between vertices in different subsets. This problem has significant applications across multiple domains. This study explores encoding methods for MAX k -CUT on qubit systems by utilizing quantum approximate optimization algorithms (QAOA) and addressing the challenge of encoding integer values on quantum devices with binary variables. We examine various encoding schemes and evaluate the efficiency of these approaches. The study presents a systematic and resource-efficient method to implement the phase separation operator for the cost function of the MAX k -CUT problem. When encoding the problem into the full Hilbert space, we show the importance of encoding the colors in a balanced way. We also explore the option of encoding the problem into a suitable subspace by designing suitable state preparations and constrained mixers (LX- and Grover-mixer). Numerical simulations on weighted and unweighted graph instances demonstrate the effectiveness of these encoding schemes, particularly in optimizing circuit depth, approximation ratios, and computational efficiency.
This paper presents an experimental study of the Quantum Approximate Optimization Algorithm (QAOA) applied to the Max-Cut problem on small graph instances. Using PennyLane and a quantum simulator, a modular QAOA framework is implemented and evaluated on a fixed set of graphs, including ring, complete, star, and random graphs. For each graph, QAOA circuits with one and two layers are optimized using gradient-based methods, and the optimized states are sampled to obtain candidate cut solutions. The analysis includes visualizations of graph structures, cut partitions, bitstring distributions, and quantum circuit diagrams. This study serves as a clear and reproducible demonstration of QAOA behavior on combinatorial optimization problems.
The Quantum Approximate Optimization Algorithm (QAOA) is one of the promising near-term algorithms designed to find approximate solutions for combinatorial optimization problems. The algorithm prepares a parametrized state that is aimed to maximize the expectation value of the objective function of the problem. The circuit for QAOA consists of p layers, and it depends on 2p parameters, the determination of which can be carried out using a variational quantum eigensolver (VQE) subroutine. This is a variational approach for finding optimal parameters, where the quantum state is prepared with a quantum computer, and a classical computer is used for calculating the corresponding objective function value and performing an outer loop optimization routine in order to find optimal parameters. Earlier work on the algorithm has shown that QAOA is closely related to the Quantum Adiabatic Algorithm, which results in noticeable patterns in the optimal parameters. To this purpose, two methods, INTERP and FOURIER, were proposed to exploit this structure by Zhou et al. (2018). In this thesis, I extend upon these results by examining the effectiveness of one of their methods, the INTERP method, on the Max-Cut problem from graph theory. The performance of QAOA using this method was studied on several classes of graphs and benchmarked against the currently best-known classical algorithm Goemans-Williamson, which achieves an approximation ratio of ρ ≈ 0.878 on arbitrary graphs. The classes of graphs examined include cyclic, weighted and unweighted 3-regular, and Erdős–Rényi graphs with edge probabilities of 0.5 and 0.75 and graph sizes ranging from 4 to 16. Moreover, it was investigated if the INTERP method offers an advantage compared to random initialization of parameters and whether or not the method is polynomial in p. It was found that different graph classes lead to similar but different parameter patterns. Furthermore, the performance of QAOA using the INTERP method was dependent on the graph class considered. For the small graphs investigated it was found that QAOA outperforms Goemans-Williamson on ER-0.50, ER-0.75 and weighted 3-regular graphs from p=7 and beyond with respect to how well the expectation of the objective function approximates the optimal value, while for unweighted 3-regular graphs p=6 suffices. However, the calculation of the parameters is costly and requires a number of function evaluations that is on the order of the number of possible bipartitions of small graph with sizes n ≤ 16. Moreover, the numerical results suggested that indeed the method is polynomial in p advocating that the algorithm might be a strong alternative to Goemans-Williamson for large graph sizes, once hardware allows it.
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.
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.