When using classical approximation algorithm to solve the max-cut problem,the time complexity increases with the complexity of graph.In order to improve the solution efficiency,this paper uses quantum adiabatic approximation algorithm to solve the ground state for Hamiltonian of max-cut problem,and the ground state value is the optimal solution.The time complexity is independent of the number of vertices and edges of graph using quantum adiabatic approximation algorithm,and the max-cut problem can be calculated within finite steps.The computing process is based on the ProjectQ software developed.By establishing the evolutionary path from initial Hamiltonian to max-cut problem Hamiltonian,the change of the expected value is analyzed to determine whether the algorithm can find the optimal solution of max-cut problem or not.Numerical analysis results show that,this algorithm can effectively improve the efficiency of solving the max-cut problem,the accuracy of solution is 0.999 9 for 3-vertex undirected graph and 6-vertex undirected sparse graph,and the results for the 6-vertex undirected complete graph is 0.969 6.
When using classical approximation algorithm to solve the max-cut problem,the time complexity increases with the complexity of graph.In order to improve the solution efficiency,this paper uses quantum adiabatic approximation algorithm to solve the ground state for Hamiltonian of max-cut problem,and the ground state value is the optimal solution.The time complexity is independent of the number of vertices and edges of graph using quantum adiabatic approximation algorithm,and the max-cut problem can be calculated within finite steps.The computing process is based on the ProjectQ software developed.By establishing the evolutionary path from initial Hamiltonian to max-cut problem Hamiltonian,the change of the expected value is analyzed to determine whether the algorithm can find the optimal solution of max-cut problem or not.Numerical analysis results show that,this algorithm can effectively improve the efficiency of solving the max-cut problem,the accuracy of solution is 0.999 9 for 3-vertex undirected graph and 6-vertex undirected sparse graph,and the results for the 6-vertex undirected complete graph is 0.969 6.
The max-cut is used to solve maximum partition for given undirected weighted graph, the sum of the weights of all sides between the subset of vertex and complementary subset obtains a maximum value. Quantum adiabatic approximation is used to design system Hamiltonian Hsys, and the ground state of system Hamiltonian corresponds to the solution of max-cut problem. System Hamiltonian Hsys changes slowly along with initial Hamiltonian Hini follow the evolution path of Hamiltonian Hmax, and then the ground state of Hamiltonian can be calculated. By analyzing the change of expected value with evolution time, one can judge whether the approximate solution is the optimal solution or not. In this paper, we test the adiabatic evolution of complete undirected graph with vertices from 6 to 14. Based on Python and Project Q package, we write a solver program that sets parameters according to the number of vertices and edges of undirected graph, and then obtain experimental results by measuring the state of qubits in quantum registers. It can be inferred from experimental results that for a complete undirected graph with less vertices, expected value can converge well, and then the optimal solution of max-cut problem can be obtained. When the number of vertices increase, the energy variation of Hamiltonian become more complex and the expected value is hard to converge.
According to the adiabatic theorem of quantum mechanics, a system initially in the ground state of a Hamiltonian remains in the ground state if one slowly changes the Hamiltonian. This can be used in principle to solve hard problems on quantum computers. Generically, however, implementation of this Hamiltonian dynamics on digital quantum computers requires scaling Trotter step size with system size and simulation time, which incurs a large gate count. In this work, we argue that for classical optimization problems, the adiabatic evolution can be performed with a fixed, finite Trotter step. This "Floquet adiabatic evolution" reduces by several orders of magnitude the gate count compared to the usual, continuous-time adiabatic evolution. We give numerical evidence using matrix-product-state simulations that it can optimally solve the Max-Cut problem on 3-regular graphs in a large number of instances, with surprisingly low runtime, even with bond dimensions as low as [Formula: see text]. Extrapolating our numerical results, we estimate the resources needed for a quantum computer to compete with classical exact or approximate solvers for this specific problem.
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.
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 MaxCut is a combinatorial problem which belongs to the NP-hard class. The algorithm to solve it can be mapped into a quantum system, where the optimal configuration matches the ground state of a target Hamiltonian. It is possible to implement the adiabatic theorem into the circuit model via the QAOA algorithm, by arbitrarily choosing from both the gate-based and the measurement-based quantum computing (MBQC) settings. Recently, the MaxCut problem has been generalized to the Max K-Cut version also in the MBQC frame, where K stands for the number of cuts to implement. Here, the QAOA algorithm for a Max K-Cut is solved for a graph with arbitrary number of vertices V and number of cuts K scaling as a power of 2, along with the circuit depth p. The problem is solved on a emulator based on classical HPC resources. In order to test the performances of a Max K-cut implemented in the MBQC frame, we report how the average runtime and the approximation ratio scale by increasing V, K and p, comparing the results from two different topologies of graphs, 2-regular graphs with null weights on the diagonals versus the the fully-connected graphs. Eventually, the approximation ratio, evaluated as a parameter of success, is shown to yield optimal performances.
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.