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 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.
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.
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 24-vertex Erdős-Rényi graphs. Our framework only consumes 18 qubits and achieves 0.9950 approximation ratio on average, which outperforms the previous methods showing 0.9778 (quantum algorithm using the same number of qubits) and 0.9643 (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.
The MaxCut is a combinatorial problem consisting of finding the best sub-partitions of a graph, in order to maximize the value of a tailored-suited cost function. Although such specific problem belongs to the NP-hard class, quantum computing could lower the underlying computational complexity. The paramount quantum algorithm in literature to tackle the MaxCut is the QAOA, a trotterized version of adiabatic computation. Within such formulation, the cost function is encoded by a target Hamiltonian, whose spectrum matches the energy landscape to be handled by an optimization process. In the field of classical machine learning, a recent tool introduced to deal with graphs is provided by Graph Neural Networks (GNNs), whose peculiar architecture allows graph data to be embedded into Euclidean spaces, exploiting their topological structure as an intrinsic feature. We exploit the predictive power of GNNs, in order to classically emulate the energy landscape of the QAOA target Hamiltonian. In order to assess the predictive capacity of GNNs, we trained our model on graphs scaling from 3 to 12 vertices and achieving an accuracy between 0.9 and 0.99. We trained and tested the models on different topologies to assure the algorithm performances do not depend on the specific instance of the problem. Furthermore, the trained model proves to extend the correctness of the predictions for graphs up to 20 vertices still keeping a high accuracy. Such result envisages further development by exploiting HPC infrastructures and adapting warm starting techniques.
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.
Variational quantum algorithms (VQAs) have demonstrated considerable potential in solving NP-hard combinatorial problems in the contemporary noisy intermediate-scale quantum (NISQ) era. The quantum approximate optimization algorithm (QAOA) is one such algorithm, used in solving the maximum cut (Max-Cut) problem for a given graph by successive implementation of L quantum circuit layers within a corresponding Trotterized ansatz. The challenge of exploring the cost function of VQAs arising from an exponential proliferation of local minima with increasing circuit depth has been well documented. However, fewer studies have investigated the impact of circuit depth on QAOA performance in finding the correct Max-Cut solution. Here we employ basin-hopping global optimization methods to navigate the energy landscapes for QAOA ansätze for various graphs, and analyze QAOA performance in finding the correct Max-Cut solution. The structure of the solution space is also investigated using discrete path sampling to build databases of local minima and the transition states that connect them, providing insightful visualizations using disconnectivity graphs. We find that the corresponding landscapes generally have a single funnel organization, which makes it relatively straightforward to locate low-lying minima with good Max-Cut solution probabilities. In some cases below the adiabatic limit the second lowest local minimum may even yield a higher solution probability than the global minimum. This important observation has motivated us to develop broader metrics in evaluating QAOA performance, based on collections of minima obtained from basin-hopping global optimization. Hence we establish expectation thresholds in elucidating useful solution probabilities from local minima, an approach that may provide significant gains in elucidating reasonable solution probabilities from local minima. Published by the American Physical Society 2024